LATTICE THEORY
First Concepts and Distributive Lattices
GEORGE GRATZER
University of Manitoba
DOVER PUBLICATIONS, INC.
Mineola, New York

Copyright
Copyright © 1971, 1999 by George Gratzer
All rights reserved.
Bibliographical Note
This Dover edition, first published in 2009, is an unabridged republication of
the work originally published in 1971 by W. H. Freeman and Company, San
Francisco.
Library of Congress Cataloging-in-Publication Data
Gratzer, George A.
Lattice theory : first concepts and distributive lattices / George Gratzer —
Dover ed.
p. cm.
Originally published: San Francisco : W.H. Freeman, 1971.
Includes bibliographical references and index.
ISBN-13: 978-0-486-47173-0
ISBN-10: 0-486-47173-X
1. Lattice theory. 2. Lattices, Distributive. I. Title.
QA171.5.G73 2009
511.3'3—dc22
2008046925
Manufactured in the United States of America
Dover Publications, Inc., 31 East 2nd Street, Mineola, N.Y. 11501

CONTENTS
PREFACE vii
ACKNOWLEDGMENTS xi
TABLE OF NOTATION xiii
1 FIRST CONCEPTS
1. Two Definitions of Lattices 1
2. How to Describe Lattices 11
3. Some Algebraic Concepts 18
4. Polynomials, Identities, and Inequalities 31
5. Free Lattices 38
6. Special Elements 56
Further Topics and References 61
Problems 66

VI
CONTENTS
2 DISTRIBUTIVE LATTICES
7. Characterization Theorems and Representation
Theorems 69
8. Polynomials and Freeness 80
9. Congruence Relations 87
10. Boolean Algebras i^-generated by Distributive
Lattices 102
11. Topological Representation 117
12. Free Distributive Product 128
13. Some Categorical Concepts 139
Further Topics and References 151
Problems 155
3 DISTRIBUTIVE LATTICES WITH
PSEUDOCOMPLEMENTATION
14. Introduction and Stone Algebras 159
15. Identities and Congruences 166
16. Representation Theorems 175
17. Injective and Free Stone Algebras 183
Further Topics and References 190
Problems 192
BIBLIOGRAPHY 193
INDEX 207

PREFACE
In the first half of the nineteenth century, George Boole's attempt to
formalize propositional logic led to the concept of Boolean algebras.
While investigating the axiomatics of Boolean algebras at the end of the
nineteenth century, Charles S. Peirce and Ernst Schroder found it useful
to introduce the lattice concept. Independently. Richard Dedekind's
research on ideals of algebraic numbers led to the same concept. In fact,
Dedekind also introduced modularity, a weakened form of distributivity.
Although some of the early results of these mathematicians and of
Edward V. Huntington are very elegant and far from trivial, they did not
attract the attention of the mathematical community.
It was Garrett Birkhoff's work in the mid-thirties that started the general
development of lattice theory. In a brilliant series of papers he
demonstrated the importance of lattice theory and showed that it provides a
unifying framework for hitherto unrelated developments in many
mathematical disciplines. Birkhoff himself, Valere Glivenko, Karl Menger, John
von Neumann, Oystein Ore, and others had developed enough of this
new field for Birkhoff to attempt to "sell" it to the general mathematical

VIII
PREFACE
community, which he did with astonishing success in the first edition of
his Lattice Theory. The further development of the subject matter can
best be followed by comparing the first, second, and third editions of the
book (G. BirkhofT [1940], [1948], and [1967]).
Distributive lattices have played many roles in the development of
lattice theory. Historically, lattice theory started with (Boolean)
distributive lattices; as a result, the theory of distributive lattices is the most
extensive and most satisfying chapter in the history of lattice theory.
Distributive lattices have provided the motivation for many results in general
lattice theory. Many conditions on lattices and on elements and ideals of
lattices are weakened forms of distributivity. Therefore, a thorough
knowledge of distributive lattices is indispensable for work in lattice theory.
Finally, in many applications the condition of distributivity is imposed on
lattices arising in various areas of mathematics, especially algebra.
The realization of the special role of distributive lattices moved me to
break with the traditional approach to lattice theory, which proceeds from
partially ordered sets to general lattices, semimodular lattices, modular
lattices, and finally distributive lattices. The goal of the present volume is
to provide a detailed presentation of the theory of distributive lattices.
General lattices will be discussed in the companion volume of the present
one, which is now in preparation.
Chapter 1 includes a concise development of the basic concepts of lattice
theory and a detailed development of free lattices. Diagrams are emphasized
because I believe that an important part of learning lattice theory is the
acquisition of skill in drawing diagrams. This chapter lays the foundation
for the material included in both first and second volumes. Chapter 2,
which develops the theory of distributive lattices, comprises the most
substantial part of the present book. The last decade witnessed the birth
of a new field in lattice theory: distributive lattices with pseudocomple-
mentation. Chapter 3 discusses the basic results of this field and focuses on
one of the major preoccupations of twentieth-century mathematicians:
structure theory.
The exercises, which number more than 500, form an integral part of
this book. Only exercises marked by an * could be left out by the reader
without any loss in comprehension of the subject matter. The Bibliography
contains only works that are referred to in the text. The sixty-seven
original research problems, as well as the " Further Topics and References"
included at the end of each chapter, should be of help to those who are
interested in further reading and research in lattice theory.

PREFACE
IX
Finally, the reader will note that the symbol % is placed at the end of a
proof; if a theorem or lemma contains more than one statement, the
proof of a part is ended with ). The abbreviation "iff " stands for "if and
only if." More difficult exercises are marked by *. "Theorem 10" refers
to Theorem 10 of the same section, whereas "Theorem 15.10" refers to
Theorem 10 of section 15. Similarly, "exercise 2.6" means exercise 6 of
section 2. References to the Bibliography are given in the form "J. Jakubik
[1957]," which refers to a paper (or book) by J. Jakubik published in 1957.
Such references as " [1957a]" and " [1957b ]" indicate that the Bibliography
contains more than one work by the author published in that year.
"R. S. Pierce [a]" refers to a paper by R. S. Pierce that had not appeared
in print at the time the manuscript of this book was submitted for
publication.
Winnipeg, Manitoba
September 1970
George Gratzer

ACKNOWLEDGMENTS
An individual takes the responsibility and gets (he hopes) the credit for
writing a book, but in reality ventures of this kind often require the
cooperation of a group of people.
The first version of this book was developed as a set of lecture notes for
my course on lattice theory at the University of Manitoba in the academic
year 1968-1969. The undergraduates and graduate students who took this
course and many of my colleagues who attended helped by criticizing my
lectures and by simplifying proofs. The lecture notes were also read by
Dr. P. Burmeister, who offered many helpful remarks.
A rewritten version of the lecture notes was read by R. Balbes, M. I.
Gould, K. M. Koh, H. Lakser, S. M. Lee, P. Penner, C. R. Piatt, and R.
Padmanabhan, who coordinated the work of this group. I rewrote a
substantial part of the manuscript as a result of the changes they suggested.
I am very thankful to the whole group, and especially to Dr. Piatt and Dr.
Padmanabhan, for their untiring work.
I would like to thank K. D. Magill, Jr., for the opportunity to conduct a

XII
ACKNOWLEDGMENTS
course on lattice theory at the State University of New York at Buffalo
in the summer of 1970, and the students of this class, especially J. H.
Hoffman, for their valuable observations.
Last but not least, the manuscript was read for the publisher by B.
Jonsson; all his suggestions were gratefully accepted.
The typing and retyping of the manuscript were done with unusual
skill and patience by Mrs. M. McTavish. The rest of the secretarial work
was capably handled by Mrs. N. Buckingham.
The galley proofs were read by R. Antonius, J. A. Gerhard, K. M. Koh,
W. A. Lampe, R. Quackenbush, I. Rival, and the group coordinator,
R. Padmanabhah. E. Fried assisted me in compiling the Index.
Thanks are also due to the National Research Council of Canada for
sponsoring much of the original research that has gone into this book and
to Professor N. S. Mendelsohn, who relieved me of all teaching and
administrative duties to allow me time to conduct research, to supervise
the research of graduate students, and to prepare the manuscript of this
book.

TABLE OF NOTATION
SETS
SYMBOL
G
c=
c
w,U
an
0
-
x, n
Ml
f4,9-xl'
•p-1
A<p~x
P(X)
CO
Ko
tn, n,...
EXPLANATION
membership sign
inclusion
proper inclusion
union
intersection
empty set
set difference
Cartesian product
cardinality of the set
A
composition of maps:
xfo/i) = (x<p)il>
inverse of a map
Ay'1 = {x | x<peA}
set of all subsets of X
first infinite ordinal;
n < w means n = 0,
1,2,...
first infinite cardinal
lower-case German
letters stand for
cardinals
POSETS
SYMBOL
<
>
<B
<A
<V
<
-<
>-
II
inf
sup
A
V
r+i
[a,b]
(«,»]
[a,b)
(a.b)
y
pm
0
1
APPEARS ON
1,2
2
8
9
9
8
12
12
2
3
2
3
2
19
20
20
20
20
10
52
56
56

XIV
TABLE OF NOTATION
LATTICES
SYMBOL
<L; A, v>
<L; A>
<L; v>
B(L)
C(L)
D(L)
E{X)
2>(L)
%{L)
/(L)
S(L)
Sub (L)
L/0
V
L x K
n(L(|Je/)
£"
<-B; a, v,'
2?//
581
58B
APPEARS ON PAGE(S)
6
9
9
105
25
162
31
23
23, 50
22
22,50
161
29
26
28
27
28
6,9
6,9
0, 1> 58
112
92
92
SPECIAL LATTICES
AND ALGEBRAS
B
B™
S32
5[C]
177
165
58
109
SYMBOL
^kW
Wl5
*5
<Sn
CLASSES
APPEARS ON PAGE(S)
52
40
13,69
13,69
168
OF LATTICES
AND ALGEBRAS
Amal (K)
B
Bo
Bn
Ba
D
Dfin
^{0,1}
L
M
191
139,140
178
179
179
179
39, 139
146
140
39
39
CONGRUENCES
CO
I
0(fl, b)
®(H)
0[7]
0>(7)
<D/0
0x0)
we
24
24
87
87
87
100
172
28
24
B[L)
En
F(P)
F(Pm)
FK(P)
115
19
17
40
52
39
SUBSETS AND FAMILIES OF
SUBSETS OF LATTICES
(a)
[a)
21
23

TABLE OF NOTATION
xv
SYMBOL
fob]
(a,b]
[a,b)
(a,b)
C°
D(L)
Fa
[H]
(H]
[H)
[H]R
[H]B
H{P)
/*
J(L)
&(L)
r(a)
r(I)
S(a)
S(I(L))
S(L)
X*
X*
APPEARS ON PAGE{S)
20
20
20
20
109
162
163
20, 138
21
23
105
141
72
107
72
76,119
72,75,119
119
61
107
161
133
133
OPERATIONS
a A b 4
a V b 4
a v b 58
x + y 84,92
a' 58
fl* 58
a+ 184
TOPOLOGY
X 125
y(L) 119
SYMBOL APPEARS ON PAGE(S)
yB 124
ASSOCIATED ALGEBRAS
A(L) 64
E(L) 64
^o.i(^) 64
PROPERTIES
(D)
(E)
(I)
(JID)
(L1)-(L4)
(Ln)
(MID)
(P1MP4)
P(m, n)
-P(o,i)(tn,n)
P„(m, n)
(SI)
(S2)
(S3)
(S4)
50
41
50
107
5
167
107
1
139
139
139
121
121
123
187
MISCELLANEOUS
W)
%ea(J)
f(k,n)
MKn)
H
I
p(n)
s
9e
134
134
64
64
175
175
32
175
26

CHAPTER
FIRST CONCEPTS
1. Two Definitions of Lattices
Whereas the arithmetical properties of the set of reals, R, can be expressed
in terms of addition and multiplication, the order theoretic, and thus the
topological, properties are expressed in terms of the ordering relation <.
The basic properties of this relation are as follows.
For all a,b,c e R we have:
(PI) a < a (reflexivity)
(P2) a < b and b < a imply that a = b {antisymmetry)
(P3) a < b and b < c imply that a < c {transitivity)
(P4) a < b or b < a {linearity)
There are many examples of binary relations sharing properties (Pl)-
(P4) with the ordering relation of reals, and there are even more enjoying
(P1)-(P3). This fact, by itself, would not justify the introduction of a new

2
FIRST CONCEPTS
concept. However, it has been shown that many basic concepts and results
about the reals depend only on (P1)-(P3), and these can be profitably used
whenever we have a relation satisfying (P1)-(P3). Relations satisfying
(P1)-(P3) are called partial ordering relations, and sets equipped with such
relations are called partially ordered sets or posets.
To make the definitions formal, let us start with two sets A, B and form
the set A x B of all ordered pairs {a, by with aeA, be B. If A = B, we
write A2 for A x A. Then a binary relation ponA can simply be defined as
a subset of A2. The elements a, b (a,b e A) are in relation with respect to p if
(a, b} e p. For (a, by e p, we will also write a p b, or a = b(p). Binary
relations will be denoted by small Greek letters or by special symbols.
This formal definition can be compared with the intuitive one: A binary
relation p on A is a "rule" that decides whether or not a p b for any given
a,be A. Of course, any such rule will determine the set {(a, by \ a p b,
a,b e A}, and this set determines />, so we might as well regard p as being
identical with this set.
A partially ordered set (A; p> consists of a nonvoid set A and a binary
relation p on A, such that p satisfies properties (P1)-(P3). Note that these
can be stated as follows: For all a,b,c e A, <a, a> e p; <a, by, (b, a) e p
imply that a = b\(a, by, (jb, c> e p imply that <a, c> e p.
If p satisfies (P1)-(P3), p is a partial ordering relation, and it will usually
be denoted by <. Also, a > b will mean b < a. Sometimes we will say
that A (rather than (A; <» is a poset, meaning that the partial ordering
is understood. This is an ambiguous although widely accepted practice.
A poset (A; < > that also satisfies (P4) is called a chain (also called fully
ordered set, linearly ordered set, and so on).
If (A; < y is a poset, a,b e A, then a and b are comparable if a < b or
b < a. Otherwise, a and b are incomparable, in notation a || b. A chain is,
therefore, a poset in which there are no incomparable elements.
Next we define inf and sup in an arbitrary poset P (that is, <P; <» the
same way as it is done for reals.
Let H ^ P,aeP. Then a is an upper bound of H, if h < a for all he H.
An upper bound a of H is the least upper bound of H or supremum of H if,
for any upper bound b of H, we have a < b. We shall write a = sup H, or
a = \J H. (The notations a = l.u.b. H, a = 2 H are also common in the
literature.) This notation can be justified only if we show the uniqueness of
the supremum. Indeed, if a0 and ax are both suprema of H, then a0 < al9
since a1 is an upper bound, and a0 is a supremum. Similarly, ax < a0; thus
fl0 = 0i by (P2).

Section 1 Two Definitions of Lattices
3
The concepts of lower bound and greatest lower bound or infimum are
similarly defined; the latter is denoted by inf H or A H. (The notations
g.l.b. H,Y[H are also used in the literature.) The uniqueness is proved as
in the preceding paragraph.
The adverb "similarly" at the end of that paragraph can be given a very
concrete meaning. Let <P; <> be a poset. The notation a > b (meaning
b < a) can also be regarded as a definition of a binary relation on P. This
binary relation > satisfies (P1)-(P3); as an example, let us check (P2). If
a > b and b > a, then by the definition of > we have b < a and a < b;
using (P2) for < we conclude that a = b. (PI) and (P3) are equally trivial.
Thus <P; >> is also a poset, called the dual of <P; <>. Now, if <I> is a
"statement" about posets, and if in O we replace all occurrences of < by
>, we get the dual of <&.
Duality Principle. If a statement <E> is true in all posets, then its dual is
also true in all posets.
This is true simply because O holds for <P; <> iff the dual of O holds
for <P; >>, which is also a poset.
As an example take for $ the statement: "If sup H exists it is unique."
We get as its dual: "If inf H exists it is unique."
It is hard to imagine that anything as trivial as the Duality Principle
could yield anything profound, and it does not; but it can save a lot of
work.
A poset <L; < > is a lattice if sup {a, b} and inf {a, b} exist for all a,b e L.
In other words, lattice theory singles out a special type of poset for
detailed investigation. To make such a definition worthwhile, it must be
shown that this class of posets is a very useful class, that there are many
such posets in various branches of mathematics (analysis, topology, logic,
algebra, geometry), and that a general study of these posets will lead to a
better understanding of the behavior of the examples. This was done in the
first edition of G. BirkhofFs Lattice Theory [1940]. As we go along we shall
see many examples, most of them in the exercises. For a general survey of
lattices in mathematics, see G. Birkhoff [1967], and H. H. Crapo and G.
C. Rota [1970].
We will take the usefulness of lattice theory for granted (this is a less
touchy subject now than it formerly was) and hope that the reader will like
it for its intrinsic beauty.

4
FIRST CONCEPTS
To make the definition of a lattice less arbitrary, we note that an
equivalent definition is the following:
A poset <L; < > is a lattice //sup H and inf H exist for any finite nonvoid
subset H ofL.
proof. It is enough if we prove that the first definition implies the second.
So let <L; <> satisfy the first definition and let H ^ L be nonvoid and
finite. If H = {a}, then sup H = inf H = a follows from the reflexivity of
< and the definitions of sup and inf. Let H — {a, b9 c}. To show that sup H
exists, set d = sup {a, b}9 e = sup {d9 c}; we claim that e = sup H. First of
all, a < d9b < d9 and d < e9 c < e; therefore (by transitivity) x < e9 for
all xeH. Secondly, if /is an upper bound of H9 then a < / b < / and
thus d < f; also c < /, so that c9d < /; therefore e < / since e = sup {d9c}.
Thus e is the supremum of H.
If H = {a09..., fln_!}, n > 1, then
sup {... sup {sup {a09 ax}9 a2}...9an.1}
is the supremum of H9 by an inductive argument whose steps are analogous
to those in the preceding paragraph.
By duality (in other words, by applying the Duality Principle), we
conclude that inf H exists. %
With regard to H = 0 (the void set), we will point out in the exercises
that inf 0 and sup 0 need not exist in a lattice.
This simple proof can be varied to yield a large number of equally
trivial statements about lattices and partially ordered sets in general. Some
of these will be stated as exercises and used later. To make the use of the
Duality Principle legitimate for lattices, note:
If <P; <> is a lattice, so is its dual <P; >>.
Thus the Duality Principle applies to lattices.
We will use the notations
a A b = inf {a9 b}
and
a v b = sup {a9 b}
and call A the meet and V the join. In lattices, they are both binary
operations, which means that they can be applied to a pair of elements

Section 1 Two Definitions of Lattices
5
a9bofL to yield again an element of L. Thus A is a map of L2 into L and
so is V, a remark that might fail to be very illuminating at this point.
The previous proof yields that
(• • • (fa, V ax) V a2) • • •) V fln-i = sup {a09..., an^}
and there is a similar formula for inf. Now observe that the right-hand side
does not depend on the way the elements at are listed. Thus A and V are
idempotent, commutative, and associative—that is,
(LI) aAa = a9a\/a — a (idempotency)
(L2) aAb = bAa9aVb = b\/a (commutativity)
(L3) (a A b) A c = a A (b A c)9
(a V b) V c = a V (b V c) {associativity)
As always in algebra, associativity makes it possible to write
flo A fli A • • • A fln_x
without using parentheses (and the same for v).
There is another pair of rules that connect A and V. To derive them,
note that if a < b9 then inf {a9 b} = a; that is, a A b = a9 and conversely.
Thus
a<biffaAb = a.
By duality (and by interchanging a and b) we have
a < b\Ra V b = b.
Applying the "only if" part of the first rule to a and a V b9 and that of
the second rule to a A b and a9 we get
(L4) a A (a V b) = a, a V (a A b) = a (absorption
identities)
Now we are faced with the crucial question: Do we know enough about
A and V so that lattices can be characterized purely in terms of properties
of A and V ?
Two comments are in order. It is obvious that < can be characterized
by A and V (in fact, by either of them); therefore, obtaining such a

6
FIRST CONCEPTS
characterization is only a matter of persistence. More importantly, why
should we try to get such a characterization? To rephrase the question,
why should we want to characterize <L; <> as <L; A, V>, which is an
algebra—that is, a set equipped with operations (in this case, two binary
operations)? Note that < is simply a subset of L2, whereas A and V are
maps from L2 into L. The answer is simple: We want such a
characterization because if we can treat lattices as algebras, then all the concepts and
methods of universal algebra will become applicable. The usefulness of
treating lattices as algebras will soon become clear.
An algebra <L; A, V > is called a lattice if: L is a nonvoid set, A and V
are binary operations on L, both A and V are idempotent, commutative,
and associative, and they satisfy the two absorption identities. The
following theorem states that a lattice as an algebra and a lattice as a poset are
"equivalent" concepts. (The word "equivalent" will not be defined.)
Theorem 1.
(i) Let the poset 2 = <L; < > be a lattice. Set a A b = inf{a9 b}9 a V b =
sup {a9 b). Then the algebra 2a = <L; A, V > is a lattice.
(ii) Let the algebra 2 = <L; A, V > be a lattice. Set a < b iff a A b = a.
Then 2P = <L; < > is a poset, and the poset 2P is a lattice.
(iii) Let the poset 2 = <L; <> be a lattice. Then (2a)p = 2.
(iv) Let the algebra 2 = <L; A, V > be a lattice. Then (2p)a = 2.
remark, (i) and (ii) describe how we pass from poset to algebra and
back, whereas (iii) and (iv) state that going there and back takes us back to
where we started.
PROOF.
(i) This has already been proved. )
(ii) We set a < b to mean a A b = a. Now < is reflexive since A is
idempotent; < is antisymmetric since a < b, b < a mean that
aAb = a,bAa = b, which, by the commutativity of A, imply that
a = aAb = bAa = b',<is transitive, since if a < b9 b < c, then
a = aAb9b = bAc, and so
a = a A b = a A (b A c) (A is associative)
= (a A b) A c = a A c9

Section 1 Two Definitions of Lattices
7
that is, a < c. Thus <L; <> is a poset. To prove that <L; <> is a
lattice we will verify that a A b = inf {a, b} and a V b = sup {a, 6}.
(This is not a definition.) Indeed, a A b < a, since
(a A b) A a = a A (b A a) = a A (a A b)
= (a A a) A b = a A b,
using associativity, commutativity, and idempotency of A ; similarly,
a A b < b. Now if c < a, c < b, that is, c A a = c, c A b = c, then
c A (a A b) = (c A a) A b = c A b = c; thus a A b = inf {a, 6}.
Finally, a < a M by b < a V Z>, because a = a A (a V b), b =
b A (a V b)by the first absorption identity; if a < c, 6 < c, that is,
a = a A c,b = b A c, then a V c = (a A c) V c = c, and b V c =
c by the second absorption identity. Thus
(fl V 6) A C = (fl V 6) A (fl V C) = (fl V b) A [a V (6 V c)]
= (a V b) A [(a V 6) V c] = a V b,
that is, a V b < c, completing the proof of a V b — sup {a, b). )
(iii) It is enough to observe that the partial ordering of £ and (£a)p are
identical to get (iii).
(iv) The proof of (iv) is similar to the proof of (iii). %
The proof of Theorem 1, and even the statement of Theorem 1, are
subject to criticism. To begin with, in the definition of a lattice as an
algebra, idempotency is redundant. The last step of the proof of (ii) can be
made neater by first proving that
a = a A b iff b = a V b.
Theorem 1 should be preceded by a similar theorem for "semilattices."
All these questions will be dealt with in the exercises that follow this
section.
Finally, note that for lattices as algebras, the Duality Principle takes on
the following very simple form.
Let O be a statement about lattices expressed in terms of A and V . The
dual of $ is the statement we get from $ by interchanging A and V. If O
is true for all lattices, then the dual of $ is also true for all lattices.
To prove this we have only to observe that if £ = <L; a , V >, then the
dualof£pis«L; V, A»p.

8
FIRST CONCEPTS
Exercises
Posets
1. Define x < y to mean x < y and x 9* y. Prove that, in a partially
ordered set, x < x for no x, and that jc < y, y < z imply that x < z.
2. Let the binary relation < satisfy the conditions of exercise 1. Define
x < y to mean x < y or x = y. Then show that < is a partial ordering.
3. Prove the following extension of antisymmetry: If x0 < xx < • • • <
jcn_i < jc0, then x0 = xx = • • • = xn-i.
4. Enumerate all partial orderings on a five-element set.
5. Let < be a partial ordering on A and let B be a subset of A. Set a <B b
for a,b e B if a < b. Prove that < B is a partial ordering on B.
6. Let A be a set and let P be the set of all partial orderings on A. For
p,o e P set p < a if a p b implies that a a b (a,b e A). Prove that <P; <>
is a poset.
7. Let 0 denote the void set. If inf 0 = a, then x < a for all elements x
of the poset. Find an example of a poset in which inf 0 does not exist.
8. Formulate and prove exercise 7 for sup 0.
Lattices as Posets
9. Prove that the following are examples of posets:
(i) Let A = P(X), the set of all subsets of a set X; let X0 < X± mean
X0 c A\ (set inclusion),
(ii) Let A be the set of all real valued functions defined on X\ for
f,g eAsetf<g iff/(*) < g(x) for all xe X.
(iii) Let A be the set of all continuous concave real valued functions
defined on the real interval; define / < g as in (ii).
(iv) Let A be the set of all open sets of a topological space; define < as
in (i).
(v) Let A be the set of all human beings; a < b means that a is a
descendant of b.
10. Which of the examples in exercise 9 are lattices? For those that are
lattices, compute a A b, a V b.
11. Show that every chain is a lattice.
12. Let A be the set of all subgroups (normal subgroups) of a group G; for
X,Ye A set X < Y to mean X c y. Prove that <^; <> is a lattice;
compute X A Y and X V Y.
13. Let <P; <> be a poset in which inf H exists for all H c P. Show that
<P; <> is a lattice. (Hint: For a,beP let i/ be the set of all upper
bounds of {a, b). Prove that sup {a, b} — inf H.) Relate this to exercise
12.
Semilattices as Posets
14. A poset is a join-semilattice (dually, meet-semilattice) if sup {a,b}
(dually, inf {a, b}) exists for any two elements.

Section 1 Two Definitions of Lattices
9
Prove that the dual of a join-semilattice is a meet-semi lattice, and
conversely.
15. Let A be the set of finitely generated subgroups of a group G, partially
ordered under set inclusion (as in exercise 12). Prove that <A; c> is a
join-semilattice, but not necessarily a lattice.
16. Let C be the set of all continuous strictly convex real valued functions
defined on the real interval [0, 1]. For f,g e C set / < g iff f(x) < g(x)
for all x e [0, 1]. Prove that <C; <> is a meet-semilattice, but not a
join-semilattice.
17. Show that the poset <P; < > is a lattice iff it is a join- and meet-semilattice.
Semilattices as Algebras
18. Let <v4; o> be an algebra with one binary operation <>. The algebra C4; <>>
is a semilattice ifo is idempotent, commutative, and associative.
Let <P; < > be a join-semilattice. Show that the algebra (A; v > is a
semilattice when a v b = sup {a, b}. State the analogous result for
meet-semilattices.
19. Let the algebra <>4;o> be a semilattice. Define the binary relations
<A, < v on A as follows: a<Ab iff a = a<> b;a<yb iff b = a<>b.
Prove that (A; <A> is a poset, as a poset it is a meet-semilattice, and
a A b = a o b; that {A; < v> is a poset, as a poset it is a join-semilattice,
and a V b = a o b; and that the dual of {A; < A> is <^4; < v>.
Theorem 1 for Semilattices
20. Prove the following statements:
(i) Let the poset 21 = </*; ^> be a join-semilattice. Set a V b =
sup {a, b}. Then the algebra 21° = (A; V > is a semilattice.
(ii) Let the algebra $1 = </*; o> be a semilattice. Set a<biffaob = b.
Then 2lp = <^4; < > is a poset, and the poset W is a join-semilattice.
(iii) Let the poset % = <A; <> be a join-semilattice. Then (Sta)p = 21.
(iv) Let the algebra 9t = <^; o> be a semilattice. Then (2lp)° = 21.
21. Formulate and prove Theorem 1 for the meet-semilattices.
Lattices as Algebras
22. Prove that the absorption identities imply the idempotency of A and V .
(Hint: simplify a v [a A (a V a)] in two ways to yield a = a v a.)
23. Let the algebra <>4; a, V> be a lattice. Define a ^A b iff a A b = a;
a <y b iff a V b = b. Prove that a <Abiffa < v b.
24. Prove that the algebra <>4; A, v > is a lattice iff (A; A > and <>4; v > are
semilattices and a = a A b is equivalent to 6 = a V 6. Verify that if
<>4; A, V i> and (A; A, v 2> are both lattices, then V i is the same as
V2.
25. Is there a result for semilattices that is analogous to that of exercise 22?
In other words, are the three conditions for semilattices independent
(that is, none follows from the others) ?

10
FIRST CONCEPTS
•26. Prove that an algebra (A; A, V > is a lattice iff it satisfies
a = (b A a) V a
and
{[(a A b) A c] V d} V e = {[(b A c) A a] V e} V [(/ V J) A d\
(see J. A. Kalman [1968]. The first definition of lattices by two identities
was found by R. Padmanabhan in 1967, Notices Amer. Math, Soc. 14,
No. 67T-468, and published in full in R. Padmanabhan [1969]. J. A.
Kalman's two identities are slightly improved versions of those of
R. Padmanabhan. R. N. McKenzie [a] proved the existence of a single
identity characterizing lattices.)
Miscellany
27. Let p be a binary relation on the set A. The transitive extension o/p is a
binary relation p defined by the following rule: a p b iff there exists a
sequence a = x0, *i,..., xn = b with xt p xi+1 for / = 0,..., n — 1.
Show that for a reflexive relation p, the transitive extension p is a partial
ordering relation iff p is antisymmetric. Express this condition in terms
of p.
28. Let p be a reflexive and transitive binary relation (quasi-ordering relation)
on the nonvoid set A. Call B c A a block if B ^ 0, a pb, and b p a for
any a,b e B, and for a e B, b e A, a p b, and 6pfl imply that 6 e /?. Let
P be the set of all blocks and set Bx < B2 (BUB2 e P) if ^x p 62 for some
(thus for all) &i e 2?i, b2 e i?2. Prove that <P; <> is a poset.
29. Let A be a set of sets. Let ap6 mean that there is a one-to-one map from
a into 6. What is <P; <>? (Notation is that of exercise 28.)
30. Let the binary operation o on the set A be associative. Give a rigorous
proof of the statement that any bracketing of a0 ° • • • ° an-i will yield
the same element.
31. Suppose that in a poset b v c, a v (b v c), and avii exist. Prove that
(a v b) V c exists and that a V (b V c) = (a V b) v c.
32. Prove that if a A 6 exists in a poset, so does a V (a A b).
33. Let H and # be subsets of a poset. Suppose that sup H, sup J£, and
sup (H u #) exist. (H u AT is the set union of /f and K.) Under these
conditions verify that (sup H) V (sup K) exists and equals sup (H u K).
34. In a poset P define the comparability relation y\ For a,beP, ayb if
a < b or b < a. Take a sequence al9.. .,ak of elements of P satisfying
the following conditions:
(i) a{ ^ tfi + i, aiyal + 1 for i = 1,.. .,£ - 1; afc ^ alf akyax.
(ii) For no elements ajb e P, and ij < k, i ^ / is a = a{ = ay,
& = «i + i = Oj + u or a = a{ = ak, b = ai + 1 = Oi.
(iii) For no 1 < / < k — 2 is a% y ai + 29 and neither ak^x y ax nor
ak y a2 holds.
Prove that k is even.

Section 2 How to Describe Lattices
11
*35. Prove that a binary relation y on a set A is the comparability relation of
some poset (A; < > iff y satisfies the condition of exercise 34 (A. Ghouila-
Houri [1962]; see also P. C. Gilmore and A. J. Hoffman [1964] and
M. Aigner [1969]).
2. How to Describe Lattices
To illustrate results and to refute conjectures we shall have to describe a
large number of examples of lattices. This can be done by basing the
examples on known mathematical structures, as illustrated in the exercises
of Section 1. In this section we list a few other methods.
A finite lattice can always be described by a meet table and a. join table.
For example: let L = {0, a, b, 1}.
A 0 a b 1 V 0 a b 1
00000 0 0 a b 1
a 0 a 0 a a a a 1 1
b 0 0 b b b b 1 b 1
1 0 a b 1 11111
We see that most of the information provided by the tables is redundant.
Since both operations are commutative, the tables are symmetric with
respect to the diagonal. Furthermore, x A x = x, x V x = x\ thus the
diagonals themselves do not provide new information. Therefore, the two
tables can be condensed into one.
A
V
0
a
b
1
0
a
b
1
a
0
1
1
b
0
0
1
1
0
a
b
It should be emphasized again that the part above the diagonal determines
the part below the diagonal since either determines the partial ordering.
To show that this table defines a lattice, we have only to check the
associative identities and the absorption identities.
An alternative way is to describe the partial ordering, that is, all pairs

12
FIRST CONCEPTS
1
0
Figure 2.1
<x, y> with x < y. In the preceding example we get < = {<0, 0>, <0, a>,
<0, b\ <0, 1>, <«, «>, <«, 1>, <&, &>, <A, 1>, <1, 1».
Obviously, all pairs of the form <#, *> can be omitted from the list,
since we know that x < x. Also, if x < y and y < z, then x < z. For
instance, when we know that 0 < a and a < 1, we do not have to be told
that 0 < 1. To make this idea more precise, let us say that in the poset
<P; <>, a covers b (b is covered by a) (in notation, a >~b(b-< a)) if
a > b and for no x, a > x > b. The covering relation of the preceding
example is simply:
-< ={<0,fl>,<0,^>,<fl,l>,<^,l>}.
Does the covering relation determine the partial ordering ? The following
lemma shows that in the finite case it does.
Lemma 1. Let <P; <> be a finite poset. Then a < b iff a = b or there
exists a finite sequence of elements x0i..., #n_i such that x0 = a, xn-± = b,
andx{-< xi+1,forO < i < n — 1.
proof. If there is such a finite sequence, then a = x0 < xx < • • • <
xn-x = b, and a trivial induction on n yields a < b. Thus it suffices to
prove that if a < b, then there is such a sequence. Fix a,b eP, a < b, and
take all subsets H of P such that H is a chain (under the partial ordering
induced by the partial ordering of P), a is the smallest element of H, and b
is the largest element of H. There are such subsets: {a, b}, for example.
Choose such an H with the largest possible number of elements, say, with
m elements. (P is finite.) Then H = {x0,..., *m-i}> and we can assume that
x0 < xx < • • • < *m_i. We claim that in <P; <> we have a = x0-<
xx-< < *m-i- Indeed, xt < xi + 1 by assumption. Thus if x{ -< xt+1
does not hold, then x{ < x < xi + 1 for some x e P, and H u {x} will be a
chain of m + 1 elements between a and b, contrary to the maximality of
the number of elements of H. %

Section 2 How to Describe Lattices
13
Figure 2.2
The diagram of a poset <P; <> represents the elements with small
circles o; the circles representing two elements x, y are connected by a
straight line if one covers the other; if x covers y, then the circle
representing x is higher up than the circle representing y. The diagram of the lattice
discussed previously is shown in Figure 2.1. Sometimes the "diagram"
of an infinite poset is drawn. Such diagrams are always accompanied
by explanations in the text. Note that in a diagram the intersection of two
lines need not indicate an element. A diagram is planar if no two lines
intersect. A diagram is optimal if the number of pairs of intersecting lines
is minimal. Figures 2.1, 2.2, and 2.3 are planar diagrams; Figure 2.4 is
an optimal but not planar diagram; Figure 2.6 is not optimal. As a rule,
optimal diagrams are the most practical to use.
The methods we will use will be combinations of previous ones. The
lattice 9^5 of Figure 2.2 has five elements: o, a, b, c, i, and b < a,c v b = i,
a A c = o. This description is complete—that is, all the relations follow
from the ones given. Uft5 has five elements: o, a, b, c, i and a A b =
aAC = bAC = o, avb = avc = bvc = i.
In contrast, we can start with some elements (say, a, b, c), with some
relations (say, b < a), and ask for the "most general" lattice that can be
formed, without specifying the elements to be used. (The exact meaning of
"most general" will be given in Section 5.) In this case we continue to
form meets and joins until we get a lattice. A meet (or join) formed is
identified with an element that we already have only if this identification is
forced by the lattice axioms or by the given relations. The lattice we get
from a, 6, c, satisfying b < a is shown in Figure 2.3.
To illustrate these ideas we give a part of the computation that goes into
the construction of the most general lattice L generated by a, b, c with

14 FIRST CONCEPTS
Figure 2.3
b < a. We start by constructing joins and meets a V c, b V c, a A c,
b A c; note that avbvc = (avb)vc = avc since a V b = a;
similarly, aAbAc = bAc. Next we have to show that the seven
elements (a, b, c, a v c, b V c, a A c, b A c) that we already have are all
distinct. Remember that two were equal if such equality would follow from
the relation (b < a) and the lattice axioms. Therefore, to show a pair of
them distinct, it is enough to find a lattice K with a,b,c e K, a < b, where
that pair of elements is distinct. For instance, to show a ^ a V c, take
the lattice {0, 1, 2} with 0 < 1 < 2, and b = 0, a = 1, c = 2. The next
step is to form further joins and meets: b V {a A c), a A (b V c). It is
easy to see that all the other joins and meets are equal to a given one—for
example, b A {a A c) = b A c, a V (a A c) = a. Now we claim that the
nine elements {a, b, c, a V c, b V c, a A c, b A c, b v {a A c), a A (b V c))
form a lattice. We have to prove that by joins and meets we cannot get
anything new. The thirty-six joins and thirty-six meets we have to check are
all trivial. For instance, a V [b V (a A c)] = a v b V {a A c) = a, by
the absorption law, since a V b = a; also c A [a A (b v c)] = (c A a) A
(b V c) = a A cy since c A a < b v c.
Exercises
1. Give the join and meet table of the lattice in exercise 1.9(a) for one-,
two-, and three-element sets X.

Section 2 How to Describe Lattices
15
2. Give the set < for the lattices in exercise 1. Which is simpler: the meet
and join table or < ?
3. Describe a practical method of checking associativity in a join (meet)
table.
4. Relate Lemma 1 to exercise 1.27.
5. Let CP; <> be a poset, a,beP, a < b, and let C denote the set of all
chains H in P with smallest element a, largest element b. Let H0 < Hx
mean H0 c Hx for Ho^eC. Show that <C; <> is a poset with
smallest element {a, b}.
6. The poset < Q; < > is said to satisfy the Ascending Chain Condition if any
increasing chain terminates, that is, if xteQ9 i = 0,1,2,..., and
x0 < *i < • • • < xn < • • •, then for some m we have xm = xm + i =
The element x of Q is maximal if x < y (y e Q) implies that x = y.
Show that the Ascending Chain Condition implies the existence of
maximal elements and that, in fact, every element is included in a
maximal element.
7. Dualize exercise 6. (The dual of maximal is minimal and the dual of
ascending is descending.)
8. If <Q; <> is a lattice and x is a maximal element, then y < x for all
j> e Q. Show that this statement is not, in general, true in a poset.
9. Give examples of posets without maximal elements and of posets with
maximal elements in which not every element is included in a maximal
element.
10. Let <P; <> be a poset with the property that for every a,b eP, a < b,
any chain in P with smallest element a and largest element b is finite.
Show that the poset <C; <> formed from <P; <> in exercise 5 satisfies
the Ascending Chain Condition (see exercise 7).
11. Extend Lemma 1 to posets satisfying the condition of exercise 10
(combine exercises 6 and 10).
12. Could the result of exercise 11 be proved using the reasoning of
Lemma 1 ?
13. Is the result of exercise 11 the best possible? (No)
14. Describe a method of finding the meet and join table of a lattice given
by a diagram.
15. Are the posets of Figures 2.4 and 2.5 lattices?
Figure 2.4 Figure 2.5

16
FIRST CONCEPTS
16. Show that Figures 2.6 and 2.7 represent the same lattice.
Figure 2.6
17. Simplify Figure 2.8.
Figure 2.7
Figure 2.8
18. Simplify Figure 2.9. What is the number of pairs of intersecting lines in
an optimal diagram ? (Zero)
19.
Figure 2.9
Draw the diagrams of P(X) (exercise 1.9(a)) for \X\ = 3 and \X\ = 4.
(\X\ is the cardinality of X.) Does P(X) have a planar diagram for
\X\ = 3?

Section 2 How to Describe Lattices
17
20. Draw the diagrams of the lattice of binary relations on X (partially
ordered by set inclusion) for lA'l < 4.
21. Describe the most general lattice generated by a, b, c such that a > b9
a V c = b V c, a A c = b A c.
♦22. Describe the most general lattice generated by a, b, c, d, such that
a > b > c. (The diagram is given in Ju. I. Sorkin [1952]; the lattice has
twenty elements.)
♦23. Show that the most general lattice generated by a, b, c, d, such that
a > b, c > d, is infinite. (This lattice is described in H. L. Rolf [1958].)
24. Let N be the set of positive integers, L = {</i, i> | n e N, i = 0, 1}. Set
<w, /> < <w,y> if n < m and / < /. Show that L is a lattice and draw the
"generalized diagram" of L.
25. Draw the diagrams of all lattices with, at most, six elements.
To dispel a false impression that may have been created by Sections 1 and 2,
namely, that the proof that a poset is a lattice is always trivial, we present
exercises 26-36 showing that the poset Tn defined in exercise 34 is a lattice.
This is a result of D. Tamari [1951], first published in H. Friedman and D.
Tamari [1967]. The present proof is based on S. Huang and D. Tamari [a].
26. Let Tn denote the set of all possible binary bracketings of x0 x± • • • xn;
for instance, T0 = {x0}, 7i = {(*<> *i)}, T2 = {((*<> XiXta). (*0(*i *2))},
7*3 = i(xo(xi(x2 x3))), (xodxx x2)x3)\ i(x0 xi)(x2 x3)), ((xo(xi x2))x3)y
(((*o -^1)^2)^3)}. Give a formal (inductive) definition of Tn.
27. Replacing consistently all occurrences of (A-B) by A(B) in a binary
bracketing, we get the right bracketing of the expression. For instance,
the right bracketing of (C*oC*i x2))x3) is Xo(xi(x2))(x3) and of
((*0 *l)(*2 *3»
is x0(*i)(*2(*3)). Give a formal (inductive) definition of right bracketing
and prove that there is a one-to-one correspondence between binary and
right bracketings.
28. Show that in a right bracketing of x0 xx • • • xny there is one and only one
opening bracket preceding any xi9 1 < i < n.
29. Associate with a right bracketing of x0 Xx • • • xn a bracketing function
E: {1,..., n} -> {1,..., n} defined as follows: For 1 < 1 < n there is, by
exercise 28, an opening bracket before xt; let this bracket close following
Xi\ set E{i) = j. Show that E has the following properties: (i) 1 < E(i)
for 1 < / < n; (ii) i ^j < E(i) imply that E(j) < E(i) for 1 < 1 <
j < n.
♦30. Show that (i) and (ii) of exercise 29 characterize bracketing functions.
31. Let En denote the set of all bracketing functions defined on {1,..., n).
For E,Fe En set E < FiffE(i) < F(i) for all /, 1 < 1 < n. Show that <
is a partial ordering, En as a poset is a lattice, and (E A F)(i) =
inf {£(/), F(i%

18 FIRST CONCEPTS
32. The semiassociative identity applied at the place / is a map at: Tn-^- Tn
defined as follows: If E = • • • (A(BC)) • • •, where the first variable in
B and C is xt and xj9 respectively, then Eat = • • • ((AB)C)' • •; if is is not
of such form, then Eat = E. Let A'and Y denote the bracketing functions
associated with E and Eai9 respectively. Show that X(k) = Y(k) for
k ^ i, i < k < n, and Y(i) = j - 1. Conclude that X > Y.
33. Show the converse of exercise 32.
34. For E,Fe Tn define E < Fto mean the existence of a sequence is = X0,
X1,...,Xk = F,XteTn,0<i<k such that A'i + i can be gotten from
Xt by some application of the semiassociative law for 0 < i < k. Let
E < F mean E = F or E < F. Show that < is a partial ordering.
Verify that Figure 2.10 is the diagram of T3 and T4. Is the diagram of T4
optimal ? (No)
Figure 2.10
*35. Let X, Y e En and X >- Y. Let is and F be the binary bracketings
associated with X and Y, respectively. Show that F = Eat for some i.
36. Show that Tn is a lattice for each n > 0. (In fact, Tn s En.)
3. Some Algebraic Concepts
The purpose of any algebraic theory is the investigation of algebras up to
isomorphism. We can introduce two concepts of isomorphism for lattices.
The lattices £0 = <A>; <>, Si = <A; <> are isomorphic, (in symbol,
£0 ^ Sx), and the map 9?:L0^-L1 is an isomorphism if <p is one-to-one
and onto, and
a < bin 20iff a<p < b<p in 2X.
The lattices £0 = <£; A, V>, fia = <L2; A, V> are isomorphic (in

Section 3 Some Algebraic Concepts
19
symbol, £0 = £1), and the map <p: L0 -+LX is an isomorphism if <p is one-
to-one and onto, and if
(a A b)<p = a<p A b<p
and
(a V b)<p = a<p V 6<p.
It is easy to see that the two concepts coincide under the equivalence of
Theorem 1.1. However, when we generalize these to homomorphism
concepts, we get various new nonequivalent notions. In order to avoid
confusion, each will be given a different name.
From now on we will abandon the precise notation <L; A, V> and
<L; <> for lattices and posets; we will simply write italic capitals,
indicating the underlying sets, unless for some reason we want to be more
exact.
Note that the first definition of isomorphism can be applied to any two
posets L09 Ll9 thus yielding an isomorphism concept for arbitrary posets.
Having this concept of isomorphism, we can restate the content of Lemma
2.1: The diagram of a finite poset determines the poset up to isomorphism.
Let (£n denote the set {0,..., n — 1} ordered by 0 < 1 < 2 < • • • <
n — 1. Then <£n is an ^-element chain. If C = {x09..., Jtn-i} is an n"
element chain, x0 < xx < • • • < xn_l9 then <p: /-> jc4 is an isomorphism
between (£n and C. Therefore, the ^-element chain is unique up to
isomorphism.
The isomorphism of posets generalizes as follows:
The map <p: P0 -> Px is an isotone map (also called monotone map) of the
poset PQ into the poset Pl9 if a < b in P0 implies that a<p < b<p in Px.
A homomorphism of the semilattice <50; •> into the semilattice <5x; •>
is a map <p\SQ-^S1 satisfying (a-b)<p = ay-by. Since a lattice 2 =
<L; a , V > is a semilattice both under a and under V, we get two
homomorphism concepts, meet-homomorphism ( a -homomorphism) and join-
homomorphism (V-homomorphism). A homomorphism is a map that is
both a meet-homomorphism and a join-homomorphism. Thus a
homomorphism <p of the lattice L0 into the lattice Lx is a map of L0 into
Lx satisfying both (a A b)<p = a<p A b<p and (a v b)<p = a<p V b<p. A
one-to-one homomorphism will also be called an embedding. (The list
of homomorphism concepts will be further extended in Section 6.)
Note that meet-homomorphisms, join-homomorphisms, and (lattice)
homomorphisms are all isotone. Let us prove this statement for meet-
homomorphisms. If <p: L0 -> Ll9 (a A b)<p = a<p a b<p for all ajb e L09 and

20
FIRST CONCEPTS
*b
::"^
Figure 3.1
"V=*?
Figure 3.2
Figure 3.3
if x9y e L0, x < y in L0, then x — x N y\ thus x<p = (x A 7)9? = x<p A 793,
and xq> < y<p in Llt Note that the converse does not hold, nor is there any
connection between meet- and join-homomorphisms.
Figures 3.1-3.3 are three maps of the four-element lattice L of Figure
2.1 into the three-element chain (£3. The map of Figure 3.1 is isotone but is
neither a meet- nor a join-homomorphism. The map of Figure 3.2 is a
join-homomorphism but is not a meet-homomorphism, thus not a
homomorphism. The map of Figure 3.3 is a homomorphism.
The second basic algebraic concept is that of a subalgebra:
A sublattice® = <K; A, V > of the lattice £ = <L; a, V > is defined on
a nonvoid subset K of L with the property that a9b e K implies that
a a b9 a V beK(a9 V taken in £), and the A and the V of ® are
restrictions to K of the a and the V of S.
To put this in simpler language, we take a subset AT of a lattice L such
that K is closed under A and V. Under the same A and V, Kis a lattice;
this is a sublattice of L.
The concept of a lattice as a poset would suggest the following sub-
lattice concept: Take a nonvoid subset K of the lattice L; if the partial
ordering of L makes K a lattice, call K a sublattice of L. This concept is
different from the previous one and will not be used at all.
Let AK9 A e A, be sublattices of L. Then f\ (Ax | A e A) (the set theoretic
intersection of AM A e A) is also closed under A and V ; thus for every
H c L, H ^ 0, there is a smallest [H] ^ L containing H and closed
under A and v. The sublattice [H] is called the sublattice ofL generated
by H9 and H is a generating set of [H],
The subset K of the lattice L is called convex, if a9b e K9 ceL9 and
a < c < b imply that ceK. For a9b e L, a < b, the interval [a9 b] =
{x I a < x < b} is an important example of a convex sublattice. For a
chain C, a9b e C, a < b, we can also define the half-open intervals:
(a9b] = {x\ a < x < b} and [a, 6) = {x | a < x < b}, and the o/?e/*
interval: (a, b) = {* | a < x < b}. These are also examples of convex
sublattices. A sublattice / of L is an ideal if ieI and aeL imply that

Section 3 Some Algebraic Concepts
21
Figure 3.4
a A i e I. An ideal / of L is proper if / ^ L. A proper ideal / of L is /?rime if
a,beL and a A bElimply that ae7or be/. In Figure 3.4, /is an ideal
and P is a prime ideal; note that /is not prime.
The concept of a convex sublattice is a typical example of the interplay
between the algebraic and order theoretic concepts. We will now examine
these concepts more closely.
Since the intersection of any number of convex sublattices (ideals) is a
convex sublattice (ideal) unless void, we can define the convex sublattice
generated by a subset H, and the ideal generated by a subset H of the
lattice L, provided that H # 0. The ideal generated by a subset H will be
denoted by (//], and if H = {a}, we write (a] for ({a}]; we shall call (a] a
principal ideal.
Lemma 1. Let L be a lattice and let H and I be nonvoid subsets of L.
(i) I is an ideal iff a,b e / implies that a V b e I, and aE I, xeL, x < a
imply that xeL
(ii) I = (H] iff for all ie I there exists an integer n > 1 and there exist
K,..., /zn-i e H such that i < h0 V • • • V hn-x.
(iii) For a e L, (a] = {x | x < a}.
PROOF.
(i) Let / be an ideal. Then a,b e / implies that a V bel, since / is a sub-
lattice. If x < a e I, then x = x A a e /, and the condition of (i) is
verified. Conversely, let / satisfy the condition in (i). Let a,b e /. Then
a V bel by definition, and, since a A b < ael, we also have
a A bel; thus / is a sublattice. Finally, if x e L and a e /, then
a A x < ael, thus a A xeI, proving that /is an ideal. )
(ii) Set 70 = {/1 / < h0 v • • • V /*„_! for h0,..., /^-i e i/ and for some
integer n > 1}. Using (i), it is clear that I0 is an ideal, and obviously
H c /0, Finally, if H ^ J and / is an ideal, then 70 ^ 7, and thus I0
is the smallest ideal containing H; that is, I — IQ. )
(iii) This proof is obvious directly, or by application of (ii). %

22
FIRST CONCEPTS
Let I(L) denote the set of all ideals of L and let I0(L) = I(L) u {0}.
We call I(L) the ideal lattice and I0(L) the augmented ideal lattice of L.
Corollary 2. I(L) and I0(L) are posets under set inclusion, and as posets
they are lattices.
In fact, / V J = (I u J], if we agree that (0] = 0. From (ii) of Lemma 1
we see that for /,/eI(L), x e/ V J iff x < i V j for some iel, je J. It
does not matter that we consider only two ideals in this formula. In general,
V(/a|AeA) = (U(/a|AeA)];
that is, any nonvoid subset of I(L) has a supremum. Combining this
formula with Lemma 1(11), we have:
Corollary 3. Let 7A, A e A, be ideals and let I = \/ (IA\ Xe A). Then
i e I iff i < j\0 v • ■ • V An-i» for some *nteger n > I and for some
A0,..., An_1GA,yAiG/A..
Now observe the formulas :
(a] a (b] = (a A b], (a] V (ft] = (a V ft].
Since a 7^ ft implies that (a] ^ (ft], these formulas yield:
Corollary 4. L az/i fte embedded in I(L) and also in I0(L), and a -> (a] is
such an embedding.
Let us connect homomorphisms and ideals (recall that (£2 denotes the
two-element chain with elements 0 and 1).
Lemma 5.
(i) / is a proper ideal ofL iff there is a join-homomorphism 9? ofL onto (£2
such that I = 0<p-1 {the complete inverse image ofO, that is,
l={x\x<p = 0}).
(ii) / is a prime ideal ofL iff there is a homomorphism <pofL onto (£2 with
7=09-1.
PROOF.
(i) Let / be a proper ideal and define 9 by x<p = 0 if x e I, x<p = 1 if x $ I;
obviously, this 9 is a join-homomorphism. Conversely, if 9 is a

Section 3 Some Algebraic Concepts
23
join-homomorphism of L onto (£2 and / = (ty-1, then for a,b e /, we
have a<p = b<p = 0; thus (a V b)<p — a<p V b<p = 0 V 0 = 0, that is,
a v 6 e /. If a e /, x eL, x < a, then x<p < a<p = 0, that is, jcp = 0;
thus x e /. Finally, 9? is onto, therefore I ^ L. )
(ii) If / is prime, take the 9 constructed in the proof of (i) and note that <p
can violate the property of being a homomorphism only with a,b $ I.
However, since / is prime, a A b $ I; consequently (a A b)<p = 1 =
a<p A b<p, and so 9? is a homomorphism. Conversely, let <p be a
homomorphism of L onto (£2 and let 7 = 0<p_1. If a,b $ I, then a<p = b<p = 1,
thus (a A b)<p = a<p A b<p = 1, and therefore a A b$I;I is prime. #
By dualizing, we get the concepts of dual ideal (also called filter1),
principal dual ideal, [a) (principal filter), the dual ideal [H) generated by H,
proper dual ideal, prime dual ideal {prime filter, or ultra filter), the lattice
3>(L) of dual ideals ordered by set inclusion, and ^0(L) = @(L) u{0}
ordered by set inclusion. Note that in @(L) (and @Q(L)) the largest element
is L; if L has 0 and 1, then L = [0) is the largest and {1} = [1) is the
smallest element of @(L). Furthermore, for a,b e L we have
[a) A [b) = [a V b), and [a) V [ft) = [a A ft).
Lemma 6. Lef / fte aw ideal and let D be a dual ideal. IflnD^ 0, then
/ n D is a convex sublattice, and every convex sublattice can be expressed
in this form in one and only one way.
proof. The first statement is obvious. To prove the second, let C be a
convex sublattice and set / = (C], D = [C). Then C c / n D. If t e
I r\ D, then t e /, and thus by (ii) of Lemma 1, f < c for some ceC; also,
teD; therefore, by the dual of (ii) of Lemma 1, t > dfor some deC. This
implies that t e C since C is convex, and so C = / n D.
Suppose now that C has another representation, C = Ix Pi Dx. Since
C c /l9 we have (C] £ /j. Let a e Ix and let c be an arbitrary element
of C. Then a V c e Ix and a V c > c e Du so a V c e Dl9 thus
0 V ce/x O Z>! = C. Finally, a < a V ceC; therefore, ae(C]. This
shows that Ix = (C]. The dual argument shows that Dx — [C). Hence the
uniqueness of such representations. %
1 In the literature, filter means one of the following four concepts: (i) dual ideal;
(ii) proper dual ideal; (iii) dual ideal of the lattice of all subsets of a set; (iv) proper
dual ideal of the lattice of all subsets of a set. Further variants allow the empty set as
a filter under (i) or (ii).

24
FIRST CONCEPTS
An equivalence relation 0 (that is, a reflexive, symmetric, and transitive
binary relation) on a lattice L is called a congruence relation of L if
a0 = &0(@) and fli = &i(0) imply that a0 a a± = b0 a &i(0) and
a0 V fli = 60 V &i(0) (Substitution Property). Trivial examples are co, i,
defined by x = y(a>) iff x = j>; * s= >>(i) for all # and j>. For a e L, we write
[a]0 for the congruence class containing a, that is, [a]S = {* | x = a(S)}.
Lemma 7. Let S be a congruence relation of L. Then for every aeL,
[a] 0 is a convex sublattice.
proof. Let x,y e [a] 0; then x = a(0) and j> = a(0). Therefore, x A j =
a a a = a(0), and xvy = ava = a(0), proving that [fl]0 is a
sublattice. If x < t < y, x,y e [a]0, then x = a(0) and >> = a(0).
Therefore, t = t a y = / A fl(0), and t=*tvx = (tr\a)vx =
(t a a) V a = fl(0), proving that [a]0 is convex. £
Sometimes a long computation is required to prove that a given binary
relation is a congruence relation. Such computations are often facilitated
by the following lemma (G. Gratzer and E. T. Schmidt [1958e]):
Lemma 8. A reflexive and symmetric binary relation 0 on a lattice L is a
congruence relation iff the following three properties are satisfied for
x,y,z,teL:
(i) x = y(S) iffx A y = x V y(S).
(ii) x < y < z, x = y(S), andy = z(0) imply that x = z(0).
(iii) x < y andx = y(S) imply that x A t s y a t(B) and x V t =
7 V f(0).
proof. The "only if" part being trivial, assume now that a symmetric
and reflexive binary relation 0 satisfies conditions (i)-(iii). Let b,c e [a, d]
and a = d(S); we claim that b = c(0). Indeed, a = d(0)anda < Dimply
by (iii) that 6 Ac = aV(bAc) = dv(bAc) = d(Q). Now b A c < d
and (iii) imply that b A c = (b A c) A (b V c) = d A (b V c) = b V c(0);
thus by (i), b = c(0).
To prove that 0 is transitive, let x == y(Q) and j> = z(0). Then by (i),
x A y s * V j>(0),andby(iii),j> Vz = (jVz)v(}'Ax) = (}'Vz)v
(; V x) = x V j' V z(0), and similarly, xAjAz = jaz(0).
Therefore, xAyAz = yAz = yVz = xVyV z(0) and ^AyAz<
yAz<yvz<xVyVz. Thus, applying (ii) twice, we get x A y A z
= x v j> V z(0). Now we apply the statement of the previous para-

Section 3 Some Algebraic Concepts
25
graph with a = x A y A z9 b = x9 c = z9 d=xvyVz to conclude
that x = z(0).
Let x = j>(0); we claim that x V t = y v f(0). Indeed, x A y =
x V ;(0) by (i); thus by (iii), (x A y) V t ~= x V j> V f(0). Since
* V f, jV/e[(^Aj)v/, x V y V t], we conclude that * V t =
y v f(0).
To prove the Substitution Property for v, let x0 = j>0(®) and
*i = J>i(®)- Then x0 V xx = x0 V ^ = y0 V j>i(0), implying that
^o V ^! = jo V ^i(®) since 0 is transitive. The Substitution Property
for a is similarly proved. £
Let C(L) denote the set of all congruence relations on L partially
ordered by set inclusion (remember that every 0 e C(L) is a subset of L2).
As a first application of Lemma 8 we prove
Theorem 9. C(L) is a lattice. For 0,O e C(L)9 0AO=0nO.r/ie
join, 0 V ®, can be described as follows:
x = y (0 V 3>) iff there is a sequence z0 = x A y9 zl9..., zn_x = x V y
of elements of L such that z0 < z1 < • • • < zn_x and for each /, 0 < i <
n - l9zt = zi+1(0) or z, s z, + 1(0>).
remark. C(L) is called the congruence lattice of L. Observe that C(L) is
a sublattice of E(L) (exercises 45, 46); that is, the join and meet of
congruence relations as congruence relations and as equivalence relations
coincide.
proof. The first statement is obvious. To prove the second statement, let
T be the binary relation described in Theorem 9. Then 0 s T and O c Y
are obvious. If T is a congruence relation, 0 c r, O c r, and x = yQ¥)9
then for each /, either z, s= zi + 1(0) or z, = zi + 1(0); thus z4 = z(il(r). By
the transitivity of T, x A y = x V y(T); thus # = y(T). Therefore,
Yc r. This shows that if W is a congruence relation, then T = 0 v $.
T is obviously reflexive and symmetric. If x < y < z, x = yQ¥)9 and
y = z(T), then # e= z(T) is established by putting together the sequences
showing x s ^(T) and y s z(T). Let * s >>0F), * < j>, with z(9 0 < / < it
establishing this, and t e L. Then x A t = y A f(Y), * v f == j> V tQ¥)
can be shown with the sequences z{ A t9 0 < / < n, z{ v t9 0 < i < n9
respectively. Thus (i)-(m) of Lemma 8 hold for *F, and we conclude that
T is a congruence relation. £
Homomorphisms and congruence relations express two sides of the

26
FIRST CONCEPTS
same phenomenon. To establish this fact we first define quotient lattices
(also called factor lattices). Let L be a lattice and let 0 be a congruence
relation on L. Let L/0 denote the set of blocks of the partition of L
induced by 0, that is,
L/0 = {[d\<d\aeL}.
Set
[a]8 a [6]0 = [a a b]G
and
[a]0 V [b]Q = [av b]G.
This defines A and V onL/0. Indeed, if [a]S = [aj0 and [A]0 = [&i]0,
then a = fli(0) and b = £>i(0); therefore, a A b = ax A &i(0), that is,
[a A 5]0 = [fli A &J0. Thus A and (dually) V are well defined on L/0.
The lattice axioms are easily verified. The lattice Lj 0 is the quotient lattice
of L modulo 0.
Lemma 10. The map
9>e:x-> [x]0 (xeL)
is a homomorphism of L onto L/0.
remark. The lattice K is a homomorphic image of the lattice L if there is
a homomorphism of L onto K. Lemma 10 states that any quotient lattice
is a homomorphic image.
proof. The proof is trivial. %
Theorem 11 (The Homomorphism Theorem). Every homomorphic image
of a lattice L is isomorphic to a suitable quotient lattice of L. In fact, if
(piL-i-Li is a homomorphism of L onto Lx and if 0 is the congruence
relation ofL defined by x == >>(0) iff xq> = y<p9 then
L/0 SI*;
an isomorphism (see Figure 3.5) is given by
i/t: [x]&->x<p, xeL.

Section 3 Some Algebraic Concepts
27
proof. It is easy to check that 0 is a congruence relation. To prove that
tft is an isomorphism we have to check that 0 (i) is well defined, (ii) is one-
to-one, (iii) is onto, and (iv) preserves the operations,
(i) Let [*]0 = [y]@. Then x s y(®);thus X(P = W>that is>
(ii) Let (M0)0 = ([7] 0)0, that is, x<p = 79?. Then * = j>(0), and so
[*]© = M8.
(iii) Let aeLx. Since 9? is onto, there is an xeL with x<p = a. Thus
([*]©)* = «.
(iv) (M8 A [>>]0)0 = ([x A >>]0)0 = (x A >fc> = xq>Ayq>
= (M©)0 A (b]©)f
The computation for v is identical. #
The final algebraic concept introduced in this section is that of direct
product. Let L, K be lattices and form the set L x K of all ordered pairs
<a, &> with aeL, ^e£ Define A and V in L x AT "componentwise":
<fl, &> A <0x, &i> = <fl A 0i, 6 A &i>
<fl, &> V <fli, ^!> = (flVfl!,4V 6i>
This makes L x /£into a lattice, called the direct product of L and K(for
an example, see Figure 3.6).
Lemma 12. Let L, Lx, #, Kx be lattices, L ^ Ll9 K ^ K±. Then
Lx K^Lxx Kx^ Kx x Lx.
remark. This means that Lx J^is determined up to isomorphism if we
know L and K up to isomorphism, and the direct product is determined up
to isomorphism by the factors, the order in which they are given being
irrelevant.

28
FIRST CONCEPTS
proof. Let (piL-^Lx and tfs'.K-^Kx be isomorphisms and for aeL,
beK define (a9b)x = <^>^0>- Then x-L x K-^LX x K± is an
isomorphism. Of course, Lx x Kx ^ #1 x Lx is proved by showing that
<a, £> -► <6, a} (a eLube K^ is an isomorphism. £
If L,, / e /, is a family of lattices, again we first form the Cartesian
product of the sets Yl (A | i e /), which is defined as the set of all functions
f: I-+{J(Li\ iel) such that/(/) e Lh for all / e /. We then define A and
V "componentwise"; that is,/ A g = h9f V g = k means:
/(0 A g(i) = h(i\ /(/) V g(i) = k(i\ for all i e /.
The resulting lattice is the direct product Yl {Lt I *e ^)- If Lt — L for all
i e /, we get the direct power U. Letting n denote the set {0,..., n — 1},
Ln is ((L x L) x • • •) x L (at least up to isomorphism). In particular, if
/i-times
we identify/: 2->L with </(0),/(l)>, then we get L2 = L x L.
A very important property of direct products is:
Theorem 13. Let L and K be lattices, let & be a congruence relation ofL,
and let $> be a congruence relation of K. Define the relation 0 x 3> on
L x Kby
<a, by ss <c, </> (0 x 0>) (#-0 = c(0) andb = </(<!>).
77zew 0 x O « a congruence relation on L x K. Conversely, every
congruence relation ofL x K is of this form.
proof. The first statement is obvious. Now let V be a congruence relation
on L x K. For a,beL define a =. 6(0) if <a, c> = <Jb9 c>(T) for some
c e K. Let J e J£. Joining both sides with <a a 6, </> and then meeting with
<fl V b9 </>, we get <a, </> == <£>, ^/>(V); thus <fl, c> s <£>, c) for .some

Section 3 Some Algebraic Concepts
29
c g K is equivalent to (ja9 c> s= <Jb9 c> for a// c e K. Similarly, define for
ajb eK,a== b(&) iff <c, fl> = <c, &>(¥) for any/for all ceL.lt is easily
seen that 0 and <E> are congruences. Let <c, b) = <c, </>(0 x $); then
<a, x> = <c, *>0F), <>>, &> = <>>, </>(T). Joining the two with y = a A c
and x = 6 A </, we get <«, &> s <c, rf>(T). Finally, let <a, b> = <c, ^>(T).
Meeting with {a V c, 6 A </>, we get <a, b A d> = <cyb A d>Q¥)'y
therefore, a = c(0). Similarly, b = </(<£), and so
<a,6> = <c,rf>(0 x O),
proving that Y=0xd). #
In conclusion, we introduce a nonalgebraic concept to balance the false
impression that might have been created in this section that all lattice
theoretic concepts are algebraic.
A lattice L is called complete if f\ H and N/ H exist for any subset
H £ L. The concept is self-dual, and half of the hypothesis is redundant.
Lemma 14. Let P be aposet in which /\ H exists for all H £ P. Then P is
a complete lattice.
proof. For H c p9 let K be the set of all upper bounds of H. By
hypothesis, /\ K exists; set a = /\ K. If h e //, then A < A: for all A: e i£;
therefore h < a9 and a e K. Thus, a is the smallest member of K9 that is,
Lemma 14 can be applied to I0(L) and C(L). For a further application,
let Sub (L) denote the set of all subsets A of L closed under A and v
partially ordered under set inclusion. In other words, if A e Sub (L),
A ^ 0, then A is a sublattice of L. Obviously, Sub(L) is closed under
arbitrary intersections.
Corollary 15. I0(L) and C(L) are complete lattices. If L has a smallest
element, I{L) is a complete lattice. The lattice Sub (L) is a complete lattice.
Exercises
1. Formalize and prove the equivalence of the two isomorphism concepts.
2. Let <p: L0 ->Li, $: Li —►Lo be (lattice) homomorphisms. Show that if
<pi/t is the identity map on L0 and fa is the identity map on Ll9 then <p is
an isomorphism and ^ = y-1. Furthermore, prove that if <p:L0-+Lu

30
FIRST CONCEPTS
*lf:L1-^L2 are isomorphisms, then so are <p_1 (the inverse map,
<P ~1: h\ ~-> L0) and <ptl*.
3. A one-to-one and onto homomorphism is an isomorphism. Is a one-to-
one and onto isotone map an isomorphism ?
4. Find a general construction of meet-homomorphisms and join-homo-
morphisms that are not homomorphisms.
5. Find a subset H of a lattice L such that H is not a sublattice of L but H
is a lattice under the partial ordering of X restricted to H.
6. Show that a lattice is a chain iff every nonvoid subset of L is a sublattice.
7. Prove that a sublattice generated by two distinct elements has two or
four elements.
*8. Find an infinite lattice generated by three elements.
9. Verify that a nonempty subset / of a lattice L is an ideal iff, for ajb e L,
a v be I is equivalent to ajy e /.
10. Prove that if L is finite, then L and /(L) are isomorphic. How about
««?
11. Is there an infinite lattice L such that L ^ /(L), but not every ideal is
principal? (There is no such lattice; see D. Higgs [1971].)
12. Prove the completeness of I0(L) without using Lemma 14.
13. Prove that C(L) is complete by showing the following description of
infinite joins: Let H c C(L), O = y H. Then x = y(<!>) iff there exists
a finite sequence x A y — z0, zi,..., zn_i = jc v j>, such that z0 <
zi < • • • < zn-i, and zf = Zi + i(0i) for 0 < i < n and for some S{ e H.
14. Let <p: L -> J£ be an onto homomorphism, let / be an ideal of L, and let
/ be an ideal of K. Show that Iy is an ideal of K, and /<p~x = {a \ a e L,
a<P e /} is an ideal of L.
15. Is the image P<p of a prime ideal under a homomorphism <p prime again ?
16. Show that the complete inverse image P<p~x of a prime ideal P under an
onto lattice homomorphism <p is prime again.
17. Show that an ideal P is a prime ideal of L iff L — P is a dual ideal—in
fact, a prime dual ideal.
18. Let H be a subset of the lattice L such that a9beH implies that
a V beH. Then
(if] = {f | t < h for some h e #};
that is,
(J5T1- U((fl|AeJED.
19. Show that if A^ is a sublattice of L, then X is isomorphic to a sublattice of
HL).
*20. Verify that the converse of exercise 19 is false even for some finite K.
21. Prove that if 9£5 (see Figure 2.2) is isomorphic to a sublattice of 7(L), then
9^5 is isomorphic to a sublattice of L.
22. Find a lattice L and a convex sublattice C of L that cannot be represented
as [a] 0 for any congruence relation 0 of L.
23. State and prove an analogue of Lemma 8 for join-congruence relations.

Section 4 Polynomials. Identities, and Inequalities
31
24. Describe 0 V O for join-congruences.
♦25. Find a lattice L such that L ^ L/0 for all 0 ^ w, and there are
infinitely many 0 ^ a>.
26. Describe the congruence lattice of -ft5.
27. Describe the congruence lattice of an //-element chain.
28. Describe the congruence lattice of the lattice of Figure 2.3, and list all
quotient lattices.
29. Construct a lattice that has exactly three congruence relations.
30. Construct infinitely many lattices L such that each lattice is isomorphic
to its congruence lattice.
*31. Can an L in exercise 30 be infinite?
32. Generalize Lemma 12 to the direct product of more than two lattices.
33. Show that % = L x K implies that L or K has only one element.
34. Show that I(L) is conditionally complete: Every nonvoid set H with an
upper bound has a supremum, and dually. I(L) is complete iff L has a
smallest element.
35. Let L and K be lattices, let <p: L —> K be an onto homomorphism, and
let K have a smallest element 0. Then 0<p" x is called the ideal kernel of the
homomorphism <p. Show that the ideal kernel of a homomorphism is an
ideal of L.
36. Find an ideal that is the ideal kernel of no homomorphism.
37. Find an ideal that is the kernel of more than one (infinitely many)
homomorphisms.
38. Prove that every ideal of a lattice L is prime iff L is a chain.
39. Under what conditions is L x K planar ?
40. Let L and K be lattices and let <p: L —> AT be one-to-one and onto
satisfying {(a A b)<p, (a V b)<p] = {a<p A by>, ay v b<p) for all a,b e L. Let
A e Sub (L). Show that Acp e Sub (K); that in fact A -> A<p is a (lattice)
isomorphism between Sub (L) and Sub (AT).
41. Prove the converse of exercise 40.
42. Generalize Theorem 13 to finitely many lattices.
43. Show that the first part of Theorem 13 holds for any number of lattices
but that the second part does not.
44. Show that the second statement of Theorem 13 fails for (Abelian)
groups.
45. For a set X, let E(X) denote the set of all equivalence relations on X
partially ordered under set inclusion. Show that E(X) is a (complete)
lattice.
46. Show that C(L) is a sublattice of E(L).
4. Polynomials, Identities, and Inequalities
From variables xQ9xl9...9xn9...9 we can form polynomials in the
usual manner using A, V, and, of course, parentheses. Examples of

32
FIRST CONCEPTS
polynomials are: x0y x3> x0 v x0> (x0 A x2) V (x3 A x0), (x0 A xx) V
(x0 a x2) V (x± A x2). A formal definition is:
Definition 1. The set P<71> o/n-ary lattice polynomials is the smallest set
satisfying (i) and (ii):
(i) x, e P<n), 0 < i < n.
(ii) //>,? e P<n), //ze/i (/? A ?), (p V q)e P<n).
remark. We shall omit the outside parentheses and write px v • • • V pn
for (• • • (j>1 V p2) • • • V /?n)> and the same for A. Thus we write x0 V xx
for (x0 V xx) and x0 V Xj V x2 for ((x0 v *i) V x2).
By Definition 1, a polynomial is just a sequence of symbols. It is defined
because in terms of such a sequence of symbols we can define a function on
any lattice:
Definition 2. An n-ary polynomial p defines a function in n-variables (a
polynomial function, or simply, a polynomial) on a lattice L by the following
rules (fl0,...,fln-iel):
(i) Ifp = xiyp(a0,..., fln-i) = aiy0 < i < n.
(ii) 7/>(0o> • • •> fln-i) = ayq(aQy..., an_0 = byandp a q = r,p V q = t,
then r(aQy..., an_i) = a A b and t(aQy..., an_i) — a \t b.
Thus if /? = (x0 A x0 V (x2 V Xx),then^(fl, by c) = (a A b) v (c V 6)
= b y c. Definitions 1 and 2 are quite formal but their meaning is very
simple.
A polynomial is an w-ary polynomial for some n. We will also use
x,y,zy... instead of the xt.
Note that if/7 is a unary (n = 1) lattice polynomial, then p(a) = a for
any a e L. If p is binary, then /?(a, 6) = a, or b, or a A b, or a V b.
We shall prove statements on polynomials by induction on the rank.
The rank of x{ is 1; that of/? A q (and of/? V #) is the sum of the ranks of
/? and q.
Now we are in a position to describe [H], the sublattice generated by H:
Lemma 3. a e [/f] iff a = p(h0>..., A»_i) /or so/we integer n > 1, /or
.swwe n-ary polynomial py and for A0,..., An_i e /f.
proof. First we must show that if a = /?(A0,..., An-i)» h{E H, then
a g [/f ], which can be easily accomplished by induction on the rank of/?.

Section 4 Polynomials. Identities, and Inequalities
33
Then we form the set {a \ a = p(hQy.. .9hn-^)9n > \yh{eH} and observe
that it contains H and that it is closed under A and V. Since it is contained
in [H]y it has to equal [H]. %
Corollary 4. \[H]\ < \H\ + K0-
proof. By Lemma 3, every element of [H] can be associated with a finite
sequence of elements of /f u { (,), A, v}, and there are no more than
\H\ 4- X0 such sequences. £
Definition 5. A lattice identity (inequality) is an expression of the form
P = q (P ^ q)> where p and q are polynomials. An identity (inequality)
holds in the lattice L if p(a0> au ...) = q(a0i aly...) (p(a0y au ...) <
qfao* <*i> • • •)) holds for any a0, al9... eL.
An identity p = q is equivalent to the two inequalities p < q and
q < p> and the inequality p < q is equivalent to the identity p V q = q
(and to p A q = /?). Frequently, the validity of identities is shown by
verifying that the two inequalities hold.
One of the most basic properties of polynomials is:
Lemma 6. A polynomial (function) p is isotone; that is, if a0 < b0,
ax < bl9..., then p(a0, alf...) < p(b0, bl9...). Furthermore,
X0 A • • • A Xn-i < P(X0,. . ., Xn_i) < X0 V • • • V Xn_i.
proof. We prove the first statement by induction on the rank of/?. The
first statement is certainly true for/? = x{. Suppose that it is true for q and
r and that p = q A r. Then
/?(fl0,...) A/>(60,...)
= (tfteo, • • •) A r(a09...)) A (?(&<>, • • •) A r(b0,...))
= (q(aQ9...) A q(b09...)) A (r(a0,...) A r(*0,.. •))
= q(a0,...) A r(a09...) = /?(fl0,...);
thus p(a09...) < /?(^0, • • •)• The proof is similar for p = q V r. Since
x0 A • • • A xn_i < Xi < x0 V • • • V xn_i for 0 < i < n — 1, using the

34
FIRST CONCEPTS
idempotency of A and V, we obtain:
X0 A • • • A Xn_! = p(xQ A • • • A Xn_i, . . ., X0 A • • • A Xn_i)
^ P\Xo> Xi,.. ., xn-i)
< p(xQ v • • • v xn-i,..., xQ v • • • v xn_i)
= X0 V XX V ••• V Xn_i,
proving the second statement. %
A simple application is:
Lemma 7. Lef/?j = ^, 0 < i < ny be lattice identities. Then there is a single
identity p = q such that all /?< = qiy 0 < i < n, hold in a lattice L iffp = q
holds in L.
proof. Let us take two identities,p0 = q0 and/?! = qx. Suppose that all
polynomials are /i-ary and consider the identity:
(N) p0(x0y..., xn_0 A /?i(xn,..., x2n_0
= tfoC^o* • • -9 xn-i) A qi(xn9..., x2n-i)'
It is obvious that if p0 = q0y px = qx hold in L, then (N) holds in L. Now
let (N) hold in L and let a0,..., an-x e L. Substitute x0 = a0y..., xn_i =
fln-i, *„ = ••• = x2n_! = fl0 V • • • V fln_i = 0. By Lemma 6,
/?0(a0,..., fln_i)< /?0(fl, • •., a) = a = px(ay.. .ya)
and similarly for qQy qx; thus (N) yields p0(a0y..., an _ x) = tfofao,..., a» -1).
The second identity is derived similarly from (N). The general proof is
similar. %
The most important (and, in fact, characteristic) properties of identities
are given by
Lemma 8. Identities are preserved under the formation of sublattices,
homomorphic images, direct products, and ideal lattices.
proof. Let the polynomials /?, q both be /i-ary and let /? = q hold in L.
If Lj is a sublattice of L, then/? = q obviously holds in Lx. Let y.L^K
be an onto homomorphism. A simple induction shows that
p(a0y..., a„_i)<p = p(a0<p,..., an-X<p)

Section 4 Polynomials, Identities, and Inequalities
35
and proves the similar formula for q. Therefore,
p(a0<p9..., 0n-i9>) = P(<h> • • • > ffn-i)9>
= q(a09..., an-±)<p = q(a0<p,..., 0rt-i9>)>
and so p = q holds in K. The statement for direct products is also obvious.
The last statement is an obvious corollary of the following formula:
Let p be an /i-ary polynomial and let 70,..., /n_i be ideals of L. Then
/0 /B_i61(L); thus we can substitute the 7y into p: p(I0y..., /n_i) is
also in I(L), that is, an ideal of L. This ideal can be described by a simple
formula:
p(I0,..., In-i) = {x\x < p(i0,..., i„-i),
for some /0 e /0,..., /n_i e In.x}.
This follows easily from Lemma 6 and the formula in Section 3 describing
/ V /, by induction on the rank of p. %
Now we list a few important inequalities:
Lemma 9. The following inequalities hold in any lattice:
(i) (x A y) V (x A z) < x A (7 V z)
(ii) x V (j> A z) < (x V y) A (x V z)
(iii) (x A y) v (y A z) v (z A x) < (x v y) A (y v z) A (z v x)
(iv) (x A y) V (x A z) < x A (7 V (x A z))
remark, (i)-(iii) are called distributive inequalities, and (iv) is the modular
inequality.
proof. We prove (iv) as an example and leave the rest to the reader.
Since x A y < x> x A z < xy we obtain (x A y) V (x A z) < x, x A y <
y < y V (x A z), and x A z < y V (x A z); therefore, (x A y) V
(x A z) < y v (x A z). The meet of the two inequalities (x A y) V
(x A z) < x and (x A y) v (x A z) < y v (x A z) yields (iv). #
Lemma 10. Consider the following two identities and inequality:
(i) (x A y) V (x A z) = x A 0> V z)
(ii) (x V y) A (x V z) = x v (j A z)
(iii) (x v j>) A z < x v (y A z)
Tfce/i (i), (ii)9 and (Hi) are equivalent in any lattice L.

36
FIRST CONCEPTS
remark. A lattice satisfying identities (i) or (ii) is called distributive. Note
that (i) and (ii) are not equivalent for fixed elements; that is, (i) can hold
for three elements a,b,c e L, whereas (ii) does not.
proof. Let (i) hold in L and let a,b,c eL; then, using (i) with x = a V b,
y = 0, z = c,
(0 V b) A (a V c) = ((0 V b) A a) V ((0 V 6) A c)
and using a = (a v b) A a and (i) for x = c, j> = a, z = 6,
= 0 v (0 A c) v (b A c)
= a V (b A c),
verifying (ii).
The proof of (ii) implies (i) is the dual of the preceding paragraph.
Let (ii) hold in L\ then x V (y A z) = (x v y) A (x V z) > (x V 7) A z,
since x V z > z, verifying (iii).
Let (iii) hold in L. Then, with x = a,y = b,z = av cm (iii),
(fl V 6) A (a V C) < 0 V (b A (fl V C)) = 0 V ((fl V c) A 6)
and with x = 0, ^ = c, z = c in (iii)
< 0 V (0 V (c A &)) = 0 V (c A b).
This, combined with the dual of Lemma 9(i), gives (ii). %
Corollary 11. The dual of a distributive lattice is distributive.
Lemma 12. The identity
(x A y) V (x A z) = x A (y V (x A z))
1 j equivalent to the condition:
x > z implies that (x A y) V z = x A (y V z).
remark. A lattice satisfying either condition is called modular.
proof. If x > z, then z = x A z; thus the implication follows from the
identity. Conversely, if the implication holds, then since x > x A z, we
have (x A y) V (x A z) = x A (y V (x A z)). #

Section 4 Polynomials, Identities, and Inequalities
37
Exercises
1. Give a formal proof of Lemma 3.
2. Let H = {hQ,..., /*n-i}. Prove that a e [H] iff a = p(h0,..., /i^-i) for
some w-ary polynomial p.
3. Show that the upper bound in Corollary 4 is best possible if \H\ > 3.
Give the best estimates for \H\ < 2.
4. Give a formal proof of Lemma 8.
5. Prove (without reference to Lemma 8) that if L is distributive (modular),
then so is I(L).
6. Show that the dual of a modular lattice is modular.
7. Prove that L is distributive iff the identity
(x A y) V (y A z) v (z A x) = (x V j>) A (y V z) A (z V x)
holds in L.
8. Prove that every distributive lattice is modular, but not conversely. Find
the smallest modular but not distributive lattice.
9. Find an identity p = q characterizing distributive lattices such that
neither p < q nor q < p holds in a general lattice.
10. Show that in any lattice
V (A (xtj I 0 < / < m) | 0 < j < n)
^ A (V (*o I 0 < j < n) | 0 < i < w).
11. Prove that the following identity holds in a distributive lattice:
V (A (*v I 0 < j < n) | 0 <> i < m)
= A (V (*i/<o 1 0 < i < m) | fe F),
where F is the set of all functions from {0, 1,. .., m — 1} into {0, 1,...,
n- 1}.
12. Derive (i)-(iii) of Lemma 9 from exercise 10.
13. Verify that any chain is a distributive lattice.
14. Let L be a lattice with more than one element, let L' = Lu {0}, and let
0 < x for all x e L. Show that U is then a lattice and that an identity
p = q holds in L iff it holds in L'.
15. Show that if the identity x0 = p(x0, xl9...) holds in the two-element
lattice, then it holds in every lattice.
16. Prove that P(X) is a distributive lattice.
17. The set of all equivalence relations on the set X partially ordered under
set inclusion forms a lattice E(X). Show that E(X) is distributive iff
1*1 < 2 and modular iff \X\ < 3.
18. Find an identity that holds in (£2 but not in 9t5.
19. Find an identity that holds in (£2 but not in 9tt5.
20. Let K and L be modular lattices, let D be a dual ideal of K, and let / be
an ideal of L such that there exists an isomorphism <p of D with /. Set
A = K u (L — /) (disjoint union) and define x < y, for x,y e A as

38
FIRST CONCEPTS
follows: For x,yeK, or x,yeL — /, x < y retains its meaning; for
xe K, yeL — I, let x < y mean the existence of a z e D such that
x < z, z<p < y; x < y for no xeL - /, ;el Is <^4; <> then a
modular lattice?
21. If K and L of exercise 20 are distributive, is A distributive?
22. In exercise 20, put I = D = {a} and show that any identity holding in K
and L holds in A as well.
23. Examine the statements of Lemma 8 for properties of the form,
"If/?o = ?o, thenpi = ft."
*24. Show that <L; A, v> is a lattice satisfying the identity p = # iff it
satisfies the identities
(w A x) V x = x
and
[((* A /?) A z) V «] V V = [((? A z) A X) V t>] V [(/ V U) A «],
where x, z, «, i?, and w are variables that do not occur in p or q (R.
Padmanabhan [1968]).
*25. Show that the result of exercise 24 is best possible: If p = q is an identity
not satisfied by some lattice, then the two identities of exercise 24 cannot
be replaced by one (R. N. McKenzie [a]).
26. Show that the lattice L is modular iff the inequality x A (y v z) <
y V ((x v y) A z) holds in L.
5. Free Lattices
Though it would be quite easy to develop a feeling for the most general
lattice (generated by a set of elements and satisfying some relation) of
Section 2 by way of some examples, a general definition seems to be hard
to accept. So we ask the reader to withhold judgment on whether
Definition 2 expresses his intuitive feelings until the theory is developed and
further examples are presented.
The most general lattice will be called free. Since we might be interested,
for instance, in the most general distributive lattice generated by a, b, c,
satisfying b < a, it seems desirable to define freeness with respect to a class
K of lattices.
Definition 1. Let px = qx be identities for i e /. The class K of all lattices
satisfying all identities px = qi9 i e /, is called an equational class of lattices.
An equational class is trivial // it contains one-element lattices only.

Section 5 Free Lattices
39
* 5*W)
L
Figure 5.1
The class L of all lattices, the class D of all distributive lattices, and the
class M of all modular lattices are examples of equational classes of
lattices.
Next we have to agree on what kinds of relations to allow in the
generating set. Can we prescribe only relations of the form b < a, or do we allow
relations of the form aAb = c or dve=fl Lemma 9 is an example
in which the four generators x, y, z, u are required to satisfy x V y —
xVz = yVz = u. Let us therefore agree that for a generating set we
take a poset P, and for relations we take all a < b that hold in P, all
a A b = c, where inf {a, b} = c in P, and all a V b = c, where sup {a, b} =
c in P. (A more liberal approach will be presented later in this section.)
Now we are ready to formulate our basic concept:
Definition 2. Let P be a poset and let K be an equational class of lattices.
A lattice FK(P) is called a free lattice over K generated by P if the following
conditions are satisfied:
(ii) P £ FK(P), and for a,b,c e P, inf {a, b) = cinP iff a A b = c
in FK(P), and sup {a, b} = c in P iff a V b = c in FK(P).
(iii) [P] = FJF).
(iv) Lef LeK and let <p: P -> L be an isotone map with the
properties that ifa,b,ceP9 inf {a, b} — c in P, then a<p A b<p = c<p in
L, and if sup {a, b) = c in P9 then a<p v b<p = c<p in L. Then
there exists a {lattice) homomorphism $: FK(P) -> L extending
<p (that is, a<p = aif* for all a e P).
Let e denote the identity map on P; then the crucial condition (iv) can be
expressed by Figure 5.1. In that and in all such similar diagrams (not in the
sense the word is used in Section 2), the capital letters represent lattices or
posets, and the arrows indicate homomorphisms, or maps with certain
properties. Such diagrams are usually assumed to be commutative, which,

40
FIRST CONCEPTS
in this particular case, means et/t = <p, which is (iv). Note that (ii) is also
included in the diagram if the arrows are supposed to represent maps as
required in (iv). In fact, we could have required in (ii) that the identity map
on P be an embedding of P into FK(P) in the sense of (iv).
If P is an unordered set, |P| = m, we shall write FK(m) for FK(P) and
call it difree lattice on m generators over K. In case K = L, we shall always
omit "over L"; thus "free lattice generated by P" shall mean "free lattice
over L generated by jP"—in notation, F(P).
It should be noted that if b e FK(P), then by (iii) and by Lemma 4.3,
b = p(aQ,..., fln-i)> where p is a polynomial and a0i..., an-x eP. Thus
if the tfs of (iv) exists, then we must have
biff = p(a0,..., fln-i)0 (since */t is to be a homomorphism)
= p(a0t/t, . . ., fln-10) = P(<*09, • • •, 0n-l9>)>
since a^ = a^&fi) = a{<p.
From this we conclude:
Corollary 3. The homomorphism $ in (iv) is unique.
This corollary is used to prove:
Corollary 4. Let both FK(P) and F£(P) satisfy the conditions of
Definition 2. Then there exists an isomorphism x' F^P) -> F£(P), and x can be
chosen so that ax = a for all aeP. In other words, free lattices (over K
generated by P) are unique up to isomorphism.
proof. Let us use Figure 5.1 with L = F^(P) and <p = e. Then there exist
0i: FK(P) -> F£(P) and 02: F£(P) -> FK(P) such that ^ - * and <-02 - c
Thus 0!02: Fk(P) -> ^kCP) is the identity map e on P. By Corollary 3, € has
a unique extension to a homomorphism FK(P) -> /^OP); tne identity map
on Fk(P) is one such extension. Therefore, 0x02 is the identity map on
FK(P). Similarly, 020i is the identity map on F£(P), and so (exercise 3.2)
*/t1 is the required isomorphism. %
This settles the uniqueness, but how about existence ? Naturally, FK(P)
need not exist. For instance, Fj>(9l5) should be 9£5, since by (ii) and (iii) of
Definition 2, FK(P) = P if P is a lattice; but 9?5 £ D, so (i) is violated.

Section 5 Free Lattices
41
Theorem 5. Let P be a poset and let K be an equational class of lattices.
Then FK(P) exists iff the following condition is satisfied:
(E) There exists a lattice L in K such that P £ L and for
a,b,c e P, inf {a, b} = c in P iff a A b = c in L, and
sup {a, b} = c in P iff a V b = c inL.
proof. Condition (E) is obviously necessary for the existence of FK(P);
indeed, if FK(P) exists, (E) can always be satisfied with L = FK(P) by (i)
and (ii) of Definition 2.
Now assume that (E) is satisfied. In this proof a map <p: P -> N(NeK)
will be called a homomorphism if it satisfies the conditions set forth in
Definition 2(iv).
Obviously, in Definition 2(iv) it suffices to consider N with N = [P<p].
Let <N, 9>> denote this situation—that is, iVeK, <p:P-+N is a
homomorphism and N = [P<p]. Then FK(P), or, more precisely, (F^P), c> has
the property that for any <L, 9?) there exists a ^: FK(P) -> L with 9? = etft.
To construct FkOP) we have to construct a lattice having this property for
all <L, 9>>. How would we construct one for only two?
Let <LX, 9>i> and <L2, 9>2> be given. Form L± x L2 and define a map
9?: P-+L± x L2 by /*p = (p<pup<P2>\ set AT = [ify]. The fact that 9? is a
homomorphism is easy to check. A simple example is illustrated in
Figures 5.2-5.4. Now we define fa: <xl9 x2> -> *< and obviously, for
p eP, /Np^, = />9>f and fa: N->Lt.
Figure 5.2
vy—-
Figure 5.3

42
FIRST CONCEPTS
£1
Figure 5.4
If we are given any number of <Lt, (ft), i e /, we can proceed as before and
get <N, <p>; if one of the <Lf, <pf> is the <L, c> given by (E), then (ii) of
Definition 2 will also be satisfied. There is only one problem: All the pairs
<Li, ft} do not form a set, so their direct product cannot be formed. The
<Z,f, 9>f> do not form a set because a lattice and all its isomorphic copies do
not form a set; therefore, if we can somehow restrict taking too many
isomorphic copies, the previous procedure can be followed. Observe that
by Corollary 4.4, in every pair <Lt, <p,> we have
\Lt\ < \P9i\ + Xo < \P\ + X0.
Thus, by choosing a large enough set S and taking only those <Lj, <p,> that
satisfy Lt £ s, we can solve our problem.
Now we are ready to proceed with the formal proof. Choose a set S
satisfying \P\ + X0 = \S\. Let Q be the set of all pairs <Af, f) where
M c 5. Form ^ = n (^ I <^ 0> G Q\ and for each/? eP let/p e ^ be
defined by
Finally, set
N=[{fp\peP}].
We claim that if, for all peP, we identify p with /p, then N satisfies (i)-
(iv) of Definition 2, and thus N = FKCP)-
(i) N is constructed from members of K by forming a direct product and

Section 5 Free Lattices
43
by taking a sublattice. By Lemma 4.8, N e K, since K is an equational
class,
(ii) Let inf {a, b} = c in P. Then for every <M, 0> e Q, atft a bip = a/*, so
that /a«M,^» A/b«M,^»=/c«M,^», that is, fa A fb = fc.
Since/? is identified with/p, we conclude that a A b = c in N.
Conversely, let a A b = c in N, that is, fa A fb = /c. Let L be the
lattice given by (E) and let c be the identity map on P; then we can
form <L, €>. By Corollary 4.4, \L\ < \S\, so there is a one-to-one
map a: L -> 5. Let Lx = La and make Lx into a lattice by defining
aa A ba = (a A b)a, act V 6a = (a V 6)a.
Then L ^ Lx and we can form the pair <LX, ax>, where ax is the
restriction of a to P (c L). Since Lx £ s, <LX, ax> e Q. Now/a Afb=fc
yields
/.«!*, «i» A /.((A, ax» = /c«Ia, ai»;
that is, aa A ba = ca, which in turn gives a A b = c, since a is an
isomorphism. By (E), a A b = cinL implies that inf {a, b} = c. The
second part of (ii) follows by duality.
(iii) This part of the proof is obvious by the definition of N.
(iv) Take <L, <p); we have to find a homomorphism if*: N-+L satisfying
a<p = aifs for a eP. Using |iV| < \S\, the argument given in (ii) can be
repeated to find <Li, <Pi>, an isomorphism a:L-+Lx such that
a?** = fl9>! for all aeP and Lx c S. Therefore, <LX, 9?x> e Q. Set
^:f^f«Lu9lyi feN.
Then for agjP,
Thus the homomorphism 0 = ^i**"1: N-+L will satisfy the
requirement of (iv). #
Two consequences of this theorem are very important:
Corollary 6. For any nontrivial equational class K and for any cardinal
m, a free lattice over K w/fA m generators, FK(m), ejoste.
proof. It suffices to find an L e K, X ^ L such that |Z| = m, and for
x,y e X, x^ y, x and j> are incomparable. This is easily done. Since K is

44
FIRST CONCEPTS
nontrivial, there exists an N e K, |iV| > 1; thus, (52 is a sublattice of N. By
Lemma 4.8, (£2 e K; by Lemma 4.8, ((£2)7 e K for any set /. Let |/| = m,
let L = ((£2)7; for i e /, define feLby f(i) = 1, ./JO") = 0 for i # j, and
set X = {/; | i g /}. Obviously, Z satisfies the condition. #
Corollary 7. For a/iy /?o.yef P, a /ree lattice (over L) generated by the
poset P exists.
proof. Take a poset P and define /0(P) to be the set of all subsets I of P
satisfying the condition: sup {a, b} g / iff ajb g /. Partially ordering I0(P)
by set inclusion makes I0(P) a lattice. Identifying a with {x | x < a}, we see
that I0(P) contains P and satisfies (E) of Theorem 5. The detailed
computation is almost the same as that for Theorem 20, so it will be omitted. #
The argument given in the proof of Corollary 3 shows that whenever L is
generated by P, any homomorphism <p of P has at most one extension to
L, and if there is one, it is given by
if>:p(a0,..., fln_0 ->p(a0<p,..., an-x<p).
This formula gives a homomorphism iff 0 is well defined; in other words,
iff p(a0,..., fln_i) = q(b0t..., ^m_0 implies that p(a0<p,..., an.&) =
q(bQq>,..., 6m_!<?), for any 0O, • • •, an-i,b0,..., bm.1ePand<p:P-^NeK.
This yields a very practical method of finding free lattices and verifying
their freeness.
Theorem 8. In the definition ofF^P), (iv) can be replaced by the following
condition:
If beFK(P) has two representations, b = p(a0,..., fln-i) and
b - q(b0,..., &m_i) (fl0, • • -, fln-i, 6o, • • -, bm.1 eP), then
ca« fe derived from the identities defining K and the relations ofP
of the form a A b = c and a V b = c.
remark. Thus, in proving/? = q, we can use only the A and V table of
jP, but we cannot use a ^ b or a ^ b V c, and so on.
We illustrate Theorem 8 first by determining Fd(3). The following
simple observation will be useful:

Section 5 Free Lattices
45
* V y V z
x v y
(x v y) A (y V z)
z V x
(>Vz)A(:V x)
(x v y) a Ck v z) a (z v *)
Figure 5.5
Lemma 9. Lef x, y, and z be elements of a lattice L and let x V y, y V z,
z \l x be pairwise incomparable. Then {x V y, y V z, z V x} generates a
sublattice ofL isomorphic to ((£2)3 (see Figure 5.5).
proof. Almost all the meets and joins are obvious; by symmetry, the
nonobvious ones are typified by [(x V y) A (y V z)] V [(x V y) A
(z v *)] = x V j> and [(* V y) A (j> V z)] V (z v x) = x V j> V z.
Since j> < (x V y) A (j V z) and x < (x V y) A (z v x), we get
x V y < [(x V .y) A (y V z)] V [(x V rf A (z V x)], and > is trivial.
The second equality follows from y < (x V y) A (y V z). Note that,
for example, (x V j>) A 0> V z) = (x V y) A (j> V z) A (z V x) would
imply, by joining both sides with y V z, that jcVj>Vz = zV#; thus
xVj><zVx, a contradiction. Therefore, all eight elements are
distinct. #
Theorem 10. A free distributive lattice on three generators Fu(3) has
eighteen elements (see Figure 5.6).
proof. Let x, y, and z be the free generators. The top eight and the
bottom eight elements form sublattices by Lemma 9 and its dual; note
that (x A y) V (y A z) V (z A x) = (x V y) A (y V z) A (z V x) by
exercise 4.7.
According to Theorem 8, we have only to verify that the lattice L of
Figure 5.6 is a distributive lattice, and that if p, q, r are polynomials
representing elements of Land/? A q = rinL, then/? A q = r in every
distributive lattice and similarly for v. The first statement is easily proved by

46
FIRST CONCEPTS
a = (x V y) A (x V z)A C V z)
= (x A jO V (x A 2) v (y A z)
x a j> a z
Figure 5.6
representing L by sets (see exercise 13). The second statement requires a
complete listing of all triples p, q> r with p A q = r. If /?, #, r belong to
the top eight or bottom eight elements, the statement follows from
Lemma 9. The remaining cases are all trivial except when p or q is jc, y,
or z. By symmetry, only p = x> q = y V z, r = (x A y) V (x A z) is left
to discuss, but then p A q = r is the distributive law. #
Theorem 11 (R. Dedekind [1900]). A free modular lattice on three
generators FM(3) has twenty-eight elements (see Figure 5.7).
proof. Let jc, y, and z be the free generators. Again modularity is
proved by a representation (see exercise 16). Theorem 10 takes care of most
meets and joins not involving xl9 yly zx. Of the rest, only one relation (and
the symmetric and the dual cases) is difficult to prove: x1 A yx = u. This
we do now, leaving the rest to the reader.

Figure 5.7
Compute:
*i A j>! = [(x A v) V u] A [(7 A v) V w] (since w < (j> A 1?) V u)
= [(* A 1?) A ((y A 1?) V w)] V u
= [(x A v) A (y v u) A v] V u (substitute u and v)
= [* A (y V z) a (j> V (x A z))] v u
= (x A y) V (x A z) V u = u. #
Consider the lattice represented by Figure 5.8. We would like to say that
it is freely generated by {0, a, b, 1} = P, but this is clearly not the case
according to Definition 2, since sup {a, b) = 1 in P, whereas in the lattice,

48 FIRST CONCEPTS
0
Figure 5.8
a V b < 1. So to get the most general lattice of Section 2 we have to
enlarge the framework of our discussion by introducing partial lattices. Of
course, the study of partial lattices is important also for other purposes.
Definition 12. Let L be a lattice, H £. L, and restrict A and v to H as
follows: For aybyc e Hy if a A b = c (dually, a V b = c)y then we say that
in Ha A b (dually, a V b) is defined and it equals c; if for ayb e H, a A b
(duallyy a v b) $ Hy then we say that a A b (dually; a V b) is not defined in
H. Thus </f; A, V > is a set with two binary partial operations. </f; A, V >
is called a partial lattice. <i/; A, V > is called a relative sublattice of L.
Thus every subset of a lattice determines a partial lattice. The second
part of this section is devoted to an internal characterization of partial
lattices, based on N. Funayama [1953].
We now analyze the way the eight identities that were used to define
lattices ((L1)-(L4) of Section 1) hold in partial lattices:
Lemma 13. Let </f; A, V > be a partial lattice, atbyC e H.
(i) a A a exists, and a A a = a.
(ii) IfaAb exists, then b A a exists, and a A b = b A a.
(iii) If a A by (a A b) A c, b A c existy then a A (b A c) existSy and
(a A b) A c = a A (b A c). Ifb A c, a A (b A c), a A b existy then
(a A b) A c exists, and (a A b) A c = a A (b A c).
(iv) IfaAb exists, then a v (a A b) exists, and a = a V (a A b).
proof. As an illustration let us prove the first statement of (iii). Let
H £ L as in Definition 12. The assumption that a A b, (a A b) A c,

Section 5 Free Lattices
49
b A c exist in H means that in L, ay b, cy a A by (a A b) A cy b A c e i/.
But in Ly (a A b) A c = a A (b A c)y and so a A (b A c) e H; that is,
a a (b A c) exists in i/, and, of course, (a A b) A c = a A (b A c). #
Lemma 13'. Lef (i')-(iv') de/iote the statements we get from (i)~0v) °f
Lemma 13 fly interchanging A and v. 77^/* (i')-(iv') hold in any partial
lattice.
proof. This proof is trivial by duality. %
Statements (i)~0v) and (i')-(iv') give the required interpretation of the
eight identities for partial lattices.
Definition 14. A weak partial lattice is a set with two binary partial
operations satisfying (i)—(iv) and (i')-(iv').
Corollary 15. Every partial lattice is a weak partial lattice.
Based on Figure 5.9, we give the following example of a weak partial
lattice that is not a partial lattice.
Let H = {0, ay b, cy d, e,fy gy h, 1}. Consider the lattice $ of Figure 5.9.
Define x A y = z in H if x A y = z in$. Define x V y = z in H if either
x < y in $ and y = z, or y < x in © and x — z\ or if {#, y) = {a9 c}y and
z = /; or {*, >>} = {&, J}, and z = g; or {*, >>} = {/, g}, and z = 1. Then
<#; a , V > is a weak partial lattice (check the axioms). Now suppose that
there exists a lattice L, H c l, such that <//; A, V > is a relative sublattice
0
Figure 5.9

50
FIRST CONCEPTS
of L. Then 1 = (a V c) V (b V d) in L, and thus 1 = sup {a, b, c, d}.
Since e > a,b, and h > c,d in L, and 1 > e,/*, we get 1 = sup {e, h} in L.
The fact that e,h,\ e H implies that e V h is defined in /f (and equals 1),
contrary to the definition of <//; A, V> (compare this with Lemma 19).
To avoid such anomalies we shall introduce two further conditions. To
prepare for them we prove:
Lemma 16. Let <//; A, V> be a weak partial lattice. Then we define a
partial ordering relation < on Hby"a < biff a A b exists and a A b = a."
If a V b exists, then a V b = sup {a, b}. If a A b exists, then a A b =
inf {a, b}. Also, a < b iff a V b = b.
proof. This proof is the same as the proof of the corresponding parts of
Theorem 1.1, except that the arguments are a bit longer. %
Note that in a partial lattice sup {a, b} may exist but a V b does not. For
instance, let L be the lattice of Figure 5.9, H = {0, a, b, 1}. Then sup {a, b)
= 1 in H, but a V b is not defined in H because a V b£H.
Definition 17. An ideal o/a weak partial lattice H is a nonvoid subset
I of H such that if a,b e / and a V b exists, then a V be I, and x < ael
implies that x e I. Again we set {x \ x < a} = (a]. Dual ideal and [a) are
defined dually. I0(H) is the lattice consisting of 0 and all ideals of H
{partial ordering is c)? @0{H) is the lattice consisting of 0 and all dual
ideals of H {partial ordering is c). For K c H, {K] is the ideal and [K) is
the dual ideal generated by K.
Corollary 18. Let HandL be given as in Definition 12. Let I be an ideal
ofL. Then I r\ H is an ideal of H provided that IO H ^ 0.
Lemma 19. Any partial lattice satisfies the following condition:
(I) If {a] V {b] = {c] in I0{H), then a V b exists in Hand equals c.
proof. Let H and L be given as in Definition 12, let a,b,c e H, and let
{a] V {b] - {c] in IQ{H). Set / = {a v b]L. Then {a]H, {b]H c In h, thus
(c]H = Mh V {b]H c {a v b]L, that is, c < a V 6. Since a < c,b < c,v/q
conclude that a V 6 = c. 0
Let (D) denote the condition dual to (I), namely:
(D) If [a) V [b) = [c) in @0{H), then a A b exists in H and
equals c.

Section 5 Free Lattices
51
Theorem 20 (N. Funayama [1953]). A partial lattice is a weak partial
lattice satisfying conditions (I) and (D).
proof. Corollary 14 and Lemma 19 and its dual prove that a partial
lattice is a weak partial lattice satisfying (I) and (D). Conversely, let
<#; A, V > be a weak partial lattice satisfying (I) and (D). Consider the
map
<?:*-><(*], [x)>,
sending H into I0(H) x 30(H), where &0(H) is the dual of @Q(H). This
map (p is one-to-one. If x V y = z, then (*] V (y] = (z] in /0(i/) and
[*) v b) = tz) m ®o{P)> thus ^Vj/?) = (xV jO?- Conversely, if
x<p V y<p = z<p, then (x] V (>>] = (z] in I0(H). Therefore, by (I), x v y
exists and equals z, so x<p V y<p = z<p implies that x V j> = z. A similar
argument shows that x A j> = z iff x<p A >><p = z<p. Thus we can identify
x with x<p, getting i/ c L = /0(/0 x&0(H). We have just proved that
<i/; A, V > is a relative sublattice of L. £
Let <jP; <> be a poset. We make P into a partial lattice as follows:
a A b is defined iff inf {a, b} exists and a A b = inf {a, 6}, and similarly
for a V b.
Lemma 21. <P; a , V > is a partial lattice.
proof. It is easy to verify that (I), (D), (i)-(rv) of Lemma 13, and (i')-
(iv') of Lemma 13' hold. Another proof identifies xeP with (jc], and thus
we get P c /(i>). Then we observe that A and V of I(P) restricted to P
give the A and V of P. #
We need some further definitions.
Definition 22. Let <^; A, v>, (B; a, v> be weak partial lattices,
<p:A-+B. We call 9 a homomorphism if, whenever a A b exists for
a,b e A, then a<p A b<p exists and (a A b)<p = a<p A b<p, and if the similar
condition holds for V . A one-to-one homomorphism <p is an embedding
provided that a Kb exists iffatp A by exists, and if the similar condition holds
for V.Ify is onto and <p is an embedding, then <p is an isomorphism.
Now we are ready again to define the most general lattices of Section 2.

52
FIRST CONCEPTS
Definition 23. Let % = (A; A, V > be a partial lattice and let K be an
equational class of lattices. The lattice FK($t) (or, simply, FK(A)) is a free
latticeover K generated by 51, if the following conditions are satisfied:
(i) FK(A)eK.
(ii) A c FK(A), and A is partial sublattice of FK(A).
(iii) [A] = F^A).
(iv) IfL e K and<p: A -> Lis a homomorphism, then there exists a
homomorphism */t: FK(A)->L extending <p (that is, a<p = a*!*
for a e A).
If P is a poset, then FK(P) is a free lattice over K generated by P, where
P is considered a partial lattice as in Lemma 21. This theory can be
developed exactly as it was in the first part of this section. The final result is:
Theorem 24. Let 21 = (A; A, v > be a partial lattice and let K be an
equational class. Then FK(9t) exists iff there exists a lattice L in K such that
21 is a relative sublattice ofL.
As an application we prove the existence of the lattice absolutely freely
generated by a poset. Let P be a poset; we define a partial lattice Pm on P
as follows:
x A y = z in Pm iff* and y are comparable and z = inf {x, y};
x V y = z in Pm iff* and y are comparable and z = sup {x, y}.
Definition 25. F(Pm) (= PL(Pm)) is called a lattice absolutely freely
generated by P.
Theorem 26. For any poset P, a lattice absolutely freely generated by P
exists.
proof. A subset A £ P is called hereditary if x e A and y < x imply that
ye A. Let H(P) be the set of all hereditary subsets of P partially ordered
under set inclusion. Let Px = H(P), P2 = H(P1), where P2 is the dual of
Pi. Identifying/? eP with (/?], we get P ^ Px; identifying/? ePx with [/>),
we get Pi c p2> thus P ^ p2. Obviously, P2 is a lattice. Let a,b,c e P,
a V b = c in P2. If fl and 6 are incomparable, then (a] U (b] (^ (c]) is an

Section 5 Free Lattices
53
upper bound for a and b, thus a V b < c in P2. A similar argument works
for a A b = c. Therefore, P2 ^ P satisfies the condition of Theorem 24. #
Exercises
1. Show that an equational class is closed under the formation of sub-
lattices, homomorphic images, and direct products.
2. Show that the class T of all one-element lattices is an equational class.
For any equational class K,K2 T.
3. Let K|, / e /, be equational classes. Show that p| (Ki I 'G O ls again an
equational class.
4. Find an equational class A with A c M, D c A, and A ^ M, A ^ D.
(Hint: Let A be defined by
i
(x V y) A (x V z) A (x V u)
= x V [(x V y) A z A u] V [(* V z) A >> A «] V [(x V «) A y A z].)
5. Let K be a nontrivial equational class. Show that K contains arbitrarily
large lattices.
6. Let P = {0, a, b}, inf {a, b) = 0, and let K be a nontrivial equational
class. Show that FK(P) s F(2).
7. Find an example of an isomorphism <p: FK(P) —> FK(P) that is not the
identity map on P.
8. Let P be a poset, let K, N be equational classes, and assume that FKCP)
and FN(P) exist. Prove that if K ^ N, then there exists a homomorphism
<P from FK(P) onto FN(P) such that <p is the identity map on P.
9. Work out a proof of the existence of F(3) without any reference to
Theorem 5.
10. Prove Theorem 8.
11. Formulate and prove the form of Theorem 8 that is used in the proofs of
Theorems 10 and 11.
12. Prove that FM(4) is infinite (G. Birkhoff [1933]). (Hint: Let R be the set
of real numbers and let L be the lattice of vector subspaces of R3. Set
a = {<*, 0, x> | x e R}> b = {<0, x9x>\xe R}>
c = m0yx>\xeR}t d = Kx>X>x>\xeR}.
Then [{a, b, c, d}] is infinite.)
13. Let A, B be disjoint three-element sets. Let L be the set of the following
subsets of A u B: all X c A, all A u Y, Y c £, all three-element sets
Z with \Zc\ A\ = 2. Prove that <L; c> is a lattice, that A and V are
intersection and union, and that thus L is distributive. Figure 5.6 is the
diagram of L (A. D. Campbell [1943]).

54
FIRST CONCEPTS
14. Represent the lattice of Figure 5.6 as a sublattice of ((£2)6-
15. Prove that six is best possible in exercise 14.
16. Represent the lattice of Figure 5.7 as a sublattice of L x 9tt5, where L is
the lattice of exercise 13 and %R5 is the lattice of Figure 5.10.
Figure 5.10
17. Show that condition (E) of Theorem 5 is equivalent to the following:
For any a,b,c e P not satisfying inf {a, b} = c, there exist a lattice L in K
and a homomorphism <p: P->L with a<p A by ^ c<p, and dually.
18. The statement "21 = (A; A, V > is a partial algebra'* means that A is a
nonvoid set and that A, V are partial binary operations on A. For an
w-ary polynomial p, a0,..., an-± e A, interpret p(a0,..., an-i). (When
is it defined and what is its value?)
19. In the weak partial lattice represented by Figure 5.9,
p = x A [(x v y) v (x V z)]
is not defined for x = 0, y — e, z = h. Verify that in every lattice
p = q, where q = x and q is defined for x = 0.
20. An identity p = q holds in the partial algebra % = <.A; A, V > if the
following three conditions are satisfied:
(i) If p(a0,...), q(a0i...) are defined (a0i... e A), then p(a0,...) =
(ii) If p(a0,...) is defined, if q = qo*qi, where * is A or V, and if
q0(a0,...), #i(ao, • • •) are defined, then q0(a0,...) * #i(a0, •..) is
defined.
(iii) This condition is the same as (ii) with p and q interchanged.
Check that Lemmas 13 and 13' give this interpretation to the lattice
axioms.
21. Let p = (((x V z) v (y v «)) V v) V w and let
q = [(v V x) V (v V >0] V [(w V z) V (w V «)].
Show that p = q in any lattice. Show that p — q does not hold in the
weak partial lattice defined in connection with Figure 5.9.
22. Let /o, h be ideals of a weak partial lattice. Set J0 = l0u Iu Jn =
{x\x <y V z.ytzeJn-t^n = 1,2,.. ..and/ = UWI» = 0,U...).
Show that / = /o V A.
23. Let the weak partial lattice L violate (/); that is, (a] V (b] = (c], but
a V b is undefined. Let 70 = («], /i = (6], and ce/„ (see exercise 22).

Section 5 Free Lattices
55
Generalizing exercise 21, find an identity p = q that holds in any
lattice but not in L. (Exercise 21 is the special case in which n = 2.)
24. Prove that a partial algebra 51 = (A; A, v > is a partial lattice iff every
identity p = q holding in any lattice also holds in 51.
25. A homomorphism of a partial algebra 5t = (A; A, V > into a lattice L is
a map <p: A -» L such that (a A £)<? = a<p A by whenever a A b exists,
and the same for v . Prove that there exists a one-to-one homomorphism
of 51 into some lattice L iff there exists a partial ordering < on A
satisfying a A b = inf {a, 6}, whenever a A 6 is defined in 51, and a v b =
sup {a, 6}, whenever a V 6 is defined in 51.
26. Show that every weak partial lattice satisfies the condition of exercise 25,
but not conversely.
27. Let A = {0, a, b, c, 1}, 0 < a, b, c < 1. For x < y define
xAy = yAx = x, xVy = yVx = y,
and define
aAb = bAa = 0y aVb = bVa=l.
Show that (A; A, v > is a partial lattice.
28. Let A = {0, a, b, c, rf, 1}, 0 < a, b, c, J < 1. For x < y define
xAy = yAx = x, xVy = yVx = y>
and define
aAb = bAa = cAd = dAc = 0,
a\lb = bva = cvd=d\lc=\.
Show that <>4; A, V > is a partial lattice.
29. Let L and K be lattices and let L n K be a sublattice of L and of K. For
x,.y eLv K define jc a ^ = z iff x,y,z e L and x A y = z in L, or if
jc^z g X and * A y = z in ^T; define x v y similarly. Is <I u ^, a , v >
a partial lattice?
30. Are the eight axioms of a weak partial lattice independent ?
31. Are the ten axioms of a partial lattice independent?
32. Define weak partial semilattice and partial semilattice; prove the
analogue of Theorem 20 for partial semilattices.
33. Let A be a weak partial lattice in which a A b exists for all a,be A. Then
A is a partial lattice iff a -> (a] is an embedding of A into I0(A).
34. Let T be the equational class of all one-element lattices. Show that
FT(A) exists iff \A\ = 1.
35. Let P = (C£2)2. Show that for any nontrivial equational class K, FK(P)
and FK0Pm) exist and that always FK(P) Z FK(Pm).
36. Determine F(P), FM(P), and FD(/>), where P = {a, b, c) and a < b.
37. Discuss the set of all weak partial lattices on A inducing a given partial
ordering on A.
38. Repeat exercise 37 for partial lattices.
39. Show that a one-to-one homomorphism of weak partial lattices need not
be an embedding.

56
FIRST CONCEPTS
40. Show that in Theorem 24, the condition " there exists a lattice L in K
such that 51 is a relative sublattice of L" can be replaced by the following
condition:
For all a,b,c e A for which a A b = c does not hold, there exists a lattice
L in K, and a homomorphism <p of A into L such that a<p A b<p ^ c<p, and
the same condition holds for v .
41. Let (A; A, v> be a partial algebra, let K be an equational class of
lattices, let L e K, and let M be a relative sublattice of L. Then M is
called a maximal homomorphic image of A in K if there is a
homomorphism <p of A onto M such that if ^ is a homomorphism of A into
iVeK, then there is a homomorphism a: M-+ N such that <pa = ^.
Prove that a maximal homomorphic image is unique up to
isomorphism if it exists.
42. Starting with an arbitrary partial algebra <>4; A, v>, carry out the
construction of Theorem 5 (Theorem 24).
43. Find examples of posets P0 <= P such that F(P%) = F(Pm).
44. After M. M. Gluhov [1960], a finite partial lattice 51 is a basis of a lattice
L if L = F(9l), but L = F($0) for no ^0 c A Show that the lattice
L = F($C) has more than one basis where 21 is defined as follows: Let
<>4; <> be the poset given by Figure 5.11; for x < y let x A y =
y A jc = jc; furthermore, the join of any two elements is defined in
{0, a, b, c, d, e,f9 1} as supremum, and 1 v ^ = ; v 1 = 1. (This
example, which is due to C. Herrmann, contradicts M. M. Gluhov's
result.)
Figure 5.11
6. Special Elements
A zero of a poset P is an element 0 with 0 < x for all x e P. A unity 1,
satisfies x < 1 for all x e P. There are at most one zero and at most one
unit. A bounded poset is one that has both 0 and 1.
A {0, \yhomomorphism (of a bounded lattice into another one) is a
homomorphism taking zero into zero and unit into unit. A {0, \}~sublattice

Section 6 Special Elements
57
of a bounded lattice L is a sublattice containing the 0 and 1 of L. Similarly,
we can define {0}-homomorphism, and so on, for lattices and sernilattices.
In a bounded lattice L, a is a complement of b if a A b = 0 and
flvi=l.
Lemma 1. In a distributive lattice, an element can have only one
complement.
proof. If b0 and bx are both complements of a, then b0 = b0 A 1 =
b0 A (a V 6i) = (60 A a) V (60 A &i) = 0 V (b0 A b±) = b0 A bx;
similarly, hi = b0 A blt thus b0 = b±. #
Let ae[6, c]; # is a relative complement of a in [b,c] if
a A x = b9 a V x = c.
Lemma 2. Zrc a distributive lattice, if a has a complement, then it also has
a relative complement in any interval containing it.
proof. Let d be the complement of a. Then x = (d V b) A c is the
relative complement of a in [b, c], provided that b < a < c. Indeed,
a A x = a A (d V b) a c = ((a A d) V (a A b)) A c
= (0 V b) A c = b,
a V x = a V ((d V b) A c) = (a V d V 6) A (a V c)
= 1 A (a V c) = c. #
Lemma 3 (De Morgan's Identities). In a distributive lattice, if a, b have
complements a' and b', respectively, then a A b and a V b have
complements {a A b)' and (a V b)', respectively, and
(a a b)' = a' V b'\
(a V b)f = a' A b'.
proof. By Lemma 1 it suffices to prove that
(a Ab) A {a' V b') = 0, (a A b) V {a' V *') = 1
to verify the first law; the second is dual. Compute:
{a Ab) A (a! V b') = (a A b A a') V (a A b A b')
-0 V 0 = 0

58
FIRST CONCEPTS
and
(a A b) V (a' V V) = (a v a' V &') A (b V a' V b')
= 1 A 1 = 1. •
A complemented lattice is a bounded lattice in which every element has
a complement. A relatively complemented lattice is a lattice in which every
element has a relative complement in any interval containing it. A Boolean
lattice B is a complemented distributive lattice. Thus, in a Boolean lattice
5, every element a has a unique complement, and B is also relatively
complemented.
A Boolean algebra is a Boolean lattice in which 0, 1, and '
(complementation) are also considered to be operations. Thus a Boolean algebra is a
system: </?; A, V,', 0, 1>, where A, V are binary,' is a unary operation,
and 0, 1 are nullary operations. (A nullary operation picks out an element
of B.) A homomorphism <p preserves 0, 1 and '; that is, it is a {0, l}-homo-
morphism satisfying (xq>)' = x'<p. A subalgebra is a {0, l}-sublattice closed
under '. 932 will denote the two-element Boolean algebra.
Note that in a bounded distributive lattice L, if b is a complement of ay
then b is the largest element x of L with a A x = 0. More generally, let L
be a lattice with 0. An element a* is a pseudocomplement of a (e L) if
0 A a* = 0, and a A x = 0 implies that x < a*. An element can have at
most one pseudocomplement. A pseudocomplemented lattice is one in
which every element has a pseudocomplement.
The concept of pseudocomplement involves only the meet operation.
Thus we can also define pseudocomplemented semilattices.
Theorem 4. Let L be a pseudocomplemented meet-semilattice, S(L) =
{a* | a e L}. Then the partial ordering ofL partially orders S(L) and makes
S(L) into a Boolean lattice. For ajb e S(L) we have a A b e £(L), and the
join in S(L) is described by
a v b = (a* A b*)*.
remark. Even if L is a lattice, the join in L need not be the same as the
join in S(L). This result was proved for complete distributive lattices by
V. Glivenko [1929]. In 1959 the result was established in its full generality
by E. T. Schmidt and the author as a method of constructing lattices from
semilattices. Since the intended application (to congruence lattices of
lattices) fell through, the result was not published. The same result was

Section 6 Special Elements
59
rediscovered and published by O. Frink [1962]. Both proofs used special
axiomatizations of Boolean algebras to get around the difficulty of proving
distributivity. The present proof is direct and is the simplest of the three.
E. T. Schmidt [1968] has succeeded in applying Theorem 4 as it was
originally intended.
proof. We start with the following observations:
a < a**. (1)
a < b implies that a* > b*. (2)
a* = a***. (3)
aeS(L)if£a = a**. (4)
aybe5(L) implies that a A beS(L). (5)
For ayb e S(L), supS(L) {a, b} = (a* A 6*)*. (6)
Formulas (1) and (2) follow from the definitions.
Formulas (1) and (2) yield a* > a***, and by (1) a* < a***, thus (3),
If a e S(L), then a = b*; therefore, by (3), a** = b*** = b* = a.
Conversely, if a *= a**, then a = b* with b = a*; thus a e S(L)y proving (4).
If a,b e 5(L), then a = a**, b = b**, and so a > (a A b)** and
b > (a A b)**y thus a A b > (a A fe)**; by (1), a A b = (a A b)**9 thus
a A beS(L). If xe5(L), x < ay and x < b, then x < a A b; therefore,
a a b = infS(L) {ay b}> proving (5).
a* > a* A b*f thus by (2) and (4), a < (a* A 6*)*; similarly, b <
{a* A b*)*. If a < xy b < x (x e S(L))y then a* > x*9 b* > x* by (2);
thus by (2) and (4), (a* A b*)* < xy proving (6).
For ayb e S(L)9 define avb = (a* A b*)*. By Formulas (5) and (6),
<5(L); A, v> is a bounded lattice. Since for a e S(L)y a v a* = (a* A a**)*
= 0* = 1, a A a* = 0, S(L) is a complemented lattice. Now we need only
prove that S(L) is distributive. For xyyyz e S(L)y xAz<xv(yAz)
and y A z < x v (y a z); therefore, x A z A (x v (y A z))* = 0 and
y A z A (xv (y a z))* = 0. Thus z A (xv (y a z))* < x* and y*y and
so z A (x v (y a z))* < x* Ay*. Consequently, z A (x v (y A z))* A
(x* A y*)* = 0, which implies that z A (x* A y*)* < (x v (y A z))**.
Now the left-hand side is z A (x v j>) by Formula (6), and the right-hand
side is x v (y A z) by Formula (4). Thus we get z A (x v y) < x v (y A z),
which is distributivity by Lemma 4.10. 0
Other types of special elements: An element a is an atom if a >— 0 and

60
FIRST CONCEPTS
a dual atom if a -< 1; it is join-irreducible if a = b V c implies that
a = b or a = c; it is meet-irreducible if a = b A c implies that a = b or
a = c. Examples are given in the following exercises.
Exercises
1. Find a homomorphism of bounded lattices that is not a {0, l}-homo-
morphism and a sublattice that is not a {0, l}-sublattice.
2. Find a modular lattice in which every element x ^ 0, 1 has exactly nt
complements.
3. Let L be a distributive lattice, a, beL. Prove that if a v £ and a A b
have complements, so do a and 6.
4. Show that Lemma 2 holds in any modular lattice.
5. In a bounded lattice L, let jcbea relative complement of a in [b, c]; let
y be a relative complement of c in [x, 1]; let z be a relative complement
of b in [0, x]; and let t be a relative complement of x in [z, >>]. Verify that
/ is a complement of a.
6. Let B0, Bx be Boolean algebras and let <p be a {0, l}-(lattice)
homomorphism of i?0 into Bx. Show that <p is a homomorphism of the Boolean
algebras.
7. Let L be a distributive lattice with 0. Show that /(L) is pseudocomple-
mented.
8. Is the converse of exercise 7 true?
9. Prove that if, in Theorem 4, L is a lattice, then a v 6 in ^(L) can be
described by a v b = (a v 6)**.
10. Let L be a pseudocomplemented lattice. Show that a** v b** =
(a v 6)**.
11. Find arbitrarily large pseudocomplemented lattices in which S(L) =
{0, 1}.
12. Prove that in a Boolean lattice, x ^ 0 is join-irreducible iff x is an atom.
13. Show that in a finite lattice every element is the join of join-irreducible
elements.
14. Verify that "finite lattice" in exercise 13 can be replaced by "lattice
satisfying the Descending Chain Condition." (A lattice L or, in general,
a poset, satisfies the Descending Chain Condition iff x0, xu x2,... eL,
x0 > Xx > x2 > • • • implies that xn = xn + 1 = • • • for some n\ see
exercise 2.7).
15. Show that some form of the Axiom of Choice must be used to verify
exercise 14.
16. Prove that if a lattice satisfies the Descending Chain Condition, then
every nonzero element contains an atom.
17. The dual of the Descending Chain Condition is the Ascending Chain
Condition (see exercise 2.6). Dualize exercises 13-16.

Further Topics and References
61
18. Show that a lattice satisfies the Ascending Chain Condition and the
Descending Chain Condition iff all chains are finite.
19. Find a lattice in which all chains are finite but that contains a chain of
n elements for every natural number n.
20. Find a lattice in which there are no join- or meet-irreducible elements.
21. Prove that the Ascending Chain Condition (Descending Chain Condition)
holds in a lattice L iff every ideal (dual ideal) of L is principal.
22. Show that the Ascending Chain Condition (Descending Chain Condition)
holds in a poset P, then every element is contained in a maximal element
(contains a minimal element).
23. Let A be a binary operation on L, let * be a unary operation on L (that
is, for every -a e L, a* e L), and let 0 be a nullary operation (that is,
0 e L). Let us assume that the following hold for all a,b,c e L: a A b =
b A a, (a A b) A c = a A (b A c), a A a = a, 0 A a = 0,
a A (a A b)* = a A b*, a A 0* = a, (0*)* = 0. Show that <L; A > is a
meet-semi lattice with 0 as zero, and for all aeL, a* is the pseudo-
complement of a (R. Balbes and A. Horn [1970a]).
24. Let L be a pseudocomplemented meet-semilattice and let a,b e L. Verify
the formulas (a A 6)* = (a** A 6)* = (a** A b**)*.
25. Let L be a meet-semilattice and let a,b e L. The pseudocomplement
a * b of a relative to b is an element of L satisfying a A x < b iff
x < a * 6. Show that a * b is unique if it exists; show that a * a exists iff
L has a unit.
26. Let L be a relatively pseudocomplemented meet-semilattice (that is, L is
a meet-semilattice and a * b exists for all a,b e L). Show that L has 0 and
1; for all a,b,c e L: a * (6 * c) = (a A 6) * c and a * (b * c) = (a * b) *
(a * c). Furthermore, if L is a lattice, then L is distributive.
27. Let L be a pseudocomplemented distributive lattice. Prove that for each
aeL, (a] is a pseudocomplemented distributive lattice; in fact, the
pseudocomplement of x e (a] in (a] is x* A a.
♦28. Using the notation of exercise 27, let S(a) denote the elements of the
form x* A a, x < a. Then S(a) is a Boolean algebra by Theorem 4.
Let va denote the join in S(a). Show that if x,y e S(a) and *,.>> e S(b)
(a,b g L), then x vay = x vb y.
♦29. Let b e S(a). Prove that S(b) c 5(a). (The results of exercises 28 and 29
appear to be new.)
30. Show that Tn (see exercise 2.36) is complemented.
31. Show that in Tn every interval is pseudocomplemented. (Exercises 30
and 31 are due to H. Lakser.)
Further Topics and References
Many of the concepts and results discussed in Chapter 1 are special cases
of universal algebraic concepts and results. To see this, the reader needs the
definition of a universal algebra. An n-ary operation f on a set A is a map

62
FIRST CONCEPTS
from An into A; in other words, if alt..., an e A, then f(al9.. .9an)e A.
If n = 1, / is called unary \ if /i = 2, / is called binary. Since .4° = {0},
a nullary operation (n = 0) is determined by/(0) e ,4, and /is sometimes
identified with/(0). A universal algebra, or simply algebra, consists of a
nonvoid set A and a set F of operations; each/e F is an w-ary operation
for some n (depending on /). We denote this algebra by $1 or (A; F>.
Many of the results of Sections 3-5 can be formulated and proved for
arbitrary universal algebras. (We shall utilize this fact in Chapter 3.) For
more details, see Chapters 1-4 of the author's book [1968].
In every poset we can introduce (as suggested by the real line) a ternary
relation r called betweenness: r(a, b, c) iff a < b < c or c < b < a.
M. Altwegg [1950] proved that the partial ordering can be defined in
terms of betweenness (naturally, up to duality).
Lattices and Boolean algebras can be defined by identities in
innumerable ways. Of the eight identities we used to define lattices ((L1)-(L4) of
Section 1), two ((LI)) can be dropped. Ju. I. Sorkin [1951] reduced six to
four, R. Padmanabhan [1969] later found two, and finally R. N. McKenzie
[a] found a single identity characterizing lattices. Ju. I. Sorkin's identities
use only three variables; the others use more. It is easy to see that two
variables would not suffice. Take the lattice of Figure 5.8 and redefine the
join of the two atoms to be 1; otherwise keep all the joins and meets.
The resulting algebra is not a lattice, but every subalgebra generated by
two elements is a lattice. Therefore, lattices cannot be defined by identities
in two variables. By means of a more complicated construction, A. H.
Diamond and J. C. C. McKinsey [1947] derived the same conclusion for
Boolean algebras.
A recent result of A. Tarski [1968] states that, given any integer n, there
exist n identities defining lattices such that no identity can be dropped.
By exercise 4.24, modular and distributive lattices can be defined by
two identities. Nicer sets of identities for these cases were found by
M. Kolibiar [1956] ((a V (b A b)) a b = b, {{a A b) a c) v (a A d) =
{{d A a) v (c A b)) A a characterize modular lattices) and M. Sholander
[1951] (a A (a v b) = a, a A (b v c) = (c A a) V (b A a) characterize
distributive lattices). See also J. Riecan [1958].
One of the most useful axiomatizations of Boolean algebras is that of
E. V. Huntington [1904],2 according to which a Boolean algebra is a
complemented lattice in which the complementation is pseudocomplemen-
2 Observe that a proof of Huntington's result is implicit in the proof of Theorem
6.4.

Further Topics and References
63
tat ion (that is, a A x = 0 implies that x < a'). It had been long
conjectured that every complemented lattice with unique complements was
Boolean; this conjecture was disproved by R. P. Dilworth [1945].
One of the briefest axiom systems of Boolean algebras in terms of A
and ' is that of L. Byrne [1946]: aAb = bAa, aA(bAc) =
(a A b) A c, a A b' = c A c' iff a A b = a. Characterization by
identities is usually longer. The observation has been made only recently
(independently, by R. N. McKenzie, A. Tarski, and the author) that
Boolean algebras can be defined by a single identity (see A. Tarski [1968],
and G. Gratzer and R. N. McKenzie [1967]). A thorough survey of the
axiom systems of Boolean algebras is given in S. Rudeanu [1963]; see also
F. M. Sioson [1964].
Identities for semilattices are easy to provide. Two identities
characterize semilattices (D. H. Potts [1965] and R. Padmanabhan [1966]); one
identity is not enough (D. H. Potts [1965]).
A diagram of a finite lattice is a graph; however, very little work has
been done in lattice theory from a graph theoretic point of view. It is
known that a finite distributive lattice is planar iff no element is covered by
more than two elements. Pairs of lattices whose diagrams are isomorphic
as graphs were investigated by J. Jakubik [1954a], [1954b]. See also
J. Jakubik and M. Kolibiar [1954].
The examples of lattice identities we have seen so far seem to suggest that
all such identities are self-dual. The identity {a A b) v (a A c) = a, where
a = x A {{x A y) V (y A z) v (z A x)\ b = (x A y) V (y A z), c =
(x A z) v (y A z), is an example of a nonself-dual identity (H. F. J.
Lowig [1943]).
Sublattices suggest a number of problems: For an equational class K of
lattices, let/K(A:, n) denote the smallest integer such that any lattice in K
having at least fK(k, n) elemeni s has at least k sublattices of exactly n
elements; write f(k, n) for/^(k, n). The function/(l, n) has been
investigated: /(1,0 = t for / < 6 (I Kaplansky);/(l,n) < «4""3 (R. Wille,
unpublished); and/(l, n) < n3n (G. Havas and M. Ward [1969]); see also
M. Curzio [1953]. The lattice cf sublattices Sub (L) and subsemilattices
of a semilattice were investigated by N. D. Filippov [1966] and L. N.
Sevrin [1964], respectively. The) gave necessary and sufficient conditions
for Sub (L) ^ Sub (L^. Filippo/'s results showed that if L is distributive
(Boolean) and Sub (L) ~ Sub (I^), then Lx is distributive (Boolean).
The lattice of subalgebras of a Boolean algebra has been characterized
in D. Sachs [1962].

64
FIRST CONCEPTS
If AT is a sublattice of L, then K is isomorphic to a sublattice of I(L);
finite lattices K for which the reverse holds were investigated by G.
Gratzer [1970].
An endomorphism <p ({0, \}-endomorphism) of the lattice L is a homo-
morphism ({0, l}-homomorphism) of L into itself. If, in addition, <p is
one-to-one and onto, then <p is an automorphism. Let E(L), E01(L)9 and
A(L) denote the set of all endomorphisms, {0, l}-endomorphisms, and
automorphisms of L, respectively. (We form EQ1(L) ifFL has 0 and 1.)
Defining the multiplication as composition of maps, E(L) and E01(L)
are monoids (semigroups with identity), and A(L) is a group. G. BirkhofT
[1946] has shown that for any group G, there exists a distributive lattice L
with A(L) ^ G; if G is finite, so is L. Small lattices L with given A(L)
were investigated by R. Frucht [1948], [1950]. For any given monoid M,
a bounded lattice L with EQtl(L) = M was constructed in G. Gratzer and
J. Sichler [a]—see also R. N. McKenzie and J. Sichler [a] and J. Sichler [a].
A very important property of complete lattices is the fixed-point theorem:
Any isotone map/of a complete lattice L into itself has a fixed point (that
is, f(a) = a for some a e L); in fact, f(a) = a if a is defined by
a = V (P\beL9b <f(b))
—see A. Tarski [1955]. A. C. Davis [1955] proved that if this theorem
holds in a lattice L, then L is complete. Various generalizations are given
in A. Tarski [1955], E. S. Wolk [1957], A. Pelczar [1961] and [1962],
S. Abian and A. B. Brown [1961], V. Devide [1963], S. R. Kogalovskii
[1964], and R. Demarr [1964].
Boolean algebras originated as an algebraic formalization of proposi-
tional logic. Most introductory logic texts give satisfactory expositions of
these ideas; see also P. R. Halmos [1963]. There is a similar relationship
between Boolean algebras and the theory of switching circuits; see M. A.
Harrison [1965].
The construction of lattices described in exercise 4.20 was investigated
by P. M.Whitman [1943].
It is hard to overemphasize the importance of free lattices; they provide
one of the most important research tools of lattice theory. Two typical
applications to modular lattices can be found in G. Gratzer and E. T.
Schmidt [1961] and G. Gratzer [1966].
P. Ribenboim [1949] first pointed out that pseudocomplementation can
be described by identities (involving *). A. Monteiro [1955] accomplished

Further Topics and References
65
the same result for relative pseudocomplementation; see also R. Balbes
and A. Horn [1970a]. This fact is applied in G. Gratzer [1969].
The connections of lattice theory with topology are manifold and have
been excluded from this book. The closed (and also the open) subsets of a
topological space X form a lattice L(X), and this lattice determines the
topological space (provided it is T±). These lattices were characterized by
H. Wallman [1938]. Thus certain parts of topology can be studied within
lattice theory—the theory of compactifications being one example (H.
Wallman [1938], O. Frink [1964]). The set of all continuous real functions
on a topological space A'also forms a lattice C(X) under pointwise
ordering. The lattice C(X) determines X if X is compact Hausdorff (I. Kap-
lansky [1947]; see also L. Gillman and M. Jerison [I960]).
All topologies on a set X form a lattice T(X) if, for topologies Sl9 S2 on
X, we set S± < S2 whenever every S^-open set is also S2-open. This point
of view permeates the topology book of R. Vaidyanathaswamy [1947] and
has influenced some parts of modern topology. A typical recent result is
that of A. K. Steiner [1966], according to which T(X) is a complemented
lattice.
A topological lattice L is a lattice equipped with a topology such that A
and V are continuous functions. The theory of topological lattices, the
study of which was started by A. D. Wallace and his students, is quite
extensive. The methods used are mostly topological rather than lattice
theoretical in nature. The reader should consult L. W. Anderson [1961]
for an early review of this field. *
In contrast with topological lattices, intrinsic topologies are considered
on arbitrary lattices. Intrinsic topologies are topologies defined on a
lattice in terms of the partial ordering. For instance, the interval topology
takes the intervals as a subbase for closed sets. For more details, see O.
Frink [1942], G. Birkhoff [1962], and B. C. Rennie [1951].
Few of the topics in this chapter have been investigated for semilattices.
Congruences are one exception (see G. Zacher [1952], D. Papert [1964],
R. A. Dean and R. H. Oehmke [1964], R. Permutti [1964], J. Varlet [1965],
and E. T. Schmidt [1969]). Congruences of partial lattices are considered
in G. Gratzer and H. Lakser [1968].

66
FIRST CONCEPTS
PROBLEMS
1. Characterize the endomorphism semigroup of a lattice.
2. Characterize E(L) and E0>1(L) for finite lattices and for lattices of finite
length (see J. Sichler [a]).3
3. For an arbitrary monoid M, find a lattice L such that the nonconstant
endomorphisms of L, En(L) form a semigroup, and En(L) £ M.4
4. Characterize the endomorphism semigroups of complemented lattices.
(The categorical versions of problems 1-4 should also be considered.)
5. Characterize the comparability relation of semilattices and of lattices
(see exercises 1.34 and 1.35).5
6. Characterize the lattices Tn and finite lattices that can be embedded in
some Tn (see exercises 2.26-2.36, 6.30, and 6.31).
7. Characterize the lattice Sub (L) of all sublattices of a lattice L.
8. For what equational classes K does LeK, Sub (L) ~ Sub (Li) imply
thatLiG^?
9. Characterize planar lattices.
10. Define the complexity of a finite lattice L as the number of intersecting
lines in an optimal diagram. Investigate this concept.
11. For an equational class K, which integers n can occur as the complexity
of a finite lattice LeK?
12. Let L = FW(A)9 where K is an equational class and A is a finite partial
lattice; L is then called finitely presented over K. Is the automorphism
group of L necessarily finite? Is this true at least if K = L?
13. Does a finitely presented lattice have only finitely many bases in the
sense of exercise 5.44? Is this true at least if K = L?
14. For which finite lattices K does it hold that if AT is a sublattice of /(L),
then K is a sublattice of L (see G. Gratzer [1970])?
15. Compare finite lattices K satisfying the condition of problem 14 with
those (i) that can be embedded in F(3), or (ii) for which {AT | N e L, K is
not a sublattice of N) is equational.
16. Describe the equational class of lattices with n < 3 identities involving
only three variables.
17. Find the shortest single identity characterizing lattices.
18. Does there exist a non-Boolean finite algebra <^; A, v,', 0, 1> such
that every two-generated subalgebra is Boolean (see A. H. Diamond and
J. C. C. McKinsey [1947])?
3 R. N. McKenzie proved that any E(L) can be represented as ZT0,i(Z,i) and that if
L is finite or of finite length, then so is Lx.
4 This problem was solved by R. N. McKenzie using the General Continuum
Hypothesis. Subsequently, J. Sichler solved it in the categorical form without the
Hypothesis.
5 R. N. McKenzie has shown that neither of the two problems has a first-order
solution.

Problems
67
19. Define and investigate weak partial lattices and partial lattices satisfying
an identity.
20. For a partial lattice A, define a congruence relation as follows: A binary
relation © on A is a congruence if there exists a lattice L and a
congruence relation O on L such that (i) A is a relative sublattice of L and (ii)
O restricted to A is @. Give an intrinsic characterization of this concept
of congruence relation.6
21. Characterize the lattice of all congruence relations of a semilattice.
22. Let L be a meet-semilattice with 0 for which [0, a] is pseudocomple-
mented for each aeL\ let S(a) denote the pseudocomplements in
[0, a]. Characterize the family of Boolean algebras {S(a) \ a e L) (see
exercises 6.28 and 6.29).
23. What is the order of magnitude of f(k, n)l (Note that f(k, n) is defined
in Further Topics and References.)7
24. For what equational classes K do all/K(A:, n) exist ?
25. Give a concrete example of a non-Boolean uniquely complemented
lattice (see R. P. Dilworth [1945]; C. C. Chen and G. Gratzer [1969a]).
26. Does there exist a complete non-Boolean uniquely complemented
lattice?
27. Investigate the lattice of join-endomorphisms of a finite lattice (see
G. Gratzer and E. T. Schmidt [1958a]).
6 This is known for universal algebras in general; see G. Gratzer [1968], Theorem
13.3.
7 B. Wolk pointed out that the argument of G. Havas and M. Ward [1969] can be
modified so as to prove that if L has more than n3n elements, then L has an n element
sublattice in which all but at most two elements are join- and meet-irreducible. This
implies the existence of /(£, n).

CHAPTER
DISTRIBUTIVE LATTICES
7. Characterization Theorems and Representation Theorems
The two typical examples of nondistributive lattices are 9tt5 and 9£5,
whose diagrams are given in Figure 7.1.
Theorem 1 is a striking and useful characterization of distributive
lattices; Theorem 2 is a more detailed version of Theorem 1.
Figure 7.1

70
DISTRIBUTIVE LATTICES
Theorem 1. A lattice L is distributive iffL has no sublattice isomorphic to
9tt5 or ^5.
Theorem 2.
(i) A lattice L is modular iff it has no sublattice isomorphic to 915.
(ii) A modular lattice L is distributive iff it has no sublattice isomorphic
toWl5.
PROOF.
(i) If L is modular, then every sublattice of L is also modular; %l5 is not
modular, thus it cannot be isomorphic to a sublattice of L. Conversely,
let L be nonmodular, let a,b,c e L, a > b, and let (a A c) V b ^
a A (c v b). The free lattice generated by a, b, and c with a > b is
shown in Figure 2.3. Therefore, the sublattice of L generated by a, b,
and c must be a homomorphic image of the lattice of Figure 2.3.
Observe that if any two of the five elements a A c, (a A c) V b,
a A (b v c), b V c, c are identified under a homomorphism, then so
are (a A c) V b and a A (b v c). Consequently, these five elements
are distinct in L, and they form a sublattice isomorphic to 9£5. )
(ii) Let L be modular, but nondistributive, and choose x,y,z e L such that
x A (y V z) 7^ (x A y) V (jc A z). The free modular lattice generated
by x, y, z is shown in Figure 5.7. By inspecting the diagram we see that
w> *i> yu Zi> v form a sublattice isomorphic to 9ft5. Thus in any
modular lattice they form a sublattice isomorphic to a quotient lattice of
9J?5. But 9J?5 has only two quotient lattices: ^5 and the one-element
lattice. In the former case we have finished the proof. In the latter case,
note that if u and v collapse, then so do x A (y V z) and (x A y) V
(x A z), contrary to our assumption. %
Naturally, these results could be proved without any reference to free
lattices. A routine proof of (ii) runs as follows: Take x,y>zeL such that
x A (y V z) ^ (x A y) V (x A z) and define w, y, xl9 }>!, zx as the
corresponding polynomials of Figure 5.7. Then a direct computation shows
that «, vy jc1s >>i, zx form a sublattice isomorphic to 9#5. There is a very
natural objection to such a proof, namely: How are the appropriate
polynomials found? How is it possible to guess the result? And there is
only one answer: by working out the free lattice.
Corollary 3. A lattice L is distributive iff every element has at most one
relative complement in any interval.

Section 7 Characterization Theorems and Representation Theorems 71
proof. The "only if" part was proved in Section 6. If L is nondistribu-
tive, then by Theorem 2 it contains 9£5 or 9tt5, and each has an element
with two relative complements in some interval. %
Corollary 4. A lattice L is distributive iff for any two ideals I, J ofL:
I V J = {i Vj\ieIJeJ}.
proof. Let L be distributive. By Lemma 3.1(ii), if tel v /, then / <
i V j for some / e /, j e /. Therefore, t = (t A /) v (t A j), t A i e /,
f A jeJ. Conversely, if L is nondistributive, then L contains elements
a, b, c as in Figure 7.1. Let / = (b] and / = (c]; observe that as I v /,
since a < b v c. However, a has no representation as required by
Corollary 4, since if a = / v y, / el, je Jy then 7 < a, j < c. Therefore, y <
a A c < b, thus 7 g /, and so a = i V j e /, a contradiction. #
Another important property of ideals of a distributive lattice is:
Lemma 5. Let I and J be ideals of a distributive lattice L. If I A J and
I v J are principal, then so are I and J.
proof. Let / A / = (*] and / V J = (y]. Then y = z v j for some
z g /, j g J. Set c = * V z and b = x V j; note that eel, be J. We claim
that / = (c], / = (A]. Indeed, if, for instance, / ^ (b]9 then there is an
a > b, aeJ, and {*, a, b, c, y} form an 9£5. #
Theorem 6. Let Lbe a distributive lattice and let aeL. Then the map
<p: x-><jc A a, x v a>, xeL
w fl/i embedding ofL into (a] x [a); it is an isomorphism if a has a
complement.
proof. 9 is one-to-one, since if x<p = y<p, then x and j> are both relative
complements of a in the same interval; thus x = j> by Corollary 3. Dis-
tributivity implies also that <p is a homomorphism.
If 0 has a complement, 6, and <w, y> e (a] x [a), then for * = (u V 6) A v,
x<p = <w, y>; therefore, 9 is an isomorphism. %
We start the detailed investigation of the structure of distributive
lattices with the finite case.

72
DISTRIBUTIVE LATTICES
Definition 7. For a distributive lattice, L, let J(L) denote the set of all
nonzero join-irreducible elements, regarded as a poset under the partial
ordering ofL. For ae L set
r(a) - {x | x < a, x e/(L)}.
Definition 8. For a poset P, call A £ p hereditary if xeA, y < x
implies that y e A. Let H(P) denote the set of all hereditary subsets partially
ordered by set inclusion.
Note that H(P) is a lattice in which meet and join are intersection and
union, respectively, and thus H(P) is distributive. The structure of finite
distributive lattices is revealed by the following result:
Theorem 9. Let L be a finite distributive lattice. Then the map
<p: a-+r(a)
is an isomorphism between L and H(J(L)).
proof. Since L is finite, every element is the join of join-irreducible
elements; thus
a = V r(a),
showing that <p is one-to-one. Obviously, r{a) n r(b) = r(a A b), and so
(a A b)<p = a<p A b<p. The formula (a v b)<p = a<p v b<p is equivalent to
r(a v b) = r(a) u r(b).
To verify this formula, note that r(a) u r(b) c r(a V 6) is trivial. Now
let xer(a v b). Then x = x A (a V 6) = (x A fl) V (x A b); therefore,
x = xAaorx~xAb, since x is join-irreducible. Thus x e r(a) or
x e r(b), that is, x e r(a) u r(^).
Finally, we have to show that if A e H(J(L)), then 09? = y4 for some
aeL. Set a = V A. Then r(a) ^ A is obvious. Let xer(a); then x =
x A a = x A \/ A = \/ (x A y \ y e A). So^ = ^A.yfor some y e A,
implying that x e A, since ,4 is hereditary. #
Corollary 10. 7%g correspondence L->7(L) makes the class of all
finite distributive lattices correspond to the class of all finite posets;
isomorphic lattices correspond to isomorphic posets, and vice versa.

Section 7 Characterization Theorems and Representation Theorems 73
proof. This is obvious from J(H(P))^ P, and H(J(L)) ^ L. #
A subset S of P(A) is called a ring of sets if X, Y e S implies that In 7,
lu Ye S. Since H(J(L)) is a ring of sets, we obtain
Corollary 11. A finite lattice is distributive iff it is isomorphic to a ring
of sets.
If the poset Q is unordered, H(Q) = P(Q); if B is Boolean, /(£) is the
set of all atoms, and therefore J(B) is unordered. Thus we get:
Corollary 12. A finite lattice is Boolean iff it is isomorphic to the
Boolean lattice of all subsets of a finite set.
A representation a = x0 V • • • V xn-x is redundant if
a = x0 V • • • V x,_i V x,+1 V • • • V xn_i,
for some 0 < / < n; otherwise it is irredundant.
Corollary 13. Every element of a finite distributive lattice has a unique
irredundant representation as a join of join-irreducible elements.
proof. The existence of such a representation is obvious. If
a = x0 V • • • V xn_!
is an irredundant representation, then r{a) = U (r(xt) \ 0 < / < n). Thus
x occurs in such a representation iff x is a maximal element of r{a); hence
the uniqueness. %
A finite chain of n elements is said to be of length n — 1.
Corollary 14. Every maximal chain C of the finite distributive lattice L
is of length \J(L)\.
proof. For a e /(L), let m(a) be the smallest member of C containing a.
Then a -^ m(a) is a one-to-one map of J(L) onto the nonzero elements of
C. Indeed, if m(a) = m(b), m(a) >- x, and xeC, then x V a = x V b;
therefore, a = (a A x) v (a A b\ implying that a < x or a < b. But
a < x implies that m(a) < x < m(a), a contradiction. Consequently,
a < b; similarly, b < a; thus a = b. Let y e C, j> >-z, z e C. Then
r(y) => r(z), and so y = m(a) for any a e r(j>) - r(z). %

74
DISTRIBUTIVE LATTICES
Figure 7.2
The crucial Theorem 9 and its most important consequence, Corollary
11, depend on the existence of sufficiently many join-irreducible elements.
In an infinite distributive lattice there may be no join-irreducible element.
Note that in a distributive lattice, a is join-irreducible iffL — [a) is a prime
ideal. In the infinite case the role of join-irreducible elements is taken over
by prime ideals. The crucial result is the existence of sufficiently many
prime ideals (as illustrated in Figure 7.2).
Theorem 15 (M. H. Stone [1936]). Let L be a distributive lattice, let I be
an ideal, let D be a dual ideal of L, and let I C\ D ~ 0. Then there exists
a prime ideal P ofL such that P ^ I and P n D = 0.
proof. Some form of the Axiom of Choice is needed to prove this
statement. The most convenient form for this proof is:
Zorn's Lemma. Let Abe a set and let %be a nonvoid subset ofP(A). Let
us assume that 2£ has the following property: If C £ & and C is a chain
(that is, for any X,YeCwe have X c Y or Y ^ X), then
U(I|IeC)£l
Then & has a maximal member (that is, an M e& such that M £ X and
Xe% imply that M = X).
Let #* be the set of all ideals of L that contain /and that are disjoint from
D. We have to verify that & satisfies the hypothesis of Zorn's Lemma.
Since /e #*, we conclude that X is nonvoid. Let C be a chain in X and let

Section 7 Characterization Theorems and Representation Theorems 75
M = U (X | X e C). If 0,6 e Af, then a e *, 6 e 7 for some JJeC;
since C is a chain, either X ^ Yor Y ^ X;if, say, Z c Y, then c,& e F,
and so a V 6e Y c Af, since 7 is an ideal. Also, if b < aeM, then
flel; since Xis an ideal, be X ^ M. Thus M is an ideal. It is obvious that
M 3 / and MnD = 0, verifying that Me°£. Therefore, by Zorn's Lemma,
9E has a maximal element, say, P. We claim that P is a prime ideal. Indeed,
if P is not prime, there exist ajb e L such that a,6 £ P but a A be P.
Because of the maximality of P, (?v (fl])nD/0,(?v (&]) n Z> ^ 0.
Let p V ae D,q V be D,pyqeP. Then x = (p V a) A (q V b) e D,
since Z> is a dual ideal. Also, x = (p A q) V (p A b) V (a A q) V
(a A b)eP; thus P n D ^ 0, a contradiction. #
Corollary 16. Lef L 6e a distributive lattice, let I be an ideal ofL, and let
aeL and a$I. Then there is a prime ideal P such that ?2/, and a$P.
proof. Apply Theorem 15 to / and D = [a). %
Corollary 17. Let L be a distributive lattice, ajbsL, a ^ b. Then there
is a prime ideal containing exactly one of a and b.
proof. Either (a] n [b) = 0, or [a) n(b] = 0. #
Corollary 18. /foery iVfei/1 of a distributive lattice is the intersection of
all prime ideals containing it.
proof. Let Ix = p| (P | P 2 7, P is a prime ideal of L). If / ^ 7lf then
there is an a e Ix - /, and so by Corollary 16 there is a prime ideal P, with
P 3 /, fl £ P. But then a £ P ^ /2 is a contradiction. %
Theorem 19 (G. BirkhofT [1933] and M. H. Stone [1936]). A lattice is
distributive iff it is isomorphic to a ring of sets.
proof. Let L be a distributive lattice and let X be the set of all prime
ideals of L. For a e L set
r(a) = {P\a$P,PeX).
Then the r(a) form a ring of sets, and a -> r(a) is an isomorphism. The
details are similar to the proof of Theorem 9, except for the first step, which
now uses Corollary 17. #

76
DISTRIBUTIVE LATTICES
This result has a very useful corollary.
Corollary 20. Let Lbe a distributive lattice with more than one element.
An identity holds in L iff it holds in the two-element chainy (£2-
proof. Let p = g hold in L. Since \L\ > 1, (£2 is a sublattice of L, and
p — g holds in (£2. Conversely, let/? = g hold in (£2- Note that (£2 = P(^0>
with \X\ = 1, and that P(A) is isomorphic to the direct power P(X)]A].
Therefore, /? = g holds in any P(A). By Theorem 19, L is a sublattice of
some P(A); thus p = g holds in L. 0
We can reformulate Theorem 19 using the concept of a field of sets: A
field of sets is a ring of sets closed under set complementation.
Corollary 21 (M. H. Stone [1936]). A lattice is Boolean iff it is
isomorphic to afield of sets.
proof. Use the representation of Theorem 19. Obviously r(a') =
X — r(a)y and thus complements are also preserved. %
Let ^(L) denote the set of all prime ideals of L, regarded as a poset
under c. The importance of ^(L) should be clear from the previous
results. A topology on ^(L) will be discussed in Section 11.
Some interesting properties of L are reflected in ^(L). An important
result of this type is the following theorem (see also L. Rieger [1949]):
Theorem 22 (L. Nachbin [1947]). Let L be a distributive lattice with 0
and 1. Then L is a Boolean lattice iff &(L) is unordered.
proof. Let L be Boolean, P,Qe &(L\ P c: Q. Choose ae Q - P. Since
fle^fl'^e, and thus a' <£ P. Thus a,a' $ P, but a A a' = 0 e P, a
contradiction, showing that &*(L) is unordered.1
Now let &(L) be unordered and aeL> and let us assume that a has no
complement. Set D == {x \ a V x = 1}. Then D is a dual ideal. Take
Dx = D V [a) = {x | x > J A a for some J e Z>}. The dual ideal Dx does
not contain 0, since 0~dAayavd=l would mean that d is a
complement of a. Thus there exists a prime ideal P disjoint to Dx. Note that
1 £ (a] v P, otherwise 1 = a V /? for some/? eP, contradicting P n D = 0.
1 This proof in fact verifies that in a Boolean algebra every prime ideal is maximal.

Section 7 Characterization Theorems and Representation Theorems 77
Thus some prime ideal Q contains (a] V P; and soPc Qt which is
impossible since P(L) is unordered. #
According to Corollary 18, every ideal is an intersection of primes.
When is this representation unique ?
Theorem 23 (J. Hashimoto [1952]). Let L be a distributive lattice with
0 and 1. Every ideal has a unique representation as an intersection of prime
ideals iffL is a finite Boolean lattice.
proof. If L is a finite Boolean lattice, then P is a prime ideal iff P = (a],
where a is a dual atom; the uniqueness follows from Corollary 13 (or is
obvious by direct computation).
Now let every ideal of L have a unique representation as a meet of prime
ideals. We claim that I(L) is Boolean. Let IeI(L),
J=C\{P\Pe0>{L\P$l).
Then lAJ=(~)(P\Pe &>(L)) = (0]. If L j- I V /, then there is a prime
ideal P0 2 / V /, and consequently J has two representations:
Thus L = / V /, and / is a complement of / in I(L).
By Lemma 5, every ideal of L is principal; therefore L ^ /(L), and so
L is Boolean. By exercise 6.21, L satisfies the Ascending Chain Condition;
thus (exercise 6.22) every element of L other than the unit is contained in
a dual atom. Since the complement of a dual atom is an atom, by taking
complements we find that every nonzero element of L contains an atom.
If Po>Pu- • ->Pti9 — - are distinct atoms in L, then the ascending chain
Po>Po V pi,.. .,p0 V/?i V • • • V pn,... does not terminate, and thus L
has only finitely many atoms, p09.. .,pn-i- Let a = p0 v • • • V pn-i- If
a' ^ 0, then a' has to contain an atom, which is impossible. Therefore,
a' = 0, a = 1, and L - P(X), with \X\ = n. #
Exercises
1. Work out a direct proof of Theorem 2(i).
2. Work out a direct proof of Theorem 2(ii).

78
DISTRIBUTIVE LATTICES
3. Let K be a five-element distributive lattice. Is there an identity p - q
such that p = q holds in a lattice L iff L has no sublattice isomorphic
toKl
4. Does the property stated in Lemma 5 characterize distributive lattices?
5. Let L be a distributive lattice with 0 and 1. Prove that the direct
decompositions L ^ L0 x Li of L are in one-to-one correspondence with
the complemented elements of L.
6. Prove that the complemented elements of a distributive lattice form a
sublattice.
7. Let L be a distributive lattice with 0 and 1. Let L ^ L0 x Lx ^ K0 x ATi.
Show that there is a direct decomposition L ^ A0 x Ax x A2 x A3 such
that >4o x /li ^ L0, /*2 x ^3 = £i, A0 x A2 = K0> Ax x A3 ^ A^.
8. Extend Theorem- 9 to distributive lattices satisfying the Descending
Chain Condition (see exercise 6.14).
9. Extend Corollary 10 to distributive lattices satisfying the Descending
Chain Condition.
10. Can exercises 8 and 9 be further sharpened?
11. Let L be a distributive lattice with 0 and 1. Let C0 and Cx be finite chains
in L. Show that there exist chains D0 2 C0, Dx ^ Cl9 such that
|A>| = |/>i|.
12. Derive from exercise 11 the result that all maximal chains of a finite
distributive lattice have the same length.
13. Find examples showing that exercise 11 is not valid if "finite" is
omitted.
14. Prove the theorem "L is modular iff /(L) is modular" by showing that
"L has 9^5 as a sublattice iff /(L) has 9£5 as a sublattice."
*15. Is the second statement of exercise 14 true for 9ft5 rather than for -ft5?
16. Let L be a distributive lattice, a,b,ceL, a < b. Show that [a, b] is
Boolean iff [a A c, b A c] and [a V c, b V c] are Boolean.
17. For a poset P let /Tf(^) denote the lattice of all subsets of the form
(a0] vj (tfi] u • • • u (tfn_i], where zz is an arbitrary integer. Show that
Theorem 9 holds for HF(P).
18. Prove that if P is a finite poset, then HF(P) = i/(P).
19. Verify that if Theorem 9 holds for a lattice L, then L s #f0°) for some
poset P.
20. Show that the Ascending Chain Condition implies the Descending
Chain Condition for Boolean algebras.
21. Show that exercise 20 fails to hold for generalized Boolean algebras
(that is, relatively complemented distributive lattices with 0).
22. Use exercise 20 to simplify the proof of Theorem 23.
23. Let L be a lattice, let P be a prime ideal of L, and let a,b,c e L. Prove that
if a v (b A c) e P, then (a v b) A (a v c) e P.
24. Using exercise 23, show that the lattice L is distributive iff, for all
x,y eL, x < y, there exists a prime ideal P with xeP, y$ P.
25. Verify the statement of exercise 24 using Theorem 6.

Section 7 Characterization Theorems and Representation Theorems 79
26. Verify the statement of exercise 24 using Theorem 1.
27. Let L be a distributive lattice. Then L is relatively complemented iff
^{L) is unordered.
28. Prove Theorem 15 by well-ordering L, L = {aY \ y < «}, and deciding
one by one for each aY whether ay e P or aY e P.
29. Let L be a distributive lattice with 1. Show that every prime ideal P is
contained in a maximal prime ideal Q (that is, P c Q, and for any prime
ideal JIT of jL, 0 <= X implies that 0 = X).
30. Let L be a distributive lattice with 0. Verify that every prime ideal P
contains a minimal prime ideal Q (that is, P 2 Q, and for any prime
ideal X of L, Q 2 X implies that Q = X).
31. Find a distributive lattice L with no minimal and no maximal prime
ideals.
32. Investigate the connections among the Ascending Chain Condition
(and Descending Chain Condition) for a distributive lattice, for 7(jL),
and for ^(L).
33. Let L be a distributive lattice with 0 and let IeI(L). Show that the
pseudocomplement of I is I* = {x \ (x] A I = (0]}.
34. Prove that / = /** for any IeI(L) of a distributive lattice with 0 iff £
is a generalized Boolean lattice satisfying the Descending Chain
Condition.
35. The congruence relations © and O permute if a = &(©), b = c(O)
imply that a = */(<&), */ = c (0) for some </. Show that the congruences
of a relatively complemented lattice permute.
36. Prove the converse of exercise 35 for distributive lattices.
37. Generalize Theorem 23 to distributive lattices without 0 and 1.
*38. Let Ibea distributive lattice, let a e L, let S be a sublattice of L, and
let a <£ S. Show that there exists a prime ideal P and a prime dual ideal
Q such that a$Pu Q ^ S provided that a is not the 0 or 1 of L (J.
Hashimoto [1952]).
39. Let L be a relatively complemented distributive lattice. A sublattice K
of L is proper \{ K ^ L. Show that every proper sublattice of L can be
extended to a maximal proper sublattice of L (K. Takeuchi [1951]; see
also J. Hashimoto [1952]; and G. Gratzer and E. T. Schmidt [1958d]).
40. Show that the statement of exercise 39 is not valid in general if L is not
relatively complemented (K. Takeuchi [1951]).
41. Generalize Corollary 13 to infinite distributive lattices, claiming the
unique irredundant representation of certain ideals as a meet of prime
ideals.
42. If P is a prime ideal of L, then (P] is a principal prime ideal of I(L). Is the
converse true ?
43. Show that Corollary 17 characterizes distributivity.
44. A chain C in a poset P is called maximal if, for any chain D in P, C e D
implies that C = D. Using Zorn's Lemma, show that every chain is
contained in a maximal chain.

80
DISTRIBUTIVE LATTICES
8. Polynomials and Freeness
We can introduce an equivalence relation = for lattice polynomials:
p = q iff p and q define the same functions in the class D of distributive
lattices. More formally, if p and q are lattice polynomials (see Section 4),
then p = q iff, for any distributive lattice L and a0, au ... e L, we have
p(a0,...) = q(a0,...) (see Definitions 4.1 and 4.2). For an /i-ary lattice
polynomial p, let [p]D denote the set of all /i-ary lattice polynomials q
satisfying p = q and let PD(«) denote the set of all these equivalence
classes, that is, PD(/i) = {[p]D \ p e P(n)}. Observe that for p,pl9q,q! e P(n)
if p = p1 and q = ql9 then p A q = p1 A q1 and p V q = Pi V q±. Thus
[/?]D A [q]D = [p a q]D and [/?]D V [?]D = [p v ?]D define A and
V on PD(«). It is easily seen that PD(/i) is a distributive lattice and [p]D <
[q]D iff the inequality/? < q holds in the class D.
To describe the structure of PD(«), let Q(n) denote the poset of all
nonvoid subsets of {0, 1,...,«— 1}.
Theorem 1.
(i) PD(/i) is isomorphic with H(Q(ri)).
(ii) Pd(/i) is a free distributive lattice on n generators.
(iii)2 2n < \P0(n)\ < 22ft.
(iv) A finitely generated distributive lattice is finite.
PROOF.
(i) A lattice polynomial p is called a meet-polynomial if it is of the form
xio a • • • A xifc. For / c {0,..., n - 1}, / ^ 0, set
/>/*= A(xt\ieJ).
We claim that [^]D < [pK]D (J,K c {0,...,«- 1}) iff/ 2 AT. The
"if" part is obvious. Now let J ^> K; then there exists an / e K such
that i $ J. Consider the two-element chain (£2 and substitute xt = 0
and #y = 1 for j ^ i. Obviously, /?7 = 1 and pK = 0; thus the
inequality pj < pK fails in (£2 and therefore in D.
Every lattice polynomial is equivalent to one of the form
VPS,
2 The problem of determining \FD(n)\ goes back to R. Dedekind [1900]; see
R. Church [1940], E. N. Gilbert [1954], G. Hansel [1967], and D. Kleitman [1969].

Section 8 Polynomials and Freeness
81
because every x{ is of this form, the join of two such polynomials is of
this form and the same holds for the meet in view of
V Ph A V^ = V (/>/, A pKj)
and
Pj> a Pk, = Pjiuk,-
Next we claim that [p]D is join-irreducible (in PD(/i)) iff it is a
[/?/]D. Since every [p]D e PD(/i) is a join of some [pj]D, it suffices to
prove that a [pj]D is join-irreducible. Let
Pj = V(Pii\ieK,Ji^{0,...,n- 1});
J^Ji follows from [/?,]D > [/?/JD. Now if [pj]D > [/?/4]D for all i,
then we have / <=■ J{. Choose jt e Ju jx $ J for all i e AT. In (£2 put
jcfe = 0 for all k = j\ and xfe = 1 otherwise. Then p} = 1, and
V (P/t I ' G ^) = 0, which is a contradiction.
To sum up, the join-irreducible elements form a poset isomorphic
to Q(n). A reference to Theorem 7.9 completes the proof of (i). )
(ii) Let L be a distributive lattice, a0,..., an.x e L. Then the map xf -> at
can be extended to the homomorphism
[>]D->/?(0o,...,tfn-i),
proving (ii). )
(iii) This proof is obvious from (i). )
(iv) This proof is obvious from (iii). %
Figure 5.6 is the diagram of PD(3).
Boolean polynomials are defined exactly as lattice polynomials except
that all five operations A, V, ', 0, 1 are used in the formation of
polynomials. A formal definition is the same as Definition 4.1 with two
clauses added: If/? is a Boolean polynomial, so is/?'; 0 and 1 are Boolean
polynomials. An /i-ary Boolean polynomial p defines a function in n
variables on any Boolean algebra B; p(a0,..., an-x) can be defined
imitating Definition 4.2.
For the Boolean polynomials /? and q, set /? = q if, for any Boolean
algebra B and a0, al9... e B, we have p(a0,...) = q(a0,...). Let [/?]B
denote the equivalence class containing/?. Observe that/? = q is equivalent
to the identity/? = q holding in the class B of all Boolean algebras.

82
DISTRIBUTIVE LATTICES
Let PB(n) denote the set of all [p]B, where p is an w-ary Boolean
polynomial. It is easily seen that [p]B A [q]B = [p A q]B, [p]B V [q]B =
[p v q]B, ([p]B)' = [p']B9 0 = [0]B, 1 = [1]B defines the Boolean
operations on PB(n), and thus PB(n) is a Boolean algebra.
Theorem 2.
(i) PB(n) is isomorphic to (932)2n-
(ii) PB(n) is a free Boolean algebra on n generators.
(iii) |i>B(«)| = 22\
(iv) A finitely generated Boolean algebra is finite.
proof. A Boolean polynomial is called atomic if it is of the form
xl0° A • • • A jfrri1,
where i) = 0 or 1, x° denotes x, x1 denotes x'. For every / c {0,...,
n — 1} there is an atomic polynomial pj for which /,• = 0 iff jeJ. The
crucial statement is: [pJo]B < [pjJB ifT/0 = J\- Indeed, let J0 ^ J±. We
make the following substitution in 932: x{ = 1 if / e J0, x{ = 0 if / £ JQ\ this
makes/?/o = \,pJx = 0, contradicting [/?/o]B < [p/JB.
Let #(/*) be the set of all Boolean polynomials that are equivalent to one
of the form V (Pjt I 'G ^0- Then B{ri) is closed under v and A, since
VPit A \/ Pik = V (/>/, A pIk),
and/?,, A /?/te = pJk if Jt = /fc, and/>7< a pJk = 0 otherwise.
Now we prove by induction on n that x^x'i e 2?(«) for / < n. For » = 1,
x0, xQ are atomic polynomials, so x0,x0 e .5(1). By induction, x0 =
V (/?/, | ' e A'), where pJt are atomic (/* — l)-ary polynomials; then
X0 = X0 A (xB_i V <_!) = (x0 A Xn_0 V (x0 A ^_i)
= V(P/,A Xn_!|l6JO V V(/>/f A ^.J/G^),
and similarly for jcq. Thus x0,x0 e i?(w), and, by symmetry, xux\ e i?(«) for
all i < n. Since (/?,)' = V (**' I *e «J) V V (** I ''$ J), we conclude that
(pj)' e B(n); therefore B(n) is closed under '. Thus B(n) is closed under
A, V,', 0, 1. Since B(n) includes all xl9 i < n, B(n) is the set of all w-ary
Boolean polynomials.
Consequently, every [p]B is a join of atomic ones, the [p]B for p atomic
polynomials are unordered and 2n in number, implying (i) and (iii). The

Section 8 Polynomials and Freeness 83
proof of (ii) is routine (same proof as that of Theorem l(ii)), and (iv)
follows trivially from (iii). %
We can use Theorems 1 and 2 to characterize free distributive lattices
and free Boolean algebras, respectively.
Theorem 3. Let L be a distributive lattice generated by to \ i e /}. L is
freely generated by the at iff the validity in L of a relation of the form
A(ai\ieI0)<\/(ai\iEl1)
implies that I0 n Ix =£ 0, for /0, h finite nonvoid subsets of L
proof. The " only if" part can be easily verified by using substitutions in
(£2- For the converse, let F be the distributive lattice freely generated by
xi9 i e I, and let <p be the homomorphism of F into (in fact, onto) L
satisfying Xi<p = ax for i g /. It suffices to prove that for the lattice polynomials
P> <1>P9 ^ <19 implies that [p]B < [q]D. (We think of the elements of Fas
equivalence classes of polynomials in the xiy i e I.)
Let
p = \J(\(x{\ieI^\jeJ)
and
q^ \{\/{xi\ieKt)\teT).
Then p<p < q<p takes the form
V (A to I /6/y) \jeJ) < A (V to I ieKd \teT)9
which is equivalent to
A(ai\ieIj)<\/(ai\ieKt)
for all jejy teT. By assumption this implies that IjCiKt^ 0 for all
ye/, teT; thus
A(*|/e/y)£ V(Xi\ieKt)
for all jeJ,teT; implying that [p]D < [q]D. #
Theorem 4. Let B be a Boolean algebra generated by to \ i e /}. Then B
is freely generated by to | i e /} iff, whenever I0, Ily J0y Jx are finite subsets

84
DISTRIBUTIVE LATTICES
of I with I0 U Ix = J0u Jx andl0 n Zx = 0, then
A (at I ie/0) A A W I ^/i) < A fa I ''e/o) A A W I »e/J
implies that I0 = /0 awrf 7x = /x*
proof. Again, the "only if" part is by substitution into S32- On the other
hand, clearly B is freely generated by {a{ \ iel} iff, for every finite subset
/ of /, the subalgebra [{at \ i e I}] is freely generated by {at \ i e I}. By
Theorem 2, the latter holds iff [{a{ \ iel}] has 22,/| elements, which, in
turn, is equivalent to [{at \ i e I}] having 2|;| atoms. Using the proof of
Theorem 2 and the present hypothesis for I0 u Ix = /, we can see that the
elements of the form A (ai I 'e h) A A W I *e A)» where 70 u Ix = /
and 70 n Ix = 0, are distinct atoms in [fa | /e/}], thus completing the
proof. %
For a simpler variant of Theorem 4 see exercise 9.43.
Now we turn our attention to an important application of polynomials:
finding homomorphisms.
Theorem 5. Let the Boolean algebra B be generated by the subalgebra Bx
and the element a. Let B2 be a Boolean algebra and let tpbea homomorphism
of Bx into B2. The extensions of <p to homomorphisms of B into B2 are in
one-to-one correspondence with the elements p of B2 satisfying the following
conditions:
(i) Ifx e Bl9 x < a, then x<p < p.
(ii) IfxeBux>a> then x<p > p.
To prepare for the proof of this theorem we verify a simple lemma, in
which + denotes the symmetric difference; that is,
x + y = (x' A y) V (x A /).
Lemma 6. Let the Boolean algebra B be generated by the subalgebra Bx and
the element a. Then every element x of B can be represented in the form
x = {a A x0) V (a' A Xx), Xo^ e Bx.
This representation is not unique. In fact,
(a A x0) V (a' A *x) = (a A y0) V (a' A yx), x0>*id'od'i e ^1
iff
a < (x0 + yoy and xx + yx < a.

Section 8 Polynomials and Freeness
85
proof. Let B0 denote the set of all elements of B having such a
representation. If x e Bu x = (a A x) v (d A x)\ thus Bx c B0. Also
C = (flAl)V(fl'A 0),
and so a e B0. Therefore, to show B0 = £, it suffices to verify that B0 is a
subalgebra, which is left as an exercise. Now note that for pyq e B> p = q
iff p A a = q A a and p A a' = q A a'\ thus (a A x0) V (a' A x±) =
(a A y0) V (a' A y±) iff a A x0 = a A y0 and a' A x± = a' A yx.
However, a A x0 = a A y0 is equivalent to (a A x0) + (a A y0) = 0; that is,
a A (x0 + ^0) = 0 (see exercise 11), which is the same as a < (x0 + y0)\
Similarly, a' A x± = a' A y iff xx + yx < a. %
proof of theorem 5. Let p be an element as specified and let the map
tfs: B-+ B2be defined as follows:
ip: (a A x0) V (a' A x±) -> (p A x0<p) v (/>' A *i<p).
By Lemma 6, iff is defined on B. It is well defined, because if (a A x0) V
(a' A xj = (a A y0) V (a' A ^0, then x1 + y± < a < (x0 + yQ)'\ thus
(*i + yi)<P < p < (x0 + y0y<p, and therefore
*i9> + yi<p < p < (x0<p + y0<p)',
implying that (p A x0<p) v (// A X&) = (p A y0(p) V (/?' A yxq>). It is
routine to check that ^ is a homomorphism. Conversely, if 0 is an extension
of 9 to By then ip is uniquely determined by/? = a*/*, and/? satisfies (i) and
(ii). •
Corollary 7. Let us assume the conditions of Theorem 5. In addition, let
B2 be complete. Set
*o = V (xq> | x e Bl9 x < a) and x1 = A (x<p \ x e Bl9 x > a).
Then the extensions of <p to B are in one-to-one correspondence with the
elements of the interval [x0> x±]. In particular, there is always at least one
such extension.
Exercises
1. Get lower and upper bounds for |PD(/i)| that are sharper than those
given by Theorem l(iii).
2. Work out the details of the last step in the proof of Theorem L

86
DISTRIBUTIVE LATTICES
3. Let pj be an atomic Boolean polynomial. Show that under the
substitution Xi = 1 for ieJ and x% = 0 for / £ /, we get pj = 1 and pJo = 0 for
all J0 * J.
4. Let /: {0, l}n -> {0, 1}. Prove that there is an /i-ary Boolean polynomial
p that defines the function f on S32 (see Definition 4.2). (Let K =
{J\J ^ {0, 1,..., n - 1} such that f(x0y..., xn-i) =? 1, where xt = 1
if ieJ and *i = 0 if /£/}; set /> = V (/>/1 /e #) and use exercise 3.)
5. Let B be a Boolean algebra. Prove that there is a one-to-one
correspondence between zz-ary Boolean polynomials over B and maps {0, l}n —>
{0, 1}. In other words, all 0, 1 substitutions take 0 and 1 as values and
determine p.
6. An algebraic function over a Boolean algebra B is built up inductively
from the variables and the elements of B using A, v, and '. Show
that the /i-ary algebraic functions are in one-to-one correspondence
with maps {0, l}n -► B.
7. Let p be an w-ary algebraic function over the Boolean lattice B and ©
a congruence relation on B. Show that at = M®)> i < n implies that
p(a0,..., an_i) = p(b0y..., 6n_i)(0).
8. Show that the property described in exercise 7 characterizes algebraic
functions (G. Gratzer [1962]).
*9. Use the property described in exercise 7 to define Boolean algebraic
functions over a distributive lattice. Show that for distributive lattices
with 0 and 1, exercise 6 holds without any change (G. Gratzer [1964]).
10. Show that a free Boolean algebra on countably many generators has
no atoms.
11. Show that a A (b + c) = (a A b) + (a A c) holds in any Boolean
algebra.
12. Let B be the Boolean algebra freely generated by {xt | / e /}. Let L be the
sublattice generated by {xt | i e /}. Prove that L is the free distributive
lattice freely generated by {xt \ i e /}.
13. Let L and Lx be distributive lattices, let L = [A], and let <p be a map of >4
into Li. Show that there is a ho isomorphism of L into Li extending <p
iff, for all pairs of /i-ary lattice polynomials p and <7 and a0, • • •, an-1 e <<4,
p(a0,..., fln-i) = <7(tfo, •.., fln-i) implies that p(a0<p,..., tf*-i9>) =
#(tf0<P> • • •> tfn-i9>)-
14. State and prove exercise 13 for Boolean algebras.
15. Interpret Lemma 6 using exercise 14.
16. Extend the last statement of Corollary 7 to the case in which Bx is
generated by B and a0 a„-iG5i,n > 1.
17. Let p and q be lattice polynomials. Since p and q can also be regarded as
Boolean polynomials, p = q was defined in two ways in this section:
with respect to D and with respect to B. Show that the two definitions
are equivalent for lattice polynomials.
18. Define = for lattice polynomials with respect to a class K of lattices.
Show that PK(n) e K iff the free lattice over K with n generators exists,
in which case Pn(n) is a free lattice with n generators.

Section 9 Congruence Relations
87
9. Congruence Relations
Let L be a lattice and let H c Z,2. We denote by S(H) the smallest
congruence relation such that a = b for all <a, by e H.
Lemma 1. For any H c L2, ®(H) exists.
proof. Let <D - A (® I 0 = *(@) ^r all <«, b> e H). Since in the lattice
C(L) the meet is intersection, it is obvious that a = b($>) for all <a, b} e i/;
thus 0) - 0(i7). #
We will use special notations in two cases: If H = {<a, &>}, we write
0(0, 6) for 0(#). If H = 72, where 7 is an ideal, we write 0[7] for 0(#).
The congruence relation 0(a, 6) is called principal; its importance is
revealed by the following formula.
Lemma 2. ®(H) = V (©(*, &) I <*> *>> e #).
proof. The proof is obvious. 0
Note that ®(a,b) is the smallest congruence relation under which
a = b, whereas 0[7] is the smallest congruence relation under which /is
contained in a single class.
In general lattices, not much can be said about 0(i/). In distributive
lattices, the following description of 0(a, b) is important (G. Gratzer and
E. T. Schmidt [1958e]):
Theorem 3. Let L be a distributive lattice, a,b,x,y e L, and let a < b.
Then x = y(9(a, b)) iff x A a = y A a and x V b = y V b.
remark. This situation is illustrated in Figure 9.1.
proof. Let O denote the binary relation under which x = y(<b) iff
x A a = y A a and x V b = y v b. Q> is obviously an equivalence
relation. If x = }>(0) and z eL, then (x V z) A fl = (x A fl) V (z A a) =
(jAfl)v(zAfl) = (}'Vz)A 0, and (x V z) V b = z V (x V b) =
z V (y v b) = (y v z) v b; thus x V z = j> V z(<D). Similarly, x A z =
j> A z(O), and so $ is a congruence relation. That a = b(<b) is obvious.
Finally, let 0 be any congruence relation such that a = 6(0) and let
x = j>(<D). Therefore, xAa = yAa,xvb = yvbyxVa = xv b(®)9

88
DISTRIBUTIVE LATTICES
and x A b = x A a(0). Then, computing modulo 0, we obtain
x = x v (x A a) = x v (y A a)
= (x V y) A (x v a) = (x V y) A (x v b)
= (x V y) A (y V b) = y V (x A b)
= y v (x a a) = y v (y A a) = y9
that is, x s j>(0), proving that O < 0. £
explanation. Since a = b implies that (a v p) A q = (b v /?) A q, we
must have x = j>(0(a, b)) if x A y = (a V p) A q, x V y = (b V p) A q.
It is easy to check that the xy y satisfying the conditions of Theorem 3 are
exactly the same as those for which such /?, q exist. Thus Theorem 3 can be
interpreted as follows: We get all pairs x>y with x = y(®(a>b)) and
x < y by applying the substitution property "twice." No further
application of the substitution property or transitivity is needed.
Some applications of Theorem 3 follow.
Corollary 4. Let I be an ideal of the distributive lattice L. Then
x = j>(0[/]) iff x v y = (x A y) V i for some is I. Therefore, I is a
congruence class modulo 0[/].

Section 9 Congruence Relations
89
(xAy)Vi
Figure 9.2
RENfARK. This situation is illustrated in Figure 9.2, in which the wavy line
indicates congruence modulo ©[/].
proof. If x v y = (x a y) V /, then x = y(Q(x A y A /, /)), x A y A /,
ie/, and so x = >>(0[7]). Conversely, 0[7] = V (0(w> ») I w,i>e/) by
Lemma 2. However, 0(w, y) v 0(wi, vx) < 0(w A v A «i A vu u V
p V «i V Di); therefore 0[7] = U(0(w» ») I uyvel). If x == J>(0(m, u))>
w,i? e /, m < v, then x V v = >> V i\ and so (x A y) V [v A (x V y)] =
x V y; thus the condition of Corollary 4 is satisfied with i = v A (x V y)
e 7. Finally, if a e 7, a = 6(0[7]), then a v b = (a A b) v i, i e I, and so
a V be I, and 6 e 7, showing that 7 is a full congruence class. #
Corollary 5. Lef Lbe a distributive lattice, xyyyayb e L, aw/ let x < y <
a<bora<b<x<y. Then x = 7(0(fl, &)) implies that x = y.
A very important congruence relation has already been used in the proof
of Lemma 3.5(ii): Given a prime ideal P of the lattice L, we can construct
a congruence relation that has exactly two congruence classes, P and
L — P. This statement can be generalized as follows: Let A be a set of
prime ideals of a lattice L and let us call two elements x and y congruent
modulo A iff, for every P g A, either x,yeP or x,y eL — P; this describes
a congruence relation on L. For instance, if A = {P, g, 7*}, Q ^ P,
7* <= P, we get five congruence classes as shown in Figure 9.3; the quotient
lattice is shown in Figure 5.8.

90
DISTRIBUTIVE LATTICES
Figure 9.3
This principle will be used often. An interesting application is :
Theorem 6. Let Kbe a sublattice of a distributive lattice L and let 0 be a
congruence relation of K. Then 0 can be extended to L; that is, there exists
a congruence relation $> onL such that, for x,y e K, x = y(<b) iffx = }>(0).
proof. Let (p be the natural homomorphism of K onto K/Q, that is,
9>: x —v [x]0; then, for every prime ideal P of K/Q, P9?"1 is a prime ideal
of K. Therefore, (P9"1] is an ideal of L and [K — P9"1) is a dual ideal of
L; thus by Theorem 7.15 we can choose a prime ideal Px of L such that
Px ^ P(p~x and Pi n (# — P<p_1) = 0. For every prime ideal P of K we
choose such a prime ideal P± of L; let A denote the collection of all such
prime ideals. Let $ be the congruence relation associated with A as
previously described. Now for x,y e K the condition x = y(S) is equivalent
to x<p = y<p, and so for every P1e A either x,y e P1 or x,y $ P1; thus
x == j>(<I>). Conversely, if x = J>(0), then for every P1e A either jc,j> e Px
or x,y$Pl9 and so either x<p,y<peP or x<p,y<p$P. Since every pair of
distinct elements of K/S is separated by a prime ideal (Corollary 7.17), we
conclude that x<p = y<p and thus x = }>(0). #
It is well known that in rings, ideals are in a one-to-one correspondence
with congruence relations. In one class of lattices the situation is exactly
the same as in the class of rings.
Theorem 7. Let L be a Boolean lattice. Then
0->[O]0
is a one-to-one correspondence between congruence relations and ideals ofL.

Section 9 Congruence Relations
91
0
Figure 9.4
proof. By Corollary 4 the map is onto; therefore, we have only to prove
that it is one-to-one, that is, that [O]0 determines 0. This fact, however, is
obvious, since a = 6(0) iff a a b = a V 6(0), which in turn is equivalent
to c = 0(0), where c is the relative complement of a A b in [0, a V b]
(see Figure 9.4). Thus a = 6(0) iff c e [O]0. #
This proof does not make full use of the hypothesis that L is a
complemented distributive lattice. In fact, all we need to make the proof work is
that L has a zero and is relatively complemented. Such a distributive lattice
is called a generalized Boolean lattice.
Theorem 8 (J. Hashimoto [1952]). Let Lbea lattice. There is a one-to-one
correspondence between ideals and congruence relations of L under which
the ideal corresponding to a congruence relation, 0, is a whole congruence
class under 0 iffL is a generalized Boolean algebra.
proof (G. Gratzer and E. T. Schmidt [1958e]). The proof of the "if"
part is the proof of Theorem 7. We proceed with the "only if" part. The
ideal corresponding to w has to be (0], and thus L has a 0. If L has Wl5 as a
sublattice, then (using the notation of Figure 7.1) (a] cannot be a
congruence class, because a = o implies that i = aVc = oVc = c9 and
b = bAi = bAc = o. But o e (a], and thus any congruence class
containing (a] contains b, and b $ (a]. Similarly, if L contains 9£5, and a
congruence class contains (b], then b = o; thus i = bVc = oVc = c, and
so a = a a / = fl A c = o. Therefore, this congruence class has to
contain a, and a$(b]. Thus, by Theorem 7.1, L is distributive. Let a < b,

92
DISTRIBUTIVE LATTICES
x V y
x A yc( Jpx + y
0
Figure 9.5
/ = [O]0(fl, b). By Corollary 4, 0[/] is also a congruence relation of L
having /as a whole congruence class; consequently, 0[/] = 0(a, b), and
so a = b(S[I]). Thus, again by Corollary 4,b = avi for some / e /, and
i s O(0(fl, £>)). The latter is equivalent to / V b = 0 V b and i A a =
0 A fl. Thus a v i = b and a A / = 0, and so / is the relative complement
offlin[0,6]. •
It is no coincidence that, in the class of generalized Boolean lattices,
congruences and ideals behave as they do in rings. Indeed, generalized
Boolean lattices are rings in disguise.
Theorem 9 (M. H. Stone [1936]).
(i) Let 93 = <5; A, V> be a generalized Boolean lattice. Define the
binary operations • and + on B by setting
x-y = x A y
and by defining x + y as the relative complement of x A y in
[0, x V y] (see Figure 9.5). Then 93* = <£; +, •> is a Boolean ring—
that is, an (associative) ring satisfying x2 = jc, for all xe B (and,
consequently, satisfying xy = yx and x + x = 0 for x,y e B).
(ii) Let 93 = </?; +, •> be a Boolean ring. Define the binary operations A
and V on B by
x A y = x-y
and
xvy = x + y + x>y.
Then 93L = <£; A, V > is a generalized Boolean lattice.
(iii) Let 93 be a generalized Boolean lattice. Then (93B)L = 93.
(iv) Let S&bea Boolean ring. Then (93L)B = 93.

Section 9 Congruence Relations
93
The proof of this theorem is purely computational. Some steps will be
given in the exercises. The correspondence between Boolean rings and
generalized Boolean lattices preserves many algebraic properties.
Theorem 10. Let 93 0 ond 93 x be generalized Boolean lattices.
(i) Let I c B0. Then I is an ideal of 930 iff I is an ideal of 93g.
(ii) Let <p: B0-^ Bx. Then q> is a {0}-homomorphism ofS8Q into 93 x iff<p is a
homomorphism of 93g into 93J.
(iii) 930 is a {0}-sublattice ofSQx iff®$ is a subring o/93?.
The proof is again left to the reader.
Congruence relations on an arbitrary lattice have an interesting
connection with distributive lattices:
Theorem 11 (N. Funayama and T. Nakayama [1942]). Let L be an
arbitrary lattice. Then C(L), the lattice of all congruence relations of L, is
distributive.
proof. Let 0,O>,T e C(L). Since 0 A (O v T) > (0 a O) v (0 A T),
it suffices to prove that a = 6(0 A (<D V ¥)) implies that a == 6((0 A <D)
V (0 A T)). So let flEi(0A((Dv T)); that is, a = 6(0) and
a == 6(<I> v *F). By Theorem 3.9, there exists a sequence a a b = z0 < • • •
< zn = a V b such that z, = zi + 1(^>)orZi s zx + X(V) for every 0 < i < n.
Since a s 6(0), we also have a A 6 s a v 6(0), and so z{ = zi + i(0) for
every 0 < / < n. Thus for every 0 < i < n, z, = zj+1(0 a O) or
z, = zi + 1(0 a V), implying that a = 6((0 a O) v (0 A T)). #
Another property of congruence lattices is given in the following
definition.
Definition 12.
(i) Let Lbe a complete lattice and let a be an element ofL. Then a is called
compact ifa<\/ Xfor some X c L implies that a < N/ Xifor some
finite Xx c X.
(ii) A complete lattice is called algebraic if every element is the join of
compact elements.
In the literature, algebraic lattices are also called compactly generated
lattices.
Just as for lattices, a nonvoid subset / of a join-semilattice F is an ideal

94
DISTRIBUTIVE LATTICES
if, for ajb e F, we have a V b e I iff a and be I. Again, 1(F) is the poset
(not necessarily a lattice) of all ideals of F partially ordered under set
inclusion. If F has a zero, then 1(F) is a lattice.
Using /(F), we give a useful characterization of algebraic lattices:
Theorem 13. A lattice L is algebraic iff it is isomorphic to the lattice of all
ideals of a join-semilattice with 0.
proof. Let F be a join-semilattice with 0; we want to prove that 1(F) is
algebraic. We know that 1(F) is complete. We claim that for aeF, (a] is
a compact element of 1(F). Let X £ 1(F) and let
(«]<= V(/|/eI).
Just as in the proof of Corollary 3.2,
V(/|/e*) = {x|x</0 V ... V tn-ntteluIteX}.
Therefore, a < t0 V • • • V tn_lt tt e /, /, e Z. Thus with
-*1 — {A)» • • • > -*n -l}»
(«]<= V(/|/6*l).
Since for any / e /(F) we have
/= \/((a]\aeI\
we see that /(F) is algebraic.
Now let L be an algebraic lattice and let F be the set of compact
elements of L. Obviously, 0 e F. Let a,b e F, a V b < \/ X, X ^ L. Then
a<aVb<\/X, and so a < V ^o> for some finite X0 ^ Z. Similarly,
b <\/ Xly for some finite A^ c jr. Thus a v b <\J (XQv Xx), and
A^ u A\ is a finite subset of A\ So a V b e F.
Therefore, <F; v> is a join-semilattice with 0. Consider the map:
<p: a -* {x \ x e F, x < a}, aeL.
Obviously, <p maps L into 1(F). By the definition of an algebraic lattice,
a = V a<P,
and thus 9? is one-to-one. To prove that <p is onto, let / e /(F), a = \/ L
Then 09? 3 /. Let x e a<p. Then x < V A so tnat by tne compactness of jc,

Section 9 Congruence Relations
95
x < V h f°r some fimte h ^ I- Therefore x e /, proving that a<p c /.
Consequently, aq> = /, and so <p is onto. Thus cp is an isomorphism. %
Now we connect the foregoing with congruence lattices.
Lemma 14. Every principal congruence relation is compact.
proof. Let L be a lattice, a,b el, X £ C(L),
0(a, 6) < V (0 | 0 e *).
Then a = 6(V (0 | 0 e A")), and thus (just as in Theorem 3.9) there exists
a sequence a = x0, xl9..., xn_x = b, in which jc£ = xi + 1(0{) for some
0, e X. Therefore, a = b(\/ (0 | 0 e JT0)), where JT0 = {0O,..., 0n-i},
and so 0(a, 6) < N/ (0 I 0 G ^o), where Z0 is a finite subset of X. %
Theorem 15. Let L be an arbitrary lattice. Then C(L) is an algebraic
lattice.
proof. For every 0 e C(L),
0 = V (0(a, b) | a = b(8)).
Consequently, this theorem follows from Lemma 14 and Corollary 3.15. #
Combining Theorems 11 and 15 we get:
Corollary 16. Let L be an arbitrary lattice. Then C(L) is a distributive
algebraic lattice.
The converse of Corollary 16 is a long-standing conjecture of lattice
theory. We shall verify the conjecture in the finite case. This was first
established by R. P. Dilworth. The present proof combines a construction
of G. Gratzer and E. T. Schmidt [1962] (Lemma 18) with a result of G.
Gratzer and H. Lakser [1968] (Lemma 19).
Theorem 17.3 Let K be a finite distributive lattice. Then there exists a
finite lattice L such that K is isomorphic to C(L).
3 The reader is advised to page over the proof of Theorem 17 at the first reading of
this book.

96
DISTRIBUTIVE LATTICES
a b
"\\ //'"
Mo \A// a9beJ(K)
0
Figure 9.6
proof. The proof of Theorem 17 is immediate from Lemmas 18 and 19.
We take L = I(M) where M is given in Lemma 18. By Lemma 19,
C(L) £ C(M); by Lemma 18, C(M) £ K. Since M is finite, so is L. %
Let M be a finite poset such that inf{a ,b} exists in Af for any a,b e M.
We define in M:a A b = inf {a, 6} and a V b = sup{a,b} whenever it
exists. This makes M into a partial lattice. An equivalence relation 0 on
M is a congruence relation if a0 = &o(®) and a^ = &i(0) imply that
a0 a fli s 60 A &i(0) and that a0 V ax = b0 V &i(0) whenever a0 V fli
and 60 V b1 exist. Then the set C(M) of all congruence relations is again
a lattice.
Lemma 18. Let K be a finite distributive lattice. Then there exists a finite
poset M such that inf {a, b} exists for any a,b e M and C(M) is isomorphic
toK.
proof. Take the set M0 = J(K) u {0} and make it a meet-semilattice by
defining inf {a, b} = 0 if a ^ b (J(K) is the set of nonzero join-irreducible
elements of K; see Section 7), as illustrated in Figure 9.6. Note that
a = 6(0) and a ^ b imply in M0 that a = 0(0) and 6 5= 0(0); therefore,
congruence relations of MQ are in one-to-one correspondence with subsets
of J(K). Thus C(M0) is a Boolean lattice whose atoms are associated
with elements of J(K); the congruence 3>a associated with a e J(K) is:
x = y(®a) if {*, y} = {a, 0}, and if {*, y) ± {a, 0}, then x = y((Da) implies
that x = y.
If /(A) is unordered in AT, then we are ready. However, if, say,
ayb e J(K)9 a > b in K, then we must have Oa > Ob. The simplest way to
make this happen is to use the lattice M(a, b) of Figure 9.7. Note that
M(a, b) has three congruence relations, namely, w, t, and 0, where 0 is the
congruence relation with congruence classes {0, bu b2, b}, {au a{b)}. Thus
0(fli> 0) = t. In other words, fl^O "implies" that bx = 0, but b± = 0
does not "imply" that a1 = 0.

Section 9 Congruence Relations
97
M(a, b) ax
Figure 9.7
We construct M by "inserting" M(a,b) in MQ whenever a > b in
J(K). Figure 9.8 gives M for the three-element chain.
More precisely, M consists of four kinds of elements: (i) 0; (ii) all
maximal join-irreducible elements of K (that is, all a e J(K) such that there
is no xeJ(K) with a < x in K); (iii) for any nonmaximal join-irreducible
element a of K, three elements: a, al9 a2\ (iv) for each pair a,b e J(K) with
a > ft, a new element, a(ft). To simplify the notation, for each maximal
join-irreducible element a, we write a = aY = a2- For 0»ft G ^(^) with
a > ft, we set M(a, ft) = {0, al9 ft, bl9 ft2, a(b)}.
Observe that M(a, b) n M(c, J) = M(a, b) if a = c and ft = </;
M(fl, ft) n Af (c, J) = {0, ft, ft1} ft2} if a ± c and ft = d; M(a, ft) n M(c, d)
= {0, flj if a = c and ft # J; M(a, ft) n Jlf (c, J) - {0, ftj if ft = c;
otherwise, M(a, ft) n M(c, </) = {0}.
For jc,j> e M, let us define x < y to mean that for some ajb e .7(^0 with
a > ft, we have ;c,j> e Af (a, ft) and * < j> in the lattice M(a, ft) as
illustrated in Figure 9.7. It is easily seen that x < y does not depend on the
choice of a, ft and that < is a partial ordering relation. Since, under this
b6
a = #! =s a2
Figure 9.8

98
DISTRIBUTIVE LATTICES
partial ordering, all M(a, b) and M(a, b) n M(c, d) are lattices and
x,y e M, xe M(a, b\ and y < x imply that y e M(a9 b\ we conclude that
inf {u, v} exist for all u,v e M.
Now we describe C(M). Let HeH(J(K)) (notation of Definition 7.8).
We define a binary relation 0H on M:
x ee y(QH) if
either x,y e \J (M(a, b)\a,b e H, a > b)u \J ({0, au a2, a}\ae H),
or x,y e {al9 a(b), a(c)}9 where a > b,a> c, b,c e H, or x = y.
In other words, [O]0H contains all al9 a2, a with a e //; and if a > 6,
a,6 e //, then it also contains a(b). Outside this class the only nontrivial
congruence is a(b) = ax s a(c), where a $ H, and b,c e H, a > b, a > c.
0H is obviously an equivalence relation. The fact that 0H restricted to
any M(a, 6) is a congruence relation easily implies that 0H is a congruence
relation. Given a SH we get
H = {a | fll = O(0H)};
thus the map
9>:#->0H
is a one-to-one order preserving map of H(J(K)) into C(M). To show that
9? is an isomorphism, we have to show that q> is onto. So let 0 be a
congruence relation of M, and
H = {a | fll » 0(0)}.
Since in Af(a, 6) every congruence 0 is determined by the atoms in [O]0,
the same holds in M. Therefore, 0 - 0H. Thus H(J(K)) ^ C(Af). By
Theorem 7.9, K ~ H(J(K)), and so tf ^ C(M). #
Lemma 19. Lef M be a finite poset with the property that inf {a, b} exists
for any a,b e M. 77*e« /or eyery congruence relation 0 f/zere ex/rf,? exactly
one congruence relation 0 of I(M) such that for ajb e Af, (a] = (6](0) #T
a s 6(0).

Section 9 Congruence Relations
99
proof. Since arbitrary meets exist in M, for every element meM, (m]is
a (finite) lattice, and so if {x, y} has an upper bound, then x V y exists.
Let 0 be a congruence relation of M. For X c M set [X] 0 =
U (M 0 | x e X); that is, [X]Q - {y \ x = X®) for some x e X). If
I,JeI{M), define / = J(0) iff [/]0 = [J]Q. Obviously, 0 is an
equivalence relation. Let / = J(0), N e 7(M), and x e I n N. Then x ~= y(B) for
some j> e/, and so x ss x A j>(0) and x A yeJn N. This shows that
[/nW]0 c [/n#]0. Similarly, [JnJV]0 c [/n#]0, so InN =
J n N(0). To show that I V N = J V Af(0), recall the description of
/ V iVgiven in exercise 5.22: Set A0 = Iu N, and for 0 < n < o>, An =
{x | x < f0 V flf f0>*i e ^n-i); then 7 V N = (J (An \ n < co). We prove
by induction on n that An c [/ v N]0. For « = 0, A0 = Iu N ^
[y]0uiVc[/v W]0. Suppose that ^n_x c [J v W]0 and let x e An.
Then x < t0 \f tx for some f0,'i e^n_i. Thus t0 = w0(©) and ^ s ^(0)
for some w0,Wi eJ V N, and so f0 = f0 A wo(0) and ^ = tx A «i(0).
Observe that tQ V ^ is an upper bound for {tQ A w0, ^ A wj;
consequently, (t0 A w0) V {t1 A «i) exists. Therefore, tQ V tx = (tQ A w0) V
(*i A «i)(0). Finally, x = x A {tQ V fx) = x A ((f0 A w0) V (h A Wi))(0)
and x A ((f0 A w0) V (t1 A «i)) e J v #.
Thus x e [7 v N]9. Since / V N = U (An \ n < *>), we conclude that
/ViVc[7viV]0. Similarly, / V N £ [/ v N]9, proving that IvN
= J V N(0). This completes the verification that 0 is a congruence
relation of I(M).
If 0 = 6(0) and xe(a\ then x = x a 6(0). Thus (a] c [(6]]0.
Similarly, (6] c [(a]]0, and so (a] = (&](0). Conversely, if (a] s (6](0),
then a = £>i(0) and a± = 6(0) for some fli < fl and 6X < 6. Forming the
join of the two congruences, we get a = 6(0). Thus 0 has all the
properties required by Lemma 19.
To show the uniqueness, let O be a congruence relation of I(M)
satisfying (a] = (6](0>) iff a = 6(0). Let I,JeI(M), I = /(0>), and xel. Then
(x] n / = (x] n /($), (x] n / = (*], and Mn/= (>;] for some ye J.
Thus (x] = (j>](0), and soxe X^)* proving that / c [JJ0. Similarly,
/ c [/]0, and so / = 7(0). Conversely, if / = /(0), then take all
congruences of the form xE=y(@)yxeI,yeJ. By our assumption regarding
O, (x] = (^](^>), and by our definition of 0, the join of all these
congruences yields / = J(®). Thus ^> = 0. #
More general results along these lines can be found in G. Gratzer and
E. T. Schmidt [1962], G. Gratzer and H. Lakser [1968], and E. T. Schmidt
[1968] and [1969].

100
DISTRIBUTIVE LATTICES
Exercises
1. Let L be a distributive lattice and let x,y,a,b e L. Prove that if a < b and
x < j>, then ^Afl = )»Afl and jc v b = j> v b is equivalent to
(a V p) A q = x and (6 V />) A q = y for some /?, # in L.
2. Verify Corollary 4 directly.
3. Let AT be a sublattice of the distributive lattice L and let P be a prime
ideal of K. Prove that there exists a prime ideal g of L with Q n K = P.
4. Prove that Corollary 5 characterizes the distributivity of L.
5. Show that Theorem 6 characterizes the distributivity of L.
6. What is the smallest n such that Theorem 6 for |AT| < n characterizes the
distributivity of LI
7. Let L be a sectionally complemented lattice; that is, L has a 0 and all
intervals [0, a] are complemented. Prove that 0 -> [0] 0 is a one-to-one
correspondence between congruences and certain ideals of L.
*8. Show that the "certain ideals" that appear in exercise 7 form a sub-
lattice of I(L).
9. Prove that every (principal) ideal of L is of the form [0] 0 iff L is
distributive.
10. Let L be a distributive lattice and let / be an ideal of L. Define a binary
relation on L: x = y(Q>(I)) iff there is no aeL with a < jc v y,
xAyAael, a$L Prove that 0(7) is the largest congruence relation
of L under which / is a whole congruence class.
11. Let L be a distributive lattice with 0. Prove that there is a one-to-one
correspondence between ideals and congruence relations (in the sense of
Theorem 8) iff 0[/] = O(Z) for all /e /(L).
12. Prove Theorem 8 using exercises 10 and 11 (G. Ja. Areskin [1953a]).
13. Let L be a lattice and let a be an element of L. Show that every convex
sublattice of L containing a is a congruence class under exactly one
congruence relation iff L is distributive and all the intervals [b, a]
(b e L, b < a) and [a, c] (c e L, a < c) are complemented (G. Gratzer
and E. T. Schmidt [1958e]).
14. Derive Theorem 8 (and, also, a variant of Theorem 8) by taking
a = 0 (arbitrary a eL) in exercise 13.
15. Let L be a relatively complemented lattice, /,/e /(L), / c /. Prove that
if / is an intersection of prime ideals, then so is /(J. Hashimoto [1952]).
16. Use exercises 14 and 15 to get the following theorem: Let L be a
relatively complemented lattice. Then L is distributive iff, for some element a
of L, (a] is an intersection of prime ideals and [a) is an intersection of
prime dual ideals (J. Hashimoto [1952]).
17. Prove that the verification of Theorem 9(i) can be reduced to the Boolean
lattice case and that in this case x + y = (x A /) V (x' A y).
18. Let B be a Boolean lattice. Verify that x + y = (x A y) v (*' a /).
19. Let B be a Boolean lattice. Verify that (jc + y) + z = (x A y A z) V
(jcx A y' A z) v (x A y' A z') v (jcx A y A z') and conclude that + is
associative.

Section 9 Congruence Relations
101
20. Prove that x(y + z) = xy + xz.
21. Prove Theorem 9(i).
22. Prove Theorem 9(ii).
23. Let S3 be a generalized Boolean lattice. Observe that for x,y e J5, x A y
is the same in S3 as in (S3R)L (namely, x-y); conclude that S3 = (S3R)L.
24. Verify Theorem 9(iv).
25. Verify Theorem 10.
26. Show that, using the concept of a distributive semilattice (see Section 11),
Corollary 16 can be reformulated as follows: Let L be an arbitrary
lattice. Prove that there exists a distributive join-semilattice Fwith 0 such
that C(L) is isomorphic to /(F).
27. Characterize the lattice of all ideals of a lattice using the concept of an
algebraic lattice.
28. Characterize the lattice of all ideals of a Boolean lattice.
29. Let M be a poset such that a A b exists for all a,b e M. Let 0 and O be
congruences of M. Find a result for M analogous with Lemma 3.8.
30. Let M be as in exercise 29. Generalize Theorem 3.9 to M.
31. Let M be as in exercise 29. Is C(M) necessarily distributive?
32. Let M be as in exercise 29. Assume that M satisfies the following
conditions:
(i) There exists H c M such that for he H, (h] is a lattice.
(ii) For each ideal / that belongs to the sublattice IP(M) of I(M)
generated by the principal ideals, there exist finite {hu .. .*hn} c H,
{*i,..., *'n} £ / with ij < hi and / = (/J u • • • u (in].
Under these conditions, prove Lemma 19 for Af, replacing /(M) with
IP(M) (G. Gratzer and H. Lakser [1968]).
Exercises 33-36 are from G. Gratzer and E. T. Schmidt [1962].
*33. Let AT be a finite distributive lattice and let L be the lattice of Theorem 17
as constructed in Lemmas 18 and 19. Show that L is sectionally
complemented.
34. Let L be given as in exercise 33. Show that the congruences of L permute.
35. Let n be the length of the longest chain in K. Show that the L of Theorem
17 can be constructed so that the length of the longest chain is 2/i — 1
(define a(b) only for a >- b).
36. Let P be a poset. Generalize Theorem 17 for K = HF(P) (notation is the
same as in exercise 7.17).
Exercises 37 and 38 are from G. Gratzer and E. T. Schmidt [1958c].
* 37. Show that a chain C is the congruence lattice of a lattice iff C is algebraic.
38. Prove that a Boolean algebra B is the congruence lattice of a lattice iff
B is algebraic.
39. Let L be a distributive lattice. Show that a-> ©[(a]] embeds L into
C(L).
40. Let L be a bounded distributive lattice. Show that for ayb e L, a < b,
©(a, b) has a complement in C(L), namely, 0(0, a) v 0(6,1).

102
DISTRIBUTIVE LATTICES
41. Let L be a bounded distributive lattice. Show that the compact elements
of C(L) form a Boolean lattice (J. Hashimoto [1952], G. Gratzer and E.
T. Schmidt [1958f]).
42. Let B be a Boolean algebra generated by X and let L be the sublattice
of B generated by X. Show that B is freely generated by Ziff L is freely
generated by X in D (see exercise 8.12).
43. Let the Boolean algebra B be generated by X. Show that B is freely
generated by X iff A X0 < V Xx implies that X0 n X1 & 0 for any
finite nonvoid X0i Xi £ X (compare this with Theorems 8.3 and 8.4).
10. Boolean Algebras /?-generated
by Distributive Lattices
The investigations of this section are based on the following result:
Theorem 1. Every distributive lattice can be embedded in a Boolean lattice.
proof. By Theorem 7.19, every distributive lattice L is isomorphic to a
ring of subsets of some set X. Obviously, L can be embedded into P(X). £
Definition 2. Let L be a sublattice of the generalized Boolean lattice B.
The L is said to i^-generate BifL generates B as a ringy and if L has a 0
(or 1) the same element is the 0 (or 1) ofB.
The last two conditions are equivalent to the following:
If A L exists, then A L = A B> and if V L exists, then \/ L = \J B.
Our goal is to show the uniqueness of the generalized Boolean lattice
/^-generated by L. The first result is essentially due to H. M. MacNeille
[1939]:
Lemma 3. Let B be R-generated by L. Then every aeB can be expressed
in the form
0o + tfi + • • • + ffn-i, a0 ^ ax ^ • • • < an_i, a0,..., an-x el.
example. Let B be the Boolean lattice shown in Figure 10.1 and let
L = {0, aQ, au a2}. Then L ^-generates B.

Section 10 Boolean Algebras /?-generated by Distributive Lattices 103
flo + 02
fli + a2
Figure 10.1
proof. Let 2?x denote the set of all elements that can be represented in
the form a0 + • • • + an_i, a09..., an-x el. Then L £ Bu and Bx is
closed under + and — (since x — y = x + y). Furthermore,
(fl0 + • • • + An-lX^O + • • • + &m-l) = 2 ^/>
and each term atbj = a{ A bjeL, so 2?x is closed under multiplication. We
conclude that B1 = B.
Note that L is a sublattice of B; therefore, for aybeLya V b in L is the
same as a V b in 2?. Thus avb = a + b + ab, and so
0 + b = fl& + (a V b) = (fl A b) + (a v 6).
Take a0 + • • • + an-x G ^- We prove by induction on /i that the summands
can be made to form an increasing sequence. For n = 1 this is obvious.
Let us assume that ax < • • • < an-v Then
a0 + ax + • • • + fln_!
= (a0 A fli) + (aQ v fli) + a2 + • • • + fln-i
= teo A fli) + ((flo V fli) A fla) + (fl0 V fl2) + fl3 + • • ■ + fln-i
= (tf0 A fli) + ((fl0 V fli) A a2) + ((fl0 V a2) A a3)
+ (fl0 V a3) + • • • + an^
= fao A aO + ((fl0 V fli) A fla) + • • • + ((a0 v an-2) A an.O
+ (a0 v fln-0,
and
flo A fli < (fl0 V fli) A fl2 ^ • * * ^ («o V fln_2) A fln_i < a0 V an_i. #

104
DISTRIBUTIVE LATTICES
Lemma 4. Let L be a distributive lattice. Then there exists a generalized
Boolean lattice B freely Regenerated by L—that is, a generalized Boolean
lattice B with the following properties:
(i) B is Regenerated by L;
(ii) If Bx is Regenerated by L, then there is a homomorphism <p of
B onto Bx that is the identity map on L.
proof. The existence of B can be proved by copying the proof of Theorem
5.5 (Theorem 5.24), mutatis mutandis. #
Lemma 5 (J. Hashimoto [1952]). Let B be a generalized Boolean lattice
generated by L. Then every congruence relation ofL has one and only one
extension to B.
proof. The existence of an extension was proved in Theorem 9.6. By
Theorems 9.7 and 9.10(0* the following statement implies the uniqueness
of the extension:
If / and / are (ring) ideals of B with / <=■ /, then there are elements
a,b eL,a # b, such that a = b (mod /) and a & b (mod I).
Indeed, let x eJ — I. By Lemma 3, x can be represented in the form
X = Xq + • • • + Xn_!, Xq < • • • < Xn_i, X0, • • •, Xn_i £L.
If n is odd, then x0 = x-x0 < xeJ> and thus x0ej; x0 + xx + x2 =
x-x2 e/, therefore xx + x2 = x0 + (x0 + xx + x2) e J. Similarly, x3 + x4,
x- + x6,... eJ. Since x0 + (xx + x2) + (x3 + x4) + • • • eJ - /, we
conclude that either x0 eJ — /, or x2i + x2i + 1 eJ — / for some 2/ < n.
If n is even, then we obtain x0 + xu x2 + x3f... eJ (by multiplying x by
xl9 x3i...), and we conclude that for some 2/ < n9 x2i 4- x2i +1 £ J — /.
Now if x2i + x2i + 1 eJ - /, then x2i s x2i + 1 (mod/), but x2i & x2i + 1
(mod 7), x2i, *2i+i eL. Finally, if x0 e J - /, then *<> = 0 (mod /) and
xQ & 0 (mod /). This completes the proof, provided that 0 e L. If 0 £ L,
then we can choose a >> 6 / with y < x0 and we obtain y = x0 (mod /),
y & x (mod 7). #
Theorem 6. #* 2?x and B2 are generalized Boolean lattices R-generated by
a distributive lattice L, then Bx and B2 are isomorphic.

Section 10 Boolean Algebras /?-generated by Distributive Lattices 105
remark. For a distributive lattice L we shall denote by B(L) a generalized
Boolean lattice ^-generated by L.
proof. Let B be a free generalized Boolean lattice ^-generated by L (see
Lemma 4). Let <p be a homomorphism of B onto Bx such that <p is the
identity on L (see Lemma 4(ii)). We want to show that <p is an
isomorphism. Indeed, if <p is not an isomorphism, then the ideal
/ = {x | x e B, x<p = 0}
is not 0. Thus by Lemma 5, a = b (mod /) for some a9b eL9 a ^ b. This
means that aq> = b<p9 contrary to our assumptions. Similarly, there is an
isomorphism */t between B and B2. Obviously, <p_10 is an isomorphism
between B1 and B2. #
Corollary 7. Let L0 and L± be distributive lattices and let <p be a
homomorphism ofL0 onto Lx preserving 0 and/or 1, if they exist in L0. Then q> can
be extended to a homomorphism ofB(L0) onto B(LX).
proof. Let 0 be the congruence relation of L0 induced by <p (x = y(S)
iff x<p = y<p) and let 0 be the extension of 0 to B(L0) (Lemma 5). Then
B(L0)/Q is a generalized Boolean lattice ^-generated by Lo/0 ^ Lx. Thus
B(L0)/<5 ^ B(LX) by Theorem 6, and using this, the proof of Corollary 7
becomes trivial. #
Corollary 8. Let L0 be a sublattice of the distributive lattice Lx and
assume that ifLx has 0 and/or 1, then so does L0, and the 0, 1 ofLx is the
0,1 of LQ. Let B denote the subalgebra ofB(Lx) Regenerated by L0. Then
B(L0) £ B.
proof. The proof is trivial. 0
Let L0 and Lx be given as in Corollary 8. It is natural to query the
conditions under which L0 /^-generates B(LX). For H c B(Lx)y let [H]R
denote the generalized Boolean sublattice of B(LX) /^-generated by H. We
can answer our query by determining [L0]R n Lx.
Lemma 9. Let L0 and Lx be given as in Corollary 8 and let L0 have a
zero. Then Lx n [L0]B is the smallest sublattice ofLx containing L0 that is
closed under taking relative complements in Lx. Therefore, LQ Regenerates

106
DISTRIBUTIVE LATTICES
0
Figure 10.2
2?(Li) iff the smallest sublattice ofLx containing L0 and closed under relative
complementation in Lx is Lx itself.
proof. It is obvious that L0 ^ ^ n [L0]R. If a,b9c e^rx [L0]R, d g Ll9
and d is a relative complement of b in [a, c], then d = a + b + ce
Lx n [L0]B, since (see Figure 10.2) d is a relative complement of a + b in
the interval [0, c]. Thus de^n [L0]R. Now suppose that L is a sublattice
of Lx containing L0 and closed under relative complementation in Lx. If
jc e Lx n [L0]B, then by Lemma 3 we can represent x as
x = a0 + ••• + an_i, flor..)VieIo,flo < ••• < a„_i.
We prove x g L by induction on n. If w = 1, x = a0 g L0 £ L. If w = 2,
then x is the relative complement of a0 in [0, a^, 0,ao,^i eL0, thus xgL.
If w = 3, then (see Figure 10.1) x = a0 + a1 + a2 is the relative
complement of a1 in [a0, a2], and so xeL. Now let w > 3 and let y g L be proved
for all y = b0 + • • • 4- 6fc_i, 60, • • •> h-i eL0, ^o ^ * * * ^ 6jt-i» and
k < n. Note that jcgLx and an_3 gL0 imply that xan-3 = a0 4- • • • +
ffn-3 + 0n-3 + ^n-3 = «o + * * * + tfn-3 eLi, and x V an_3 = x +
an_3 + xan-3 = a0 + • • • + an-i + an_3 + a0 + • • • + an_3 = an_3 +
an_2 4- an_i GLa. By the induction hypothesis, a0 + • • • + an_3 eL and
an_3 + tfn_2 + an_i gL; therefore, x is the relative complement in Lx of
an element (namely, an_3) of L in an interval (namely, [a0 + • • • 4- an_3,
an_3 + an_2 + an_i]) in L, and so, by assumption, jcgL. Thus
^in[L0]BsL. •
In Theorem 1 we embedded L into P(X)9 which is a complete Boolean
lattice. The question arises whether we can require this embedding to be
complete—that is, to preserve arbitrary meets and joins, if they exist in L.
It is easy to see that not every complete distributive lattice has a
complete embedding into a complete Boolean lattice.

Section 10 Boolean Algebras fl-generated by Distributive Lattices 107
Lemma 10 (J. von Neumann [1936]). Let Bbea complete Boolean lattice.
Then B satisfies the Join Infinite Distributive Identity (J ID)*
x A V(*i I iel) = V(* A Xi | ie/),
and its dual, the Meet Infinite Distributive Identity (MID).
proof.* A xt < xandx A x{ < V (*i I /e/);therefore,x A V (*i I i^l)
is an upper bound for {x A x{ \ i e /}. Now let u be any upper bound,
that is, x A x{ < u for all i e /. Then
Xi = JCt A (* V *') = (JC( A X) V (^ A X') < U V X'.
Thus
X A \/ (Xi\iel) < x A (u V x') = (X A u) V (x A x') = X A u < u,
showing that x A \/ (Xi\ iel) is the least upper bound for {x A xt \ i e /}.
(MID) follows by duality. #
Corollary 11. Any complete distributive lattice that has a complete
embedding into a complete Boolean lattice satisfies both (JID) and (MID).
Easy examples show that (JID) and (MID) need not hold in all complete
distributive lattices.
Our task now is to show the converse of Corollary 11 (N. Funayama
[1959]). The construction depends on a property of B(L) and on Theorem
6.4.
Lemma 12 (V. Glivenko [1929]). Let L be a distributive lattice with 0.
Then I(L) is a pseudocomplemented lattice in which
I* = {x | x A i = Oforalliel}.
Let S(I(L)) = {/* | IgI(L)}. If L is a Boolean lattice, then S(I(L)) is a
complete Boolean lattice and the map a -> (a] embeds L into S(I(L)); this
embedding preserves all existing meets and joins.
4 Of course, (JID) is not an identity in the sense of Section 4 but is only an infinitary
analogue of an identity.

108
DISTRIBUTIVE LATTICES
proof. The first statement is trivial. Now let L be Boolean. It follows
from Theorem 6.4 that S(I(L)) is a Boolean lattice. Furthermore, it is
easily seen that for any X ^ I(L), the inf and sup of X in S(I(L)) are
A X and (V X)**, respectively, where A and V are tne meet an<* join
of X in /(L), respectively. Since A ((*] \ xe X) = (inf X], whenever
inf X exists in L, the mapping a ->■ (a] preserves all existing meets in L.
Observe that, for x,aeL, x A a' = 0 iff x < a, and so (a] = (a']* g
S(/(L)). Now let a = sup X in L and set / = (X] (= V(W I * e *)). To
show that x -> (*] is join-preserving, we have to verify that /** = (a], or,
equivalently, that /* = (a']. Indeed, if b e /*, then b A x = 0 for all
x g /, and thus x < b'. Therefore, a = sup X < b\ proving a' > b, that is,
b g (a']. Conversely, let b e (a']. Then b' > a; therefore, b' > a =
sup X > x for all xel, and so b A x = 0 for all x e X. This shows that
b g /*, proving that J* = (a']. #
Lemma 13. Let L be a complete lattice satisfying (JID) and (MID). Then
the identity map is a complete embedding ofL into B(L).
proof. Let us write a e B(L) in the form
a = a0+ ••• +an-i, a0 < a± < • • • <an-i> a0, .. .,an-i el.
Since L is complete, it has a 0, and thus (writing a0 as 0 4- a0) we can
assume without loss of generality that n is odd. We claim that for x e L and
a g 5(L), we have x < a iff
x A a0 = x A fli,
x A a2 = x A a3, ...,
x A an_3 = x A an_2, x < an_i.
Indeed, let x < a. Then
*#! = xa^^ + • • • + an-i) = x(a0 + a1 + aa + • • • +aa) = xa0;
therefore, x A a0 = x A a^ Thus x(a2+ • • • + fl»-i) = (xa0 + x^) +
x(a2 + • • • + an-i) = xa = x, and so x < a2 + • • • + an.x. Conversely,
if x A a0 = x A ax and x < a2 + • • • 4- an_l5 then xa = xa0 + xax +
x(a2 + • • • + an_i) = x, proving that x < a. A simple induction
completes the proof of the claim.

Section 10 Boolean Algebras /?-generated by Distributive Lattices 109
Let X £ L, y = sup X in L, and a e B(L). If x < a for all xe X, then
the formulas of the preceding claim hold for all x and a and, by (JID), for y
and a, proving that y ^ a. Thus y = sup X in 5(L). The dual argument,
using (MID), completes the proof. %
Theorem 14 (N. Funayama [1959]). A complete lattice L has a complete
embedding into a complete Boolean lattice ijfL satisfies {MID) and (JID).
proof. Combine Lemmas 10-13. O
The representation for asB{L) given in Lemma 3 is not unique in
general; the only exception is when L is a chain. Since this case is of
special interest, we shall investigate it in detail.
Repeating the definition, a Boolean lattice B is R-generated by a chain if
B = B{C) for some chain C c B. This concept is due to M. Mostowski
and A. Tarski [1939] and can be extended to distributive lattices as
follows:
A distributive lattice L is R-generated by a chain C (£ L) if C i?-generates
B(L).
The following notation will facilitate the proof of the next lemma as well
as the discussion at the end of this section. For a chain C in a lattice L,
write C° to denote the chain C if L has no zero and let C° denote C u {0} if
L has a zero.
Lemma 15. Let Lbe a distributive lattice and let C be a chain in L. Then C
R-generates L iff L is the smallest sublattice of itself containing C° and
closed under the formation of relative complements.
proof. Apply Lemma 9 to C°. 0
An explicit representation of 5(C) is given as follows:
For a chain C, let B [C] be the set of all subsets of C of the form
teo] + (fli] + • • • + (an_J, 0 < a0 < a1 < • • • < an.l9
a0,.. .,an_! eC,
and 0, where + is the symmetric difference. We consider B[C] as a poset,
partially ordered by c. The clause "0 < a0" means that 0 < a0 if 0
exists; otherwise there is no restriction on a0.
We identify aeC with (a], for a ^ 0, and 0 (if it exists) with 0. Thus
C s B[C].

110
DISTRIBUTIVE LATTICES
Lemma 16. B[C] is the generalized Boolean lattice R-generated by C.
proof. The proof is obvious, by construction and by Theorem 6. %
Note that every nonvoid element a can be represented in the form
a = (a0] u (6lf aju.-.u (bn.u an^]9
0 < a0 < bx < ax < • • • < 6n_i < an.u
where the union is disjoint union, the first term (a0] may be missing, and
(x9 y] stands for {/1 x < t < y}. An element of the form (x, y] is nothing
but x + y. Thus,' a = a0 + bx + ax + • • • 4- bn_1 + an_l9 and so we
conclude:
Corollary 17. In B(C) every nonzero element a has a unique
representation in the form
a = a0 + ax + - • • 4- an-i>
0 < a0 < ax < • • • < an.l9 a0,..., an-± e C.
The following results show that many distributive lattices can be
^-generated by chains.
Lemma 18. Every finite Boolean lattice B can be R-generated by a chain;
in fact, B = [C]Rfor any maximal chain C ofB.
proof. Let Bx be the subalgebra of B ^-generated by C. Using the
notation of Corollary 7.14, the length of C = \J(B)\; also, the length of
C = |/(5x) |; thus |/CB)| = |/(5i)l = n. We conclude that both B and B1
have 2n elements, proving that B = Bx. %
Corollary 19. Every finite distributive lattice L can be R-generated by a
chain—in fact, by any maximal chain ofL.
proof. Let C be a maximal chain in L and let B = B(L). Then | J(L)\ =
\J(B)\. By Corollary 7.14, C is maximal in B. Thus, B = B(C) 2l. |
Theorem 20. Every countable distributive lattice L can be R-generated by a
chain.

Section 10 Boolean Algebras /?-generated by Distributive Lattices 111
proof. Let L = {a0i al9 a2i..., a»,...} and let Ln be the sublattice of L
generated by <z0,..., an. Let C0 be a maximal chain of L0, and, inductively,
let Cn be a maximal chain of Ln containing Cn_i. Set C = (J (Q | / < o>).
We claim that C generates L. Take aeB(L); a = x0 -f • • • + xm-l9
x0i..., jcm_! eL. L = (J (Li | i < o>); thus for some w, x0,..., xm-i e^n,
and so aeB(Ln). Since Ln is finite, we get B(Ln) = B(Cn); therefore
a e B(Cn) c 5(C), proving that L s 5(C). #
Corollary 21. 77ie correspondence C -> .5(C) ma/w //?e cto of countable
chains onto the class of countable generalized Boolean lattices. Under this
correspondence, subchains and homomorphic images correspond to sub-
algebras and homomorphic images.
Note, however, that C0 £ Cx is not implied by B(C0) £ 2*(Ci).
Much is known about countable chains. Utilizing the previous results,
such information can be used to prove results on countable generalized
Boolean lattices. To help distinguish an important class of chains, we
introduce the concept of prime interval. An interval [a, b] is prime if a -< b.
Lemma 22. Every countable chain C can be embedded in the chain Q of
rational numbers. Every countable chain not containing any prime interval is
isomorphic to one of the intervals (0, 1), [0, 1), (0, 1], and [0, l]ofQ.
proof. Let C = {x0, xl9..., xn_! • • •}. We define the map <p inductively
as follows: Pick an arbitrary r0e Q and set x0q> = r0. If x0q>9..., xn_i<p
have already been defined, we define xn<p as follows: Let
Ln = U ((xw] | xt < xni i < n)
Un = U (tew) I xt > xni i < n)
(Ln = 0 or Un = 0 is possible). Note that if Ln ^ 0, then it has a
greatest element /n, and if Un ^ 0, then it has a smallest element un. If
both are nonempty, then ln < un. In any case, we can choose an rn e Q
with rn $ Ln u Un. We set xn<p = rn. Obviously, <p is an embedding.
The second part of the proof reduces to the following statement:
Let C and D be bounded countable chains with no prime intervals.
Then C ^ D.
To prove this, let C = {c0, cl9...} and D = {d0> dl9...}. We define two
maps: <p: C-> D9 ^: D -> C. Let us assume that c0 = 0, c± = 1, and that

112
DISTRIBUTIVE LATTICES
d0 = 0, d1 = 1. For each n < w, we shall define inductively finite chains
Cn, Dn (Cn ^ c, Dn ^ D) and an isomorphism <pn: Cn -> Dn with inverse
<£n: £n -> Cn. Set C0 = {Co, cj = {0, 1}, Z)0 = {4>, rfj = {0,1} and *V0 =
i, i0o = h for / = 0, 1. Given Cn, Z)n, <pn, 0n, and w even, let k be the
smallest integer with ck $ Cn. Define uk = A (fac) n Q) and 4 =
V dck] n ^n)- Then lk < ck < uki and so lk<pn < uk<pn. Since D contains no
prime intervals, we can choose a de D satisfying lk<pn < d < uk<pn. Since
0n is isotone, d$ Dn. Define Cn+1 = Cn u {cfc}, Z)n+1 = Z)n u {d}, <pn+i
restricted to Cn to be <pn, and C/c<pn+x = d9ifsn+1 restricted to Dn to be i/*n and
#n+i = ck. If w is odd we proceed in a similar way, but we interchange
the role of C and Z), Cn and Z)n, <pn and 0n, respectively.
Finally, put 99 = (J (<pn | w < a>). Clearly, C = (J (Cn | « < o>), Z> =
U (Z)n I ai < o>), and <p is the required isomorphism. 0
Corollary 23. L//> to isomorphism there is exactly one countable Boolean
lattice with no atoms and exactly one countable generalized Boolean lattice
with no atoms and no unit element, B(D)9 where D is the [0, 1) rational
interval,
proof. Take the rational intervals [0, 1] and [0,1). The generalized
Boolean lattices in question are B([Q, 1]) and B([0, 1))- This follows from
the observation that [a, b] is a prime interval in C iff a -f b is an atom in
B(C). The results follow from Lemmas 16 and 22 and Theorem 20. %
Theorem 24. Let B be a countable Boolean algebra. Then B has either X0
or 2*o prime ideals.
remark. This is obvious if we assume the Continuum Hypothesis.
Interestingly enough, we can give a proof without it.
proof. For a Boolean algebra B and an ideal I of B, we shall write B/I for
£/©[/]. If/is an ideal of B with / 2 /, then /// = {[*]©[/] | x eJ} is an
ideal of B/I. (This is the usual notation in ring theory.) Let B be a Boolean
algebra. We define the ideals Iy by transfinite induction. I0 = (0]; I± is the
ideal generated by the atoms of B; given Iyi let / be the ideal of B/ly
generated by the atoms of B/Iy and let <p:x-+x 4- Iy be the homo-
morphism of B onto B/Iy; we set Jy + 1 = Tip-1. Finally, if y is a limit
ordinal, set Iy = ij (J6 | 9 < y). The rawA; of B is defined to be the smallest
ordinal a such that Ia = /a+1. Obviously, a < |2?|.

Section 10 Boolean Algebras /?-generated by Distributive Lattices 113
claim 1. Let B be countable. Ifla # B, then \0>(B)\ = 2«o.
Indeed, if Ia ^ B9 then Bjltt has no atoms, and thus B/Ia ~ B(C),
where C is the rational interval [0, 1]. By Lemma 5 (see exercise 27),
\0v*iq\ = mc)\ = |/(oi = 2*.
claim 2. Let B be countable. IfIa = B, then \&(B)\ = X0-
Indeed, for y < a, let &y(B) be the set of prime ideals P of B for which
/ycp,/r+1$?. Since a < X0> it suffices to show that \&y(B)\ = xo-
If Pe&y(B)9 then, by Corollary 7.16 and Theorem 7.22, we have
P V 7y+i = B. It follows that for P,Q e &y(B\ P ^ Q9 we have
P n Iy+1 ^ gn/y+1. Thus P->(P n [/y + i]B)//y is a one-to-one
correspondence of 0y(2?) into (in fact, onto) ^([/y + iW/y); but [7y + 1]B//y is just
the generalized Boolean lattice of all finite subsets of a countable set.
Therefore, \&y(B)\ = X0. •
In order to avoid giving the impression that most Boolean algebras
can be ^-generated by chains, we state
Lemma 25. Let Bbea complete Boolean algebra Regenerated by a chain C.
Then B is finite.
proof. Let B = [C]R and let the chain C be infinite. We can assume that
C contains a subchain {x0i xl9..., xni...}, x0 < xx < • • • < xn < • • • (or
dually, in which case we dualize the proof). Then define an = x0 + xx +
• * * + x2n + x2n + i for each n < w. We claim that V (fk I ' < "0 does not
exist. Indeed, let a be an upper bound for {at | i < a>}. By the remarks
immediately following Lemma 16, we can represent each an by a set
(*o> Xi]u (x2i x3] u • • • u (x2n, Jc2n+i], and we can represent a in the form
a = (b0] u (bu b2] u • • • u (6m_2, bm.x]9
where 0 < 60 < *i < * • • < 6m-i» 6i e C for i < m, and the first term,
(b0]9 may be missing. Since a contains each an, there must exist an n and a
7 < m such that both (x2n, x2n + 1] and (x2n+2, x2n+3] are contained in
(bj-ub,] (or in (b0] if j = 0). Therefore, the interval (x2n+1, x2n+2]
can be deleted from a, and it will still contain all the at—that is,
a + x2n+1 +x2n+2 is an upper bound for {a{ \ i < a>}, and
a + x2n + 1 + x2n+2 < a. We conclude that {a{ \ i < w} does not have a
least upper bound. %

114
DISTRIBUTIVE LATTICES
Next we consider which chains can be ^-generating chains of a given
distributive lattice.
Definition 26. Let L be a distributive lattice and let C be a chain in L.
The chain C is called strongly maximal in L iff, for any homomorphism <p
of L onto a distributive lattice Ll9 the chain (C<p)° is maximal in Lx. (The
notation C° was introduced preceding Lemma 15.)
Lemma 27. If the distributive lattice L is R-generated by a chain C £ L,
then C° is maximal in L.
proof. If C° is not maximal in L, then we can find a e L, a $ C, a ^ 0,
such that C u {a} is a chain. Write a = a0 -f ax -f • • • -f an_x with
0 < a0 < a± < • • • < an-1 and ate C for / < n. Since a $ C, n > 1. Now
a A a0 = a0 -f a0 -f • • • -f a0, which is a0 if n is odd and 0 if n is even.
But a0 ^ a and 0 ^ a, so, since a and a0 are comparable, a A a0 = a0i
and rt is odd. Then a A a± = a0 + #i + • • • + a1 = a0i contradicting the
comparability of a and ax. %
The converse of Lemma 27 is false. The following theorem settles the
matter.
Theorem 28. Let Lbea distributive lattice and let Cbea chain in L. Then C
Regenerates LiffC is strongly maximal in L.
proof. If C P-generates L, then, for any homomorphism <p, C<p R-
generates Ltp. By Lemma 27, (C<p)° is maximal in Dp, so C is strongly
maximal in L.
Next assume that C is strongly maximal in L but does not /^-generate L.
Without any loss of generality we can assume that L and C have a greatest
element. (Otherwise, add one. Then C u {1} is strongly maximal in L u {1}
but does not /^-generate L u {1}.) Let B± = B(L) and let B0 = [C]R. By
hypothesis, B0 ^ Bl9 so there exists anae^ — B0. We claim that there
exist prime ideals P1 ^ P2 of B1 with B0 n P1 = B0 n P2. With J =
((a] n B0] and Z> = [a), we have / n D = 0, so, by Theorem 7.15, there
is a prime ideal P1 such that / ^ Pi and P1r\ D = 0. Then let /x = (a]
and D1 = [B0 - Pj). Since (a] n £0 c pl9 it follows that 71nD1 = 0.
Let P2 be a prime ideal with Ix ^ P2> ?2 n Z>i = 0. Then aeP2 — Pi, so
Pi ^ P2. Because P2 n (B0 - Px) = 0, P2r\B0^P1r\ B0. Then by

Section 10 Boolean Algebras /?-generated by Distributive Lattices 115
Theorem 7.22 (prime ideals of a Boolean lattice are unordered), P1 r\ B0 =
P2 n B0i proving our claim.
Now (again by using Theorem 7.22) we can map B1 onto ((£2)2 by a
homomorphism ^: For xgP1c\ P2, xi/* = <0, 0>; for xgP1 - P2i
xi/* = <0, \}; for xgP2 -Pux^ = <1, 0>; for x^P1 u/>2, xi/s = <1,1>.
Since Op = {<0, 0>, <1, 1» is not maximal and G/> = (Ctyr)0, we conclude
that C is not strongly maximal in L. #
Corollary 29. Let C and D be strongly maximal chains of the distributive
latticeL. Then \C\ = \D\ and |/(C)| = \I(D)\.
proof. If L is finite, these conclusions follow from Corollary 7.14. If L is
infinite, then [C] = [D] = B(L\ and so |C | = |D| = |L|. Also, by Lemma
5,|^(C)| = \&(B(L))\ = |^(D)|,and^(C) = I{C\&(D) = 1(D); hence
the second statement. 0
Corollary 29 is the strongest known extension of Corollary 7.14 to the
infinite case. The second part of Corollary 29 is from G. Gratzer and E. T.
Schmidt [1957a].
Boolean algebras generated by chains were first investigated by M.
Mostowski and A. Tarski [1939]. Theorem 20 for Boolean lattices and
Theorem 24 were communicated to the author by J. R. Buchi. These results
have been known for some time in topology (via the Stone topological
representation theorem, see Section 11). Some of the other results are
apparently new.
Exercises
1. Give a detailed proof of Lemma 4.
2. Try to describe the most general situation to which the idea of the proof
of Theorem 5.5 (Theorem 5.24) could be applied.
3. Show that Lemma 5 does not remain valid if the word "generalized" is
omitted.
4. Find necessary and sufficient conditions on L in order that L have a
Boolean extension B to which every congruence of L has exactly one
extension.
5. Let L be a distributive lattice and define Lx to be the lattice L if L has a
unit element; let L± be L with a unit added if L does not have a unit
element. The Boolean lattice B[L] Regenerated by L is defined to be

116
DISTRIBUTIVE LATTICES
B(LX). Show that if B is any Boolean lattice, containing L as a sublattice,
and B is generated by L under A, V, and', then B is isomorphic to the
Boolean lattice iJ-generated by L.
6. Work out Corollaries 7 and 8 for the Boolean lattice ^-generated by L.
*7. The Complete Infinite Distributive Identity is (for /,/ ^ 0):
A (V fa \jeJ) | iel) = V (A fa,* \iei)\9:1->J).
Show that this holds in a complete Boolean lattice B iff it is atomic
(A. Tarski [1930]). (Hint: apply the identity to
A {a V a' | a e B) = 1).
8. Prove that the Complete Infinite Distributive Identity is self-dual for
Boolean lattices.
9. For a subset A of a lattice L, set
>4U = {* | x e L, * is an upper bound of A}
A1 = {x | x e L, * is a lower bound of >4}.
Prove that (>4U)' 5 ,4, (Al)u ^ A9 Au = ((Au)l)\ and ^J = ((Al)u)K
10. Call an ideal / of a lattice L normal if / = (/U)J. Show that every principal
ideal is normal.
11. Let /jv(L) denote the set of all normal ideals of L. Show that IN(L) is a
complete lattice but that it is not necessarily a sublattice of Io(L\
12. Show that the map: x -> (x] is an embedding of L into IN(L\ preserving
all meets and joins that exist in L. (IN(L) is called the MacNeille
completion of L; see H. M. MacNeille [1937].)
13. Let B be a Boolean lattice and let / be an ideal of B. Show that / is normal
iff/ = /** (for the concepts, see exercise 10 and Lemma 12).
14. Prove that the Boolean lattice S(I(L)) of Lemma 12 is the MacNeille
completion of the Boolean lattice L.
♦15. Show that the MacNeille completion of a distributive lattice need not
even be modular.
16. Let L be a distributive algebraic lattice. Show that L satisfies the Join
Infinite Distributive Identity. (Thus, for any lattice K> C{K) satisfies
(JID).)
17. Let L be a distributive lattice, aiyb{eL for i < w and [a0, b0] z>
[aly b±] => •• •. Define
0 = V (0(ao, at) v 0(6O, bd \ i < o).
Show that
0 v A (0(tft, bt) |/<w)^A(0v 0(a(, bi) | i < a>).
18. Use exercise 17 to show that, for a distributive lattice L, the Meet
Infinite Distributive Identity holds in C(L) iff every interval in L is finite
(G. Gratzer and E. T. Schmidt [1958c]).
19. For a chain C, introduce and describe B(C) using Corollary 17.

Section 11 Topological Representation
117
*20. Prove the converse of Lemma 18: If every maximal chain generates the
Boolean lattice By then B is finite.
21. Why is it not possible to use transfinite induction to extend Theorem 20
to the uncountable case ?
22. Let C be a chain with 0 and 1 and let a e C. Define a new order on C:
For x,y < a> and a < x,y, let x < y retain its meaning; for x < a <* yy
let y < x; let Ci be the set C with the new order. Then Cx is a chain, and
B(C) £ Bid), but in general C ^ d does not hold.
23. Describe a countable family of pairwise nonisomorphic countable
Boolean algebras.
24. Let C be a maximal chain of the distributive lattice L. Prove that C is a
maximal chain in B(L).
25. Let L0 be the [0, 1] rational interval and let Lx be the [0, 1] real interval.
Let C = {<*, *> | 0 < x < 1, x rational}. Then C is a maximal chain in
L0 x Li. Show that C is not strongly maximal (G. Gratzer and E.
T. Schmidt [1957a]).
26. Find in L0 x L± of exercise 25 maximal chains of cardinality N0 and
2«o. What are the cardinalities of strongly maximal chains?
27. Let B = B(L). Show that P->P n L for ?e 0>(B) is a one-to-one
correspondence between the prime ideals of L and B (use Lemma 5).
*28. Let A be an infinite set, B — P(A). Prove that B has maximal chains of
cardinality \A\ and 2|i41.
29. Construct an example in which the sequence of ideals Iy of Theorem 24
does not terminate in finitely many steps.
30. Let C be the [0, 1] interval of the rational numbers. Show that B(C) is
FB(K0).
11. Topological Representation
The poset ^(L) of prime ideals does give a great deal of information about
the distributive lattice L, but obviously it does not characterize L. For
instance, for a countably infinite Boolean algebra L, 0*(L) is an unordered
set of cardinality X0 or 2*S whereas there are surely more than two such
Boolean algebras up to isomorphism.
Therefore, it is necessary to endow 0*(L) with more structure if we want
it to characterize L. M. H. Stone [1937] endowed ^(L) with a topology
(see also L. Rieger [1949]). In this section we shall discuss his approach in a
slightly more general but, in our opinion, more natural framework.
A join-semilattice L is called distributive5 if a < b0 V bx (aib0ib1 gL)
5 This concept appeared quite naturally in the research of G. Gratzer and E. T.
Schmidt on congruence lattices of lattices in 1960-1961; because the research was
not very successful, this concept did not appear in print until the late sixties.

118
DISTRIBUTIVE LATTICES
b0 V &!
b0
*0
*1
01
Figure 11.1
implies the existence of a0>tfi GL,a{ < bi9 i = 0,1, with a = aQ V ax (see
Figure 11.1). Note that a0 and ax need not be unique.
Some elementary properties of a distributive semilattice are as follows:
Lemma 1.
(i) If(L\ A, V> is a lattice, then the join-semilattice <L; V y is distributive
iff the lattice <L; A, V > is distributive.
(ii) If a join-semilattice L is distributive, then for any a9b e L there is a
dsL with d < a, d < b. Consequently, I(L) is a lattice.
(iii) A join-semilattice L is distributive iffI(L)9 as a lattice, is distributive.
PROOF.
(i) If <L; A, V> is distributive, and a < b0 v bl9 then with a{ =
a A bi9 i = 0,1, a = a0 V ax. Conversely, if <L; v> is distributive,
and the lattice L has 90*5 or 9£5 as a sublattice, then a < b v c (see
notation of Figure 7.1) leads to a contradiction. )
(ii) a < a V b9 thus a = a0 V 60, where a0 < a, b0 < b. Since, in
addition, b0 < a, b0 is a lower bound for a and 6. )
(iii) First we observe that, for /,/ e I(L),
/ V /= {/ V j\ieIJeJ}
follows from the assumption that the join-semilattice L is distributive.
Therefore, the distributivity of I(L) can be easily proved. Conversely,
if I(L) is distributive, and a < b0 V bl9 then
(a] = (a] A ((&„] V (*J) = ((«] A (6J) V ((a] A (AJ),
and so a = a0 V o^ a0 e (6<J> fli e (ft^, which is distributivity for

Section 11 Topological Representation
119
A subset D of a join-semilattice L is called a dual ideal if a e D and
x > a imply that xe D, and a,b e D implies that there exists a lower bound
d of {a, b} such that d e D. An ideal / of L is prime if / ^ L and L — / is a
dual ideal. Again, let ^%L) denote the set of all prime ideals of L.
Lemma 2. Let I be an ideal and let D be a dual ideal of a distributive
join-semilattice L. If I n D = 0, then there exists a prime ideal P ofL with
P ^ /, P n D = 0.
proof. The proof is a routine modification of the proof of
Theorem 7.15. #
In the rest of this section, let L stand for a distributive join-semilattice
with zero.
In ^(L), sets of the form r{a) = {P\ a$P) represent the elements ofL.
We will make all these sets open.
Let &{£) denote the topological space defined on ^(L) by postulating
that the sets of the form r(a) be a subbase6 for the open sets; we shall call
«2%L) the Stone space ofL.
Lemma 3. Let I be an ideal ofL,
r(l)={P\PeSr(L)9P$I}.
Then r{T) is open in ^(L). Conversely, every open set U of ^(L) can be
uniquely represented as r(I)for some ideal I ofL.
proof. We simply observe that r(I) n r(J) = r(I A J),
r(V(//|yeA0)-Ufr(/i)|yeJO,
and r((a]) = r(a), from which it follows that the r(I) form the smallest
collection of sets closed under finite intersection and arbitrary union
containing all the r(a). Observe that a elif[r(a) s r(/). Thus r(I) = r(J)
iff a e I is equivalent to a e /, that is, iff / = /. 0
Lemma 4. The subsets of 5f(L) of the form r(a) can be characterized as
compact open sets.
6 Exercises 1-22 review all the topological concepts that are used in this and the
next section.

120
DISTRIBUTIVE LATTICES
Figure 11.2
proof. Indeed, if a family of open sets {r(Ik) \ k e K) is a cover for r(a),
that is, r (a) £ (J (r (4) \keK) = r(\/(Ik\ke K)), then a e V(4 | * e JQ.
This implies that ae\/ (Ik\ke K0) for some finite K0 c AT, proving that
r(a) £ (J (r(/fc) | k e J£0). Thus r(a) is compact. Conversely, if / is not
principal, then /•(/) s (J (r(a) | a e /), but r(/) $ (J (r(a) | a e /0) for
any finite I0 £ /. 0
From Lemma 4 we immediately conclude:
Theorem 5. The Stone space «£%L) determines L up to isomorphism.
If we want to use Stone spaces to construct new ones in order that new
distributive lattices can be constructed from given ones, then we have to
know what Stone spaces look like. Stone spaces are characterized in
Theorem 8. To prepare for the proof of Theorem 8, we prove Lemma 6.
Let P be a prime ideal of L. Then P is represented as an element of £f(L)
and also by r(P). The connection between P and r(P) is given in Lemma 6
and is illustrated by Figure 11.2.
Lemma 6. For every prime ideal P ofL, {P) = «9%L) - r(P)9 where {P} is
the topological closure of{P).
proof. By definition of closure,
W} = {Q\Qer(a) implies that Per{a)} = {Q \ Q 2 P}
= &{L) - {Q | Q $ P} = ST{L) - r{P). %
Corollary 7. If P ^ Q, then {P} # {Q}; in other words, Sf{JO) is a
T0-space.
proof. Combine Lemmas 3 and 6. 0

Section 11 Topological Representation
121
Lemma 6 also shows that if P is a prime ideal, then Sf(L) — r(P) must
be the closure of a singleton. In other words:
(C) If U is an open set with the property that, for the compact
open sets U0 and Ul9 UQr\U1^: U implies that U0 c u or
Ux s £/, then <^(L) - U = {P} for some element P.
Now we can state the characterization theorem:
Theorem 8. The Stone space £f ofa distributive join-semilattice with zero
can be characterized (up to homeomorphism) by the following two properties:
(51) Sf is a T0-space in which the compact open sets form a base
for the open sets.
(52) If F is a closed set in &?,{Uk\keK} is a dually directed1
family of compact open sets of £f> and Uk n F ^ 0, then
ft(Uk\keK)c\F* 0.
remark. The meaning of condition (SI) is clear. Condition (S2) is a
complicated way of ensuring that (C) holds and that Lemma 2 holds for the
join-semilattice of compact open sets of &(£).
proof. To show that (SI) holds, we have to verify that the r(a), aeL,
form a base (not only a subbase) for the open sets of ^(L). In other words,
for a,b gL9Pg r{a) n rib), we have to find a c e L with P e r(c)9 r(c) c
r(a) n r(b). By assumption, a$P9b $P\ thus, if P is prime, there exists a
ceL,c$P,c<a,c<b. Then P e r(c\ r(c) c r(a\ and r(c) c r(b), as
required. To verify (S2) for &>(L\ let F = Sf(T) - r(I) and Uk = r(ak).
Thus F = {P | P 2 /} and Uk = {P \ ak $ P). The assumptions on the ak
mean that D = {x \ x > ak for some k e K) is a dual ideal; since
Ukr\ F ^ 0, we have r(ak) $ r(I); that is, ak$I9 showing that
D r\I = 0. Therefore, by Lemma 2, there exists a prime ideal P with
P^> I9Pr\D = 0. Then ak$P, and so Psr(ak) for all A;e#. Also
P2/, thus P $ r(I\ and so PeF, proving that P e F n f| (*4 I * e AT),
verifying (S2).
Conversely, let Sf be a topological space satisfying conditions (SI) and
(S2) of the theorem. Let L be the set of compact open sets of £f. Obviously,
0 gL and if A,BeL9 then A U BeL, and thus L is a join-semilattice
with zero. Let
A ^ B0U Bt with A.Bq.Bx g L.
7 K & 0 and for fc0, #i e JST, there exists & k2e K such that C/fc2 c uk0 n C/fcl.

122
DISTRIBUTIVE LATTICES
Figure 11.3
Then A n Bt is open, and therefore A n Bt = (J (A) \jEjt)9 i = 0, 1,
where the A\ are compact open sets. Since A = (A r\ B0) u (A r\ B^) ^
\J(A)\jeJq\j Jui = 0,1), by the compactness of A we get A^
(J {A) \jeJ0 orje Ji), where Jt is a finite subset of Ji9 i = 0, 1. Set
4 = UWJI./e/«), i = 0,l.
Then A0iA1 eL, A = A0 u y^, and ^0 ^ ^o> ^i ^ ^i, showing that L is
distributive.
It follows from (SI) that the open sets of Sf are uniquely associated with
ideals of L: For an ideal /of L, let U(I) = (J (a | a e /) (keep in mind that
an a eL is a subset of ^ as illustrated by Figure 11.3). Note that for
flGL,flG/iffflC U(I).
Now let P be a prime ideal of L, F = S? - U(P), and let {Uk \ k e K}
be the set of all compact open sets of Sf that have nonvoid intersections
with F. Thus, the Uk are exactly those elements of L that are not in P.
Therefore, by the definition of a prime ideal, given k,le K9 there exists
t e K with Ut S Uk9 Ut c £/,, proving that F and {Uk \ k e £} satisfy the
hypothesis of condition (S2). By (S2) we conclude that there exists a
pE F nf} (Uk | k e K). If q e F, then for every compact open set U with
q e U, we have U n F ^ 0; thus /?e[/, proving that Jp} = F. Note that
Sf is r0; therefore /> is unique.
Conversely, if p e ^ let / = {a | a e L, a c ^ - {p}}. Then / is an
ideal of L, and ^ — {/?} = £/(/). We claim that / is prime. Indeed, if
U,VeL,U$I,V$I, then U r\{p} ^ 0, V r\{p}^ 0, and therefore
PeU,peV. Thus, pEUr\V and so C/n K$ £/(/). By (SI) there
exists slWeL with JF c V c\V,W % U(I). Therefore W$I, and so /is
prime.
Summing up, the map <p: P->p is one-to-one and onto between £%L)

Section 11 Topological Representation
123
and ^ To show that 9? is a homeomorphism, it suffices to show that U is
open in SP(L) iff U<p is open in SP. Since a typical open set in SP(L) is of the
form r(I) (7e 7(L)), and an open set of SP is of the form U(I), we need
only prove that r(I)q> = U(I) and £/(7)9>-1 = /*(/)—in other words,
that Per(I) iff (Pq> = )peU(I). Indeed, Per(I) means that P $ 7,
which is equivalent to U(P) $ U(I); this, in turn, is the same as
U(I) c\(SP - U(P)) ^ 0. Since SP - U(P) = {p} with p = P<p, the last
condition means that U(I) n {/>} ^ 0, which holds iff/? e £/(7). (Indeed,
ifp$ 1/(7), then 1/(7) <= £/(/>), and so 1/(7) n{p}= 0.) #
Corollary 9 (M. H. Stone [1937]). 77*e Stone space of a distributive
lattice is characterized by (SI), (S2), and
(S3) The intersection of two compact open sets is compact.
Corollary 10 (M. H. Stone [1936]). The Stone space SP of a Boolean
lattice (called a Boolean space) can be characterized as a compact Haus-
dorff space in which the closed open (clopen) sets form a base for open sets.
{In other words, SP is totally disconnected.)
proof. Let SP = SP(B), where B is a Boolean lattice. Then SP = r(l),
and thus SP is compact. Let P,Qe SP and P ^ Q and take aeP - Q.
Then Q e r(a), P e r(a'); therefore, every pair of elements of SP can be
separated by clopen sets, verifying that SP is Hausdorff. Obviously, SP is
totally disconnected. Conversely, let SP be compact, Hausdorff, and
totally disconnected. Then (SI) is obvious. (S2) follows from the
observation that F and the Ut,ie 7, are now closed sets having the finite
intersection property; therefore, by compactness, they have an element in
common. The compact open sets of SP form a Boolean lattice B, and thus
SP is homeomorphic to SP(B). 0
As an interesting application we prove:
Theorem 11. Let B be an infinite Boolean lattice. Then \&(B)\ > \B\.
proof. Let SP be a totally disconnected compact Hausdorff space. For
a,b e SP with a ^ b, fix a pair of clopen sets Uatb and Ubta such that
a e Uatb, b e Ubta, and Ua%b n Ub%a = 0. Now let U be clopen and aeU.
Then SP - U c \J (Ubta \ b e SP - U\ and so, by the compactness of
SP - U, 9 - U ^\J (Ub,a \beB) for some finite B c SP - U. Then

124
DISTRIBUTIVE LATTICES
Va = fWa.J b e £) is open and 0 e Va c £/. Thus, £/ c (J(^a|«e^),
so by the compactness of U, U ^ U (Ka | a e ^4) for some finite A ^ jj.
Therefore, U = \J(Va\aeA). Thus every clopen set is a finite union of
finite intersections of Uatb, and so there are no more clopen sets than there
are finite sequences of elements of Sf\ this cardinality is \$f\ provided that
|&\ is infinite. %
It might be illuminating to compare this to an algebraic proof; see
exercise 36.
Theorem 8 and its corollaries provide topological representations for
distributive join-semilattices, distributive lattices, and Boolean lattices,
respectively. It is possible to give topological representation for homo-
morphism. We do it here only for {0, l}-homomorphisms of bounded
distributive lattices.
Lemma 12. Let L0 and Lx be bounded distributive lattices and let <p be a
{0, \}-homomorphism ofL0 into Lx. Then
maps Sf(L^) into S?(L0); 5f(<p) is a continuous function with the property that
if U is compact open in 9(L0), tnen U(&*($>))'* is compact in 9(L^).
Conversely, ifi//: 9(L^) -> 9(L0) has these properties, then */» = 9(<p) for
exactly one 9\L0->L1.
proof. If U = r(a), a e L0, then
USr{9)-i ={p\Pg se{Ld9 P<p~i gr(a)}
= {P\Pe<7(L1),a$P<p-i}
= {P\Pe<?(L1),a<PiP} = r(a<p),
and so 9(<p) is continuous, having the desired property.
Conversely, if such a 0 is given and U = r(a), aeL0, then Uip'1 is
compact open, and so Uip'1 = r(b) for a uniqueb eLx. The map <p: a^>b
is a {0, l}-homomorphism, and */» = 9{cp). 0
The following interpretation of (S2) will be useful. Let 9 be a
topological space. The Booleanization of Sf is a topological space £fB on 9*
that has the compact open sets of 9 and their complements as a subbase
for open sets.

Section 11 Topological Representation
125
Lemma 13. The compact topological space Sf satisfies (SI), (S2), and (S3)
iff <?b is a boolean space.
proof. Let Se satisfy (SI), (S2), and (S3). Then 9*B is obviously Haus-
dorff and totally disconnected. To verify the compactness of £fBy let ^
be a collection of compact open sets of Sf and let ^ be a collection of
complements of compact open sets of 9* such that in & = ^ u ^ no
finite intersection is void. Because of (S3) we can assume that ^ is closed
under finite intersection. Since members of ^ are closed in 9 and 9 is
compact, p| (XI XG ^i) = F IS nonvoid. Also, for any U e ^0i and
Xe&u Un Jif is closed in U, and thus U n F = C\(Un X\ le^) ^
0. Applying (S2) to F and c^, we conclude that Q <^~ ^ 0, which, by
Alexander's Theorem (see exercise 14), proves compactness.
Conversely, if 9B is Boolean, the compact open sets of £fB form a
Boolean lattice L. We can easily verify that the compact open sets of 9
form a sublattice Lx of L. Thus Lx is a distributive lattice, and so, by
Corollary 9, 9> = 9(LX) satisfies (SI), (S2), and (S3). #
Exercises
The following exercises review the basic concepts and theorems of topology
that are utilized in Sections 11 and 12.
1. A topological space is a set A and a collection T of subsets of A closed
under finite intersections and arbitrary unions; a member of Tis called
an open set. Call a set closed if its complement is open. Characterize the
family of closed sets.
2. A family of nonvoid sets B s. T is a base for open sets if every open set
is a union of members of B. Show that for a set A, B c P(A) is a base for
open sets of some topological space defined on A iff \J B = A, and for
X, YeB and pe Xn y, there exists a. Z e B with peZ, Z ^ X, and
Zjz y
3. A family of nonvoid sets C c i>(,4) is a subbase for open sets if the finite
intersections of members of C form a base for open sets. Show that
C c P^) is a subbase of some topology defined on A iff \J C = A.
4. Let A be a topological space and let A' c A Then there exists a smallest
closed set X containing X; A" is the closure of X. Show that for JJc ,4,
Z s y implies that jc F;Ig l;Ju y= lu F;andl= I

126
DISTRIBUTIVE LATTICES
5. Prove that the four properties of X given in exercise 4 characterize it.
6. Show that ae X iff every open set (in a given base) containing a has a
nonvoid intersection with X_
7. A space A is a T0-space if {x} = {y} implies that x = y. Show that A is a
To-space iff, for every x,y e A, x ^ y, there exists an open set (in a given
base) containing exactly one of x and y.
8. A is a Ti-space if {*} = {*} for all x g A. A 7\-space is a r0-space. Show
that A is a 7\-space iff, for x,y e A, x i=- y, there exists an open set (in a
given base) containing x but not y.
9. Let A and 5 be topological spaces and /: A —> B. Then / is called
continuous if, for every open set U of B, f~ x((7) is open in A. /is a homeo-
morphism if/is one-to-one and onto and if both /and/-1 are
continuous. Show that continuity can be checked by considering only those
f~\U) where U belongs to a given subbase.
10. Show that/: A -> B is continuous iff/(J) £ /(*) for all JT c ,4.
11. A subset A" of a topological space A is compact ii X ^ \J (£/< | t/( open,
i e /) implies that X £ U (Ut I ' e A) for some finite A £ /.
The space A is compact if X = A is compact. Show that A is compact iff,
for every family F of closed sets, if f\ F1 ^ 0 for all finite Fi c F, then
n^* 0.
12. Let ^4 be a compact topological space and let A" be a closed set in A.
Show that X is compact.
13. Prove that a space /4 is compact iff, in the lattice of closed sets of A, every
maximal dual ideal is principal.
*14. Show that a Ti-space A is compact iff it has a subbase C of closed sets
(that is, {A — X \ Xe C} is a subbase for open sets) with the property:
If p) D = 0 for some D £ c, then p) #i = 0 for some finite Dx ^ D
(J. W. Alexander's Theorem [1939]).
15. Let At, i g /, be topological spaces and set A = Yl C^t I 'e /)• For U c
^i, set E(U) = {/|/ei4,/(/)e £/}. The product topology on /I is the
topology determined by taking all the sets E(U) as a subbase for open
sets, where U ranges over all open sets of A{ for all / g /. Show that the
projection map e{: /—► / is a continuous map of A onto A{.
16. Show that if the At are r0-spaces (7\-spaces), so is A = n (/4, | i g /).
17. A map f: A -^ B is 0/?e« if /(£/) is open in 5 for every open U c A.
Show that the projection maps (see exercise 15) are open.
18. Prove that a function f:B—>YlAt is continuous iff, for each iel,
e{'f: B —> Ai is continuous.
19. A space /4 is a Hausdorff space (T2-space) if, for x,^ g v4 with x ^ y> there
exist open sets £/, Vsuch that xe(/,;eK, Un V = 0. Show that:
(i) A is Hausdorff iff A = {O, x> | x g A] is closed in ,4 x A
(ii) A compact subset of a r2-space is closed.
20. Prove that a product of Hausdorff spaces is a Hausdorff space.
21. Show that a product of compact spaces is compact (Tihonov's Theorem).
(Hint: use exercise 14.)
22. A space A is totally disconnected if, for x,y g A, x ^ y, there exists a

Section 11 Topological Representation
127
closed open set U with xe U, y $ U. Show that the product of totally
disconnected sets is totally disconnected.
Following are some exercises for Section 11.
23. Let / and / be ideals of a join-semilattice. Verify that
/ V /= {t | t < i V j,iel,jej}.
24. Show that for a join-semilattice L, I(L) is a lattice iff any two elements of
L have a lower bound.
25. Give a detailed proof of Lemma 2.
26. Prove that every join-semilattice can be embedded in a Boolean lattice
(considered as a join-semilattice).
27. Show that a finite distributive join-semilattice is a distributive lattice.
28. Let L be a join-semilattice and let 0 be an equivalence relation on L
having the Substitution Property for join. Then L/0 is also a join-
semilattice. Show that the distributivity of L does not imply the
distributive of L/0.
29. Let F be a free join-semilattice; let F0 be F with a new 0 added. Show
that F0 is a distributive join-semilattice.
30. Let <p be a join-homomorphism of the join semilattice F0 into the
join-semilattice Fi. Show that, for distributive join-semilattices F0i Fl9
the proper homomorphism concept is the one requiring that if P is a
prime ideal of Fl9 then P<p~1 is a prime ideal of F0.
31. Show that there is no "free distributive join-semilattice" with the
homomorphism concept of exercise 30.
32. Does Theorem 7.22 generalize to distributive join-semilattices?
33. Characterize the Stone spaces of finite Boolean lattices and of finite
chains.
34. Let S?0 and S/{ be disjoint topological spaces; let Sf -S^0 u ^ and call
U c ^ open if U n Sf0 and U n 6^ are open. Show that if ^ and STX
are Stone spaces, then so is Sf.
35. If, in exercise 34, ^ = ^(L,), / = 0, 1, then & = Sf(L0 x LJ.
36. To prove Theorem 11, pick an element a(P,Q)eP- Q for all
P,Qe0>(B) with P * Q. Show that the a(P, Q) 7?-generate all of B.
(Corollary 8.7 or Lemma 13.3 can be used.)
37. In Lemma 12, characterize «^(<p) for one-to-one and for onto <p.
38. Determine the connection between the Stone space of a lattice and the
Stone space of a sublattice.
39. Call the Stone space of a generalized Boolean lattice a generalized
Boolean space; characterize it. (Compactness of Sf should be replaced by
local compactness: For every p e Sf there exists an open set U with pe U
and a set Kwith U c K, such that Kis compact.)
40. Show that the product of (generalized) Boolean spaces is (generalized)
Boolean.
41. Call the join-semilattice L modular if, for a9b9c eL9a < b and b < a v c

128
DISTRIBUTIVE LATTICES
imply the existence of c± e L with c± < c, b = a V d. Show that a
distributive join-semilattice is modular.
42. Show that Lemma 1 remains valid if all occurrences of the word "
distributive" are replaced by the word "modular."
43. Show that the set of all finitely generated normal subgroups of a group
(and also the finitely generated ideals of a ring) form a modular join-
semilattice.
44. The lattice of congruence relations of a join-semilattice L is distributive
iff any pair of elements with a lower bound are comparable (D. Papert
[1964], R. A. Dean and R. H. Oehmke [1964]).
12. Free Distributive Product
Let Lu i g /, be pairwise disjoint distributive lattices. Then Q =
U (Li | / g /) is a partial lattice. A free lattice generated by Q over the class
D of all distributive lattices is called a free distributive product of the Lu
i g /. To prove the existence of free distributive products, it suffices by
Theorem 5.24 to show that there exists a distributive lattice L containing Q
as a relative sublattice. This is easily done: Let L be the direct product of
the Lt u {0}, i £ /, where 0 is a new zero element of Lt. Identify xeLt with
/eL defined by/(/) = x,f(j) = 0 fory # i. Then Q becomes a relative
sublattice of L.
An equivalent definition is:
Definition 1. Let Kbe a class of lattices and let Li9 i e I, be lattices in K.
A lattice L in K is called a free K-product of the Li9 i e I, if every Lt has an
embedding et into L such that
(i) L is generated by U (L,€( | / e /).
(ii) IfKis any lattice in K and <pt is a homomorphism ofL{ into K9for i e /,
then there exists a homomorphism <pofL into K satisfying <pt = €tq>for
all i g /(see Figure 12.1).
For distributive lattices, this is equivalent to the first definition (see
exercises at the end of this section). In most cases we will assume that
each Lt is a sublattice of L and that et is the inclusion map; then (ii) will
simply state that the q>t have a common extension. Note that in all the cases
we shall consider, (i) can be replaced by the requirement that the <p in (ii) be
unique.
If, in Definition 1, K is the class of bounded distributive lattices and all

Section 12 Free Distributive Product
129
Lt *- L
\
K
Figure 12.1
homomorphisms are assumed to be {0, l}-homomorphisms, we get the
concept of a free {0, \}-distributive product.
Our first result is the existence and description of a free {0,
^-distributive product of a family of bounded distributive lattices.
Theorem 2 (A. Nerode [1959]). Let Li9 i e I, be distributive lattices with
Oandl. Let ST = Yl (&(!«) \*G I) (see exercise 11.15). Then & is a Stone
space, and thus Sf % £f(L)for some distributive lattice L. Any such lattice L
is a free {0, 1}- distributive product of the Lx, i e /.
The proof of Theorem 2 will be preceded by two lemmas. In these two
lemmas a Stone space is a topological space satisfying (SI), (S2), and (S3)
of Section 11.
Lemma 3. Let «^J, i e /, be compact Stone spaces. Then
n(^fi^/) = (n(^|/e/)r.
(The notation yB was introduced in Section 11.)
proof. For £/<=^ let E(U) = {f\feU &uf(j) e U} (see exercise
11.15). The compact open sets form a base for open sets in S^; therefore,
{E(U) | U compact open in some ^} is a subbase for open sets in
n (&\ I iel)> Note that all the sets E(U) are compact open in n &tl
therefore, V ^ n &i is compact open iff it is a finite union of finite
intersections of some of the E(U). Consequently, declaring the complements of
compact open sets to be open (in forming Qfl &if) is equivalent to making
the complements of the sets E(U) open. But the complement of E{U) is
E(&[ - U), and 9[ - U is a typical open set of S?f. Thus n &? and
(EI &if have the same topology. %
<Pt

130
DISTRIBUTIVE LATTICES
*x«^ shq = n *w>
f
4(tf)
Figure 12.2
Lemma 4. 77ie product of compact Stone spaces is again a compact Stone
space.
proof. Let &1, i e I, be Stone spaces. Then Sf = fl ^ is r0 and compact
(exercises 11.16 and 11.21). Since Sff is Boolean (Lemma 11.13), so is
n &* (exercises 11.20, 11.21, and 11.22). By Lemma 3, S?B = n ■??. and
thus £fB is Boolean. Therefore, by Lemma 11.13, $f is a Stone space. 0
proof of Theorem 2. Let et be the /th projection (et: ^(L) -> ^(LO,
/ef =/(/)). By Lemma 11.2, there is a unique {0, l}-homomorphism
Ci'.Li-^L satisfying ^fe) = e{. It is easy to visualize €f; think of the
elements of Lt as compact open sets of #?; then U€t = E(U) = C/ef1. It
is obvious from this that ^ is an embedding.
By applying ^ to Figure 12.1, we obtain Figure 12.2. Thus the method
of defining &*(?) is obvious. For x e ^(K), xSf(<p) is a member of
n <y(Li), and x£?(<p)(i) = xSffa) for / e /. To show that this
correspondence is indeed of the form ^(<p) for some homomorphism <p:L->* K,
we have to verify that (a) ^(<p) is continuous (this statement follows from
exercise 11.18), and that (b) if V is compact open in 6f(L\ then VS^^)'1
is compact open in Sf{K). It is enough to verify (b) for V = E(U) where U
is compact open in some ^{L^. Then
V<?(<p)-i = E(U)9>{<p)-i = UerWfryi = UWrfed-1 = ^fo)"1,
and therefore K^(<p)-1 is compact open since <9*(<pt) satisfies the condition
of Lemma 11.12. 0
This result not only establishes the existence of free {0, l}-distributive
products but gives a useful description of them. Some applications will be
given in the exercises at the end of this section and other applications will
be shown in Chapter 3.
Let L be a free {0, l}-distributive product of the Lh iel, and let us

Section 12 Free Distributive Product
131
assume for the rest of this section that L, ^ L. Thus L = [UAL and so
every asL can be written in the form
a = V(AX\XeJ)
(and dually) where every X e J is a finite subset of (J L,, and / is finite.
(This is obvious directly, but it also follows from Lemma 4.3 and the
representation of distributive lattice polynomials given in the proof of
Theorem 8.1.) Such representations are not unique. Therefore, the
question arises as to how we can determine whether two such expressions
represent the same element. This is called the word problem, and we will
give a solution to it based on a characterization of free products. (All the
subsequent results of this section are from G. Gratzer and H. Lakser
[1969a].)
Theorem 5. Let Lbe a distributive lattice with 0 and 1, let the Lt be {0,1}-
sublattices of L for i e 7, and let L = [IJ Lt]. Then L is a free {0, \}-distribu-
tive product of the Lt iff, for finite nonvoid 70, 7X ^ 7, for x{ e Li9 i e 70, for
y, e L}J e Il9 satisfying xki yk $ {0, 1} in Lkifor all kel,
A(xt\iGl0)< VO^Ue/i)
implies the existence ofi e 70 n 7X with x{ < yt.
remark. If there is an iel0n 7X with xt < yi9 then A (xt | i e I0) <
V (yj | j e 7i). Thus the condition should be interpreted as requiring that
A (xt | / e 70) < V (yj I j e h) holds only in the trivial case. This result is
a direct generalization of Theorem 8.3.
proof. Let L be a free {0, l}-distributive product of the L„ / e 7,
A (xt | i e 70) < V (yj I j e 7X) and let xt £ yt for every i e 70 n 7^ For
every / e 7 we choose a prime ideal P( of L, as follows: If / e 70 n 71$ then
*t ^ #, so we can choose P( with xt $Pt and ^gP,; for /e70 - /^ we
assume that x* ^ Pt; for / e 7i - 70, we assume that yt e Pt. Now we define
homomorphisms <p,:7,,-^(£2 by x<pt = 0 for xePi9x<pi = 1 for x^P*.
Since L is a free product, there is a common extension <p: L ->• (£2 of the <p,.
But this is impossible because A (*i I i e ^o)<P = A (*i<P |/e/0) = A1 = 1»
and V (yj 17 e A)<P = V ° = °»contradicting /\xt < \/ y,.
Now let L satisfy the condition of the theorem. Let L* be a free {0, 1}-
distributive product of the Lh i e I (we assume Lt c £,*), let 9, be the

132
DISTRIBUTIVE LATTICES
identity map on L( as a map from Lt into L, and let <p be the common
extension of the <pi9 i e /. We claim that <p is an isomorphism. Since L*<p
includes all L^L^) and the Lx generate L, <p is onto. So to show that <p is
an isomorphism we have only to verify that for a,b eL* — {0, 1},
a<p < b<p implies that a < b.
Let
*= V(A(*iUe/,)|./eJ)
and
b = AO/(y*\keK,)\meM)9
where It and Km are finite nonvoid subsets of /, x{ e L*, yk e Lki and / and
M are finite and nonvoid. If a<p < by, then for eachy e J and me M,
A(xt\iel,)< \/(yk\keKm)
in L; thus, by assumption, for some ielt n Afm, x, < }>t. This implies that
A(*|/e/,)£ VO^I*^
in L*; therefore a < b. %
As an immediate corollary, we get the following result of B. Jonsson
[1961] (see also Section 13).
Corollary 6. Let Kt be a {0, \}-sublattice of the distributive lattice Lt
with 0 and 1, for i e /. Let L be a free {0, \}-distributive product of the Lt
(assume that Lt c L) and K = [U (Kt | i e /)]. 7%e« A: w a /ree {0, 1}-
distributive product of the Ki9 i e I.
Another consequence is:
Corollary 7. With notation as in Theorem 5, let I0i Ix be finite nonvoid
subsets ofL Let xt e Lf - {0, \}for i e l0 and let yseLj — (0, \}forj e Ix.
Then
Aixtlielo) < A(yj\jeli\
ifflo 2 h and for i e Il9 x{ < y{.

Section 12 Free Distributive Product
133
proof. The "if" part is trivial. Conversely, if A xt ^ A^i* tnen
f\x{ < yj for every j g 71# Applying Theorem 5, we conclude that I0 n {7}
is nonvoid, that is, thaty e I0. Thus /0 ^ h> and JCt < ^ for / e Ix. %
Every element a of a free {0, l}-distributive product L of the Lt has many
representations of the form V (A (*« I * G Jy) 17 G J)- Using Theorem 5,
we can decide whether two such expressions represent the same element.
Thus Theorem 5 solves the word problem. It is desirable, however, to
select one of these expressions (which will be called normal form). We start
with a few definitions and notations.
Let L be a free {0, l}-distributive product of Li9 ie I, and let
g = U(A-{0, 1}|/g/)u{0,1}.
We identify Oe Q with 0 g L, for each / g /, and similarly for leg. A
finite nonempty subset X ^ Q is said to be reduced if | X n Lt\ < 1 for all
/ g I. It should be noted that if X is reduced and Oel, then X = {0}, and
dually. If X ^ Q is finite and nonempty, we can define a reduced subset
Xh of Q9 the A -reduct of Z, by the conditions:
(i) If /' = {/ g /1 X n Lx * 0}, then
Za ={AOrnL0|/G/'},
provided that A (^ n A) # 0 for all i e /'.
(ii) If there is an 1 e I' such that A (X n A) = 0, then ZA = {0}.
The v -reduct of A", denoted Xw, is defined in the dual manner. We note
that, in L, A X = A (^A) and V ^ = V (*v).
Definition 8. A finite nonempty family J of finite reduced subsets of Q is
said to be a V-representation of a eLif
a = V(AX\XeJ).
The family J is said to be a A-representation of a eLif
a = /\(V X\XeJ).
Obviously, every element a of L has both V" and A_rePresentations-
Given a A_representation J of an element aeL, we can write, using
distributivity (see exercise 4.11),
fl = V(AWI^?W)f

134
DISTRIBUTIVE LATTICES
where #(/) denotes the set of choice functions on /—that is, the set of
functions F:J^>(JJ such that F(X) e X for each Xe J. As a result of
our previous discussion, we find that a = V (A (F(J)h) I Fe^(J)).
Since the set ^(/) is finite, we can consider a subset ^red(^) ^ ^(J), the
set of reduced choice functions such that the set
{A(F(JY)\FeVUJ)}
is the set of all maximal elements of the set
{/\{F{JY)\FeV{J)}.
Thus the family {F(jy \ FE^red(J)} is a V_representation of a; it is said
to be a normal V-representation of a. There is, of course, dually a normal
^-representation of a.
Theorem 9.
(i) Each aeL has a normal V-representation.
(ii) Let ajb £ L and let Jx be a V -representation of a and J2 a normal V~
representation ofb. Then a < b iff the following condition holds:
(R) For each XeJx there is a Ye J2 such that for each y e Y there
is an x e X satisfying x < y.
proof. Since
a= \f{/\X\XeJx)
and
b = V(AY\ YeJ2\
condition (R) is clearly sufficient for a < b.
Now let a < b and let A' be a A-representation of b such that
J2 = {F(KY\Fe<#UK)}.
Therefore,
\/{/\X\XeJ1)< /\(VZ\ZeK).
Thus if X e Jl9 then A* ^ V Z for each ZeK. Since both X and Z are
reduced for each Ze^we conclude, by Theorem 5, that for each ZeK
there is an element G(Z) e Z such that A X < G(Z). Thus
A X < A (G(Z) | Z g K) = A (<W),

Section 12 Free Distributive Product
135
where, clearly, Ge^(K). By the definition of #redC*0> there is an Fe
#red(/0 such that A (G(KY) < A (H^Y)- The rest of the condition
follows by Corollary 7. %
Since in a normal V_rePresentation «A the elements of {A X I A" g /} are
mutually incomparable, we conclude:
Corollary 10. The normal V-representation of any element of L is
uniquely defined.
To show how normal representations can be used to investigate the
structure of free {0, l}-distributive products, we prove:
Theorem 11. Let the Li9 ie I, be distributive lattices with 0 and 1 and let L
be a free {0, \}-distributive product of Li9 is I. If all the L{ satisfy the
Countable Chain Condition (any chain in Lt is of cardinality < X0), then L
satisfies the Countable Chain Condition.
In the proof we will need:
Lemma 12. Let A be a chain and let Jf? = (Hx | A g A) be a family of
finite nonvoid sets. For each pair A, /x g A such that A < ^ let there be a
relation <pAu ^ Hx x Hu satisfying these three conditions:
(i) For every xeHx there is a y g Hu with O, y} g <pAtf, whenever
A < fJL.
00 <Paa is equality for all A g A.
(iii) If A < p < v9 then (x, z> g <pAv whenever there is a y with
<x, y} g <pAu and <>>, z> g <puv.
Then there is a family (xx \ xA g Hm A g A) such that <xA, xu} e <pXll if
A < fl.
proof. Consider Hx to be a discrete topological space. (A topological
space Sf is discrete if every subset of Sf is open.) Then HK is a compact
Hausdorff space, and therefore (by exercise 11.21) so is H = n (Hh | A g A).
For A,)it e A, A < fi, let S(A, ju) denote the set of all fe H satisfying
</(A),/G")>G?Atf. Note that S(A, /x) ^ 0 by (i). If /£ S(A, ^), then
{# I #W = /M» £(a0 = /(m)} is an °Pen set of H containing /, disjoint
from 5(A, p). Thus S(A, /x) is a nonvoid closed set. It follows from (iii) that
the S(A, )li) have the finite intersection property, and thus (exercise 11.11)

136
DISTRIBUTIVE LATTICES
there exists fe H belonging to all S(\, ^). Then xA = /(A) defines the
required family. 0
To prove Theorem 11 we need some further notation. If J is a V"
representation of a eL, we call |7| the rank of the representation and
2(|^| | X g 7) the length of the representation.
If H ^ L, then a V -representation of i/, 7(7f), is a family (7a | a e i/),
where 7a is a V-representation of a. If n is an integer and rank Ja = n for
each ae H9 then7(/7) is said to have rank n. A N/-rePresentati°n J(fl) °f
77 is said to be special if:
(i) aei7, Ar,re7a, and A ^^ A Y imply that X = Y.
(ii) a,6 e i7 and a < b imply that for each XeJa there is a YeJb such
that A X < A r.
Each H c L has a special V_rePresentati°n 5 by Theorem 9, 7a need
only be chosen as a normal V-representation of a for each ae£
proof of theorem 11. Let us assume that the conclusion of the theorem
fails to hold—that is, that there exists an uncountable chain C in L. Let
7(C) be a special V-rePresentati°n of C and let
Cn = {a | a e C, rank 7a = «}.
Then C = \J (Cn | 1 < n < a>), and thus some Cn is uncountable. In other
words, we can choose C so that rank Ja = n for all agC. We prove that
this is impossible by induction on n.
Let n = 1. For each integer k, let
C(fc) = {a\aeQ length 7a = fc}.
For each a e C<fc), let /(a) = {Jifa} and let
Ia = {i\iGliXar\Li* 0}.
Since any a,b e C(fc> are comparable and \Ia\ = \Ib\ = k9 we conclude by
Corollary 7 that 70 = 7b. Set Ik = 7a for all a e Cm. For each i e 7fc the set
77f = {x|xGlanI(,aeC('c)}
is a chain in L{ and is therefore countable. Since C(fc) is isomorphic to a
chain in n (H{ \ i e Ik) and Ik is finite, C(k) itself is countable. Because
C = (J (C(fc) | k < a>), C is countable.
Now let us assume that if C is a chain in L, and 7(C) is a special V-

Section 12 Free Distributive Product
137
representation of C such that rank Ja = k for all a e C, k < n9 then C is
countable. Let C be a chain in L and let 7(C) be a special V"rePresentati°n
of C such that rank Ja = n for all ae C. For a,b e C, a < b, we define a
relation <pab c ya x Jb: For Ig70, re/b, <X, 7> e ?ab iff A X £ A Y.
Then <pab satisfies (i) of Lemma 12 by the second clause in the definition of
special V-representations; (ii) and (iii) of Lemma 12 are trivially satisfied.
Thus, by Lemma 12, there is a family % = (Xa \ a e C, Xa e Ja9 Ja e 7(C))
such that /\ Xa < /\ Xb for a < b. Then % is a V_rePresentati°n of
rank 1 of a chain, and therefore 2£ is countable. Thus there are uncountably
many elements in C but only countably many Xa; consequently, there is an
uncountable subchain C0 of C such that Xa = Xb for a,b e C0. The family
# = {Ja — {Xa} | a e C0) is uncountable and of rank n — 1. # is clearly a
V-representation of some uncountable subset H ^ L, and the validity of
clause (i) of the definition of a special representation is clear. Now let
a,be C0, a < b, and let XeJa — {Xa}. Then there is a r£/b such that
A X < A Y. If y = JTb, then, since Xa = Xb,/\ X < /\ Xa,
contradicting the fact that 7(C) is special. Thus YeJb — {Xa} and so H is a chain
with a special V-rePresentati°n </• However, rank ,/ = n — 1 and i/ is
uncountable, contradicting the induction hypothesis. 0
Exercises
In the first paragraph of this section it was shown how Theorem 5.24 can be
used to show the existence of free distributive products. The same method,
however, does not apply to distributive lattices with 0 and 1. Nevertheless, the
idea of the proofs of Theorems 5.5 and 5.24 can be used to get a much stronger
result on the existence of free products; it is easiest to formulate this result
(G. Gratzer [1968], Theorem 29.2) for universal algebras. To do so we have to
introduce some concepts.
A type r of algebras is a sequence <«0, «i,..., nY,.. .> of non-negative
integers, y < o(r\ where o(j) is an ordinal called the order of t. An algebra 51
of type t is an ordered pair (A; F>, where A is a nonvoid set and F is a sequence
</o,.. .,/y,.. .>, y < 0(t), where f is an «y-ary operation on A. If o{j) is
finite, o(t) = h, then we write <^4;/0,. . .,/n-i> for 04; F>.
1. Define the concepts of subalgebra, polynomial, identity, and equational
class for algebras of a given type t. Show that if K is an equational class,
51 is an algebra in K, and S3 is a subalgebra of 31, then S3 is in K.
2. Define the concepts of homomorphism, homomorphic image, and direct
product for algebras of a given type. Show that an equational class is

138
DISTRIBUTIVE LATTICES
closed under the formation of homomorphic images and direct products.
3. Let 21 = 04; F> be an algebra, let H c A, and let H * 0. Show that
there exists a smallest subset [H] of A, [H] 2 H such that <[//"]; F} is a
subalgebra of %. (This subalgebra is said to be generated by H.)
4. Show that \[H}\ < \H\ + \F\ + K0.
5. Modify Definition 1 for algebras. Show that the y in (ii) is unique.
6. Let 93 and (£ be free K-products of %, i g /, with the embeddings e,,
/' g /, and Xf»' G A respectively. Show that there exists an isomorphism
«: B —> C such that e,« = xi and xt«-1 = e, for all / e /.
7. Let K be an equational class of algebras and let % e K for / e /. Choose a
set £ satisfying
|5| > l\At\ + \F\ + K0.
Let Q be the set of all pairs <93, (<pj | / g /)> such that 5 c 5, ^ is a
homomorphism of % into S3, and B — [U 04*9 I 'e /)]• Form 21 =
FI08 | 08, (9, | i e /)> g Q) (direct product), and for a <= At define
/fle^ by /a«S3, (<p, | / e /)» =" a<p,. Finally, let 9? be the subalgebra
generated by the fa, a g Au i g /. Show that 31 g K,a —>/0 is a homomorphism
c, of 21, into 9t for / g / and that 31 is generated by U (At€t I ' e /)•
8. Show that e, is one-to-one iff, for / g /, 0,6 g Au a ^ b, there exists an
algebra SeK and homomorphisms ip3: 21, --> (£ such that a^ ^ 6</»j.
9. Combine the previous exercises to prove the following result.
Existence Theorem for Free Products: Let K be an equational class
of algebras, % g K for i e I. A free K-product of%,ie I, exists iff, for
i e /, ajb e Al9 a ^ 6, ///ere exist aQ,e K, homomorphisms fa: %j —> © /or
;'g/such that a^x i=- b^.
10. Show that in proving the existence of free distributive products and free
{0, l}-distributive products, we can always choose (£ = (£2, the two-
element chain, in applying exercise 9.
11. Show that the two definitions of free K-product are equivalent for
any class K of lattices.
12. Show that the free Boolean algebra on m generators is a free {0, 1}-
distributive product of m copies of the free Boolean algebra on one
generator.
13. Prove that the free Boolean algebra on m generators can be represented
by the clopen subsets of {0, l}m where {0, 1} is the two-element discrete
topological space.
14. Find a topological representation for the free distributive lattice on
m generators (G. Ja. AreSkin [1953b]).
The remaining exercises in this Section are based on G. Gratzer and H.
Lakser [1969a].
15. Let L0 and Lx be bounded distributive lattices and let aQibQ e L0, aubi e
Li, and a0 || b0, #i II bx. Let L be a free {0, l}-distributive product of L0
and Li. Define x e L by
x = (a0 A tfi) V (b0 A &i).

Section 13 Some Categorical Concepts
139
Find the normal V-representation and also the normal A-representation
of x.
16. Let Li, i g /, be distributive lattices. For each / e /, let Lt be the result of
adjoining 0 and 1 to Lf. Let £ be a free {0, l}-distributive product of the
Lu ie I. Prove that L = L - {0, 1} is a distributive lattice and that it
is a free distributive product of the Li9 i e I.
17. A lattice L satisfies the m-chain condition if all chains in L have
cardinality < m. For an infinite cardinal m, and a cardinal n > 0, let us say
that the condition P(m, n) is satisfied if, whenever Li9 i e /, are
distributive lattices and the L, satisfy the m-chain condition, |/| = n, then a free
distributive product L of the Lu i e /, satisfies the m-chain condition.
Let P{0t i}(m, n) denote the same condition for bounded distributive
lattices and the free {0, l}-distributive product and let PB(m9 n) be the
same condition for Boolean algebras. Observe that Theorem 11 states
that P{0.i>(Ki,n) holds.
Generalize the proof of Theorem 11 to show that P{0, i}(m, n) holds
for any regular cardinal m > N0. (m is regular if mi < m for / e / and
|/| < m imply that 2 (m, | i e /) < m.)
18. Let m be an infinite cardinal not cofinal with cu (that is, m = 2 (Wi I ie I)
and |/1 = N0 imply that m = mf for some / e I). Prove that there exists a
Boolean lattice B such that B satisfies the m-chain condition but for all
n < m there exists in B a chain of cardinality n.
19. Let m be singular (that is, not regular) and not cofinal with w. Prove that
there exist Boolean lattices B0 and 2?i satisfying the m-chain condition
such that a free {0, l}-distributive product of B0 and B1 has a chain of
cardinality m.
20. Let m be cofinal with a>. Show that PB(m, N0) does not hold.
*21. Let m be cofinal with to and let L be a distributive lattice that has a chain
of cardinality m1 for all mx < m. Verify that L has a chain of cardinality
m (A. Hajnal).
22. Prove that P(m, n) holds iff either n > K0, m is regular, and m > K0,
or 1 < n < K0, and m is either regular or cofinal with <o.
23. Prove exercise 22 for P{0, i}(m, n).
24. Prove exercise 22 for PB(m, n).
25. Establish Theorem 11 without using the Axiom of Choice.
13. Some Categorical Concepts
The category of distributive lattices D is the class of all distributive lattices
together with the class of all lattice homomorphisms amongst them.
We use the word "category" to emphasize that not only the elements
(called objects) of D, but also the maps (called morphisms) that belong to the
category, are specified. To illustrate this idea, consider the category B

140
DISTRIBUTIVE LATTICES
a ^\ a
A0 ■—► Ai - A0 -* Ai
a mono: /fe = ya implies 0 = y a epi: a/? = ay implies /? = y
Figure 13.1
of Boolean algebras and the Boolean algebra homomorphisms. A related
category is BL, the elements of which are Boolean lattices and the maps of
which are lattice homomorphisms. We will also refer to the category
D{0,i} of bounded distributive lattices and {0, l}-homomorphisms. For
instance, the object (S2 belongs to D and D{0,i}; in D there are three maps
from (S2 to (S2; in D{0, i} there is only one.
Category theory is the study of those properties of algebras (and other
objects) that can be expressed in terms of maps. In this section we shall
discuss a few such concepts and investigate them in the categories D,
D{0, i}, and B.
For our purposes, in the subsequent definitions the clauses "let K be
a category" and "let K be a category of algebras" shall mean: "Let K be
one of D, D{0,i}, B." "An object of K" shall mean "a distributive lattice,"
"a distributive lattice with 0 and 1," and "a Boolean algebra,"
respectively. "A map" shall mean "a homomorphism" and "a {0, l}-homo-
morphism," respectively. Those who know the definition of a general
category or of a category of algebras will realize that the easy
observations of this section, but none of the more profound ones, apply to the
general case. Some categories of pseudocomplemented lattices will be
considered in Chapter 3.
In a general category, the morphisms need not be maps, and therefore
such concepts as one-to-one maps and onto maps are not meaningful.
Definitions 1 and 2 are obtained by abstracting certain properties of
one-to-one and onto maps, respectively.
Definition 1. A map a\ A0 -> A± in the category K is called mono, or a
monomorphism, iff, for objects B, C ofKandmaps j8: B->A0iy. C-> A0i
if pa = ya, then p = y (andB = C) (see Figure 13.1).
Definition 2. A map a:A1->A0 in the category K is called epi, or an

Section 13 Some Categorical Concepts
141
epimorphism, iff, for objects B, CofK and maps P: A0 -
if ap = ay, then p = y (and B = C) (see Figure 13.1).
B, y:AQ->C,
Lemma 3. Let Kbe a category of algebras, and let us assume the existence
of the free algebra F over K with one free generator x0. Then monomorphisms
in K are exactly the one-to-one maps in K.
proof. The reader can easily verify that a one-to-one map is a mono-
morphism. Now let a: A0-> A± be not one-to-one—that is, let aa = ba
for some a ^ b, a,b e A0. Let p and y be the homomorphisms of F into A0,
extending the maps x0 -> a and x0 -> b, respectively. Then /ta = ya but
p ^ y, so a is not a monomorphism. 0
Since each category of D, D{0, i}, and B satisfies the condition of Lemma
3, monomorphisms are one-to-one homomorphisms in each of these
categories.
Epimorphisms, however, behave in a very interesting way. Let 4^eD
and a:A0->A1 be given as in Figure 13.2. Obviously, a is a homo-
morphism and a is not onto. Now let A2 e D, P: Ax -> A2, and y: A1 -> A2
such that ap = ay. Then bp = a(aP) = a{ay) = by, similarly, Op = Oy and
\p = \y. Therefore, cp and cy both satisfy in A2 the equations bp A x =
Op and bp V x = IP; thus, by Corollary 7.3, cp = cy. This verifies that
P = y—in other words, that a is epi. The next result shows that this
example is typical.
For a lattice L and H ^ L, let [H]B denote the smallest sublattice of L
closed under relative complementation in L. Note that if L is a generalized
Boolean lattice, then [H u {0}]B = [H]R.
Theorem 4. In the categories D and D{0, i>, the map a: A0-> A1 is epi iff
[A0a]B = Ax.

142
DISTRIBUTIVE LATTICES
proof. Let p0: A1 -> C0, px: A1 -> Q. Then ap0 = «& means that p0
and fix agree on A0a; but then they agree on A2 = M0a]B. Therefore, if
A2 = Al9 then p0 = pl9 and a is epi. Conversely, if A2 =£ Al9 then adjoin
0, 1 (if there were none) and form the Boolean algebras Bx and B2
.R-generated by A± and A29 respectively. A± ^ A2 implies that Bx ^ B2 by
Lemma 10.9. Take anae^- B2. By Lemma 3.5 and Theorem 7.15,
we can take a homomorphism p of B2 into (£2 such that for x < a, xp =
0, and for x > a, xp = 1. By Theorem 8.5, p can be extended to
Pi'. [B2 u {a}]B -> (£2 such that arft = /, / = 0, 1. If we can extend the ft to
pt: B±-> (£2, then apo = apl9 p0 ^ pl9 showing that a is not epi. This will
be accomplished by the following lemma:
Lemma 5. Let Blf B2, B be Boolean algebras, let B2 be a subalgebra ofBu
and let B be complete. Let <p: B2-> B be a homomorphism. Then <p can be
extended to a homomorphism ip: B±->B.
proof. Let us form the partially ordered set Q whose elements are ordered
pairs <C, y>, where C is a subalgebra of Bl9 C ^ B29 y is a homomorphism,
y: C->* B, and y extends <p. Define <C0, y0> ^ <Ci, yi) to hold iff C0 is a
subalgebra of Cx and yi extends y0. Zorn's Lemma can be applied to Q,
thus yielding a maximal element <Af, 0>. Since Corollary 8.7 would
contradict M ^ 2?i, we must have M = B±. £
Theorem 4 yields
Corollary 6. /« the category B f/ie epimorphisms are exactly the onto
maps in B.
Besides selecting interesting maps, category theory also singles out
interesting objects.
+ B

Section 13 Some Categorical Concepts
143
Figure 13.4
Definition 7. Let Kbe a category of algebras. An object PofK is called
projective //, for any onto map y: A-^B, and any map fi:P-> B9 there is a
map a\P -> A with ay = j8 (see Figure 13.3).
Projective algebras generalize a property of free algebras. Note that the
categorical definition of projectivity requires y to be epi; this definition is
not interesting in D (see exercise 4).
Lemma 8. Let F be a free algebra in K. Then F is projective.
proof. Let xi9 i g I, be the free generators of F. Let at e (xfiyy'1 and let a
be the homomorphism extending xt -> ai9 iel. 0
Life would be much simpler if the converse of Lemma 6 were also true.
However, it does not hold in any of our categories.
Let us call a homomorphism p of A into A an endomorphism; let us
call an endomorphism p a retraction if p2 = p—in other words, if p is the
identity map on Ap. Ap is then called a retract of A.
Lemma 9. Let F be a free algebra in K and let G be a retract of F. Then G
is projective.
proof. The situation is shown in Figure 13.4. p is the retraction, / is the
inclusion map of G into F, y is a map of A onto B. Since pfi: F->B and F
is projective, there is a map a: F-> A with ay = pf$. Therefore, (ia)y =
/(ay) = i(pP) = (ip)f} = /?, since ip is the identity map on G. Thus
(ia)y = j8, and G is projective. 0
The converse is equally trivial. If G is projective, then if we take a free
algebra F with \G\ free generators xt, is I, we can find an onto map
{xt | / g 7} -> G that extends to a homomorphism y. Let £ be the identity

144
DISTRIBUTIVE LATTICES
map on G. Since G is projective, there is a homomorphism a: G -> F, such
that ay = identity on G. Therefore, G is isomorphic to a retract Ga of F,
and by stretching this concept a bit, we can say that G is a retract of F.
Lemma 10. If free algebras exist on any set of generators, then only
retracts of free algebras are projective.
It is easy to construct endomorphisms and retractions in distributive
lattices; thus Lemma 10 is only the beginning of the real work in
determining the projectives.
Theorem 11 (R. Balbes [1967]). A finite distributive lattice P is projective
(in D) iff the join of any two meet-irreducible elements is meet irreducible.
proof.8 Let P be projective. By Lemma 10 we can assume that P ^ F,
where F is free on xi9 i £ /, and p is a retraction: F^>P. Let a and b be
meet-irreducible and let a v b be meet-reducible—that is, for some c,d e P,
a V b > c A dy a V ft £ c, a V b £ d. Let c = \/ (/\Ck\ ke K) and
let d = V (A A I /G L)> where Ck and Z>z are finite subsets of {xt \ i £ /}.
Then c = cP = V (A CkP | k e#), J = </P = N/ (A A/> I /^L).
Therefore, A: and / exist such that A Ckp £ a V b and A A/3 i fl v ft. At the
same time, /\ Ck A /\ Dt < c A d < a V b; therefore, A Ck A A A ^
a or /\ Ck A /\ Dt < b by Theorem 8.3. Applying p we get A CkP A
A A/3 < a or A Cfc/3 A A A/3 ^ ft» which means that either a or ft is
meet-reducible, a contradiction.
Conversely, let P satisfy the condition of the theorem and let mu i £ /,
be the meet-irreducible elements of P. Let jFbe the free distributive lattice
on xi9 i e /, and let a be the homomorphism of F onto P extending xt -> mi9
i £ /. Let G be the join-subsemilattice of F generated by the xi9 i £ /. Now
we define a map <p: P^>Fby
<*9 = A (x | x £ G, xa > a) for a £ P.
In other words, a<p is the smallest element in the subset aa'1 of F.
Distributivity shows that (a V b)<p = a<p v b<p, since any x in G such
that xa > a V ft is of the form x = y v z, y e G9 ya > a, z e G9 za > b9
and conversely. To show that <p preserves meets, note that, by the assump-
8 This is a simplified version of R. Balbes [1967] taken from G. Gratzer and B.
Wolk [1970]. However, the crucial map <p is, by necessity, the same as that of Balbes.

Section 13 Some Categorical Concepts
145
P
Figure 13.5
tions of the theorem, xa is meet-irreducible for all x in G. Thus xa > a A b
is equivalent to the condition xa > a or xa > b. This proves that <p is a
homomorphism of P into F.
Finally, we have a = /\(mi\mi> a) for every element a of P.
Consequently, a<pa = a; therefore, P is a retract of Fand is thus projective by
Lemma 9. £
Not much is known about projectives in the category B.
Lemma 12. Every countable Boolean algebra is projective.
proof. Let B be a countable Boolean algebra. Then by Theorem 10.20,
B = B(C), where C is a countable chain with 0 and 1. Let C — {0, 1} =
{c0, Ci,..., cn,...} and let Fbe the free Boolean algebra on X0 generators,
x0i xl9..., xn9 We define a map />: C -> F. Let c0p = x0, 0/> = 0> and
1/)=1. Suppose that c0p, • • •, cn _ i/> have already been defined and choose
d,e e {0, 1, c0,..., cn_J such that in the sublattice {0, 1, c0,..., cn},
d -< cn -<: e. Set cn/> = (<//> v xn) A e/>. Let p be the extension of p to a
homomorphism: B-> F. Let 9? be the homomorphism of F onto B
satisfying xt<p = ct. Then p<p is the identity on B, and thus 5 is a retract
ofF. 0
Reversing the arrows in Definition 7 and replacing "onto" by "one-to-
one," we get another interesting concept:
Definition 13. Let K be a category of algebras. The object I is called
injective //*, for any one-to-one map y\B^ A and map /?: B -> /, there is a
map a: A -^ I with ya = £ (see Figure 13.5).
Intuitively, a homomorphism into / from a subalgebra can be extended
to the whole algebra. Injectives were first studied for modules; however,
they turn out to be very interesting in our categories. In fact, injective
objects in all three categories are the same:

146
DISTRIBUTIVE LATTICES
Theorem 14. In the categories D, D{0,i}, and B, the infective objects are
the same: They are the complete Boolean lattices {algebras).
The proof of this theorem is based on a categorical triviality and a
simple lattice theoretic lemma.
Lemma 15. Let I be injective. Then I is a retract of every extension ofl.
proof. Let / be a subalgebra of A, let B — I, and let jS and y be the
identity map of /. Then there exists an a: A -> /such that ya = jS, that is,
for x g /, xa = x. In other words, a is a retraction. 0
Lemma 16. Let A be a retract of a complete Boolean lattice B. Then A is a
complete Boolean lattice.
proof. Let a be the retraction, u = la, and v = Oa. Then 0 < x < 1 for
all xe A, and therefore v < xa = x < u\ thus u is the largest and v is the
smallest element of A. Now let x e A, x A y = 0, and x v y = 1 (y e B).
Then x A ya = v9 x v ya = w, and thus A is a Boolean lattice.
Now let X^ A and a = \/ X in B. Then x < a for all xel; thus
xa = x < aa. If, for y e A, x < y for all xe X, then a < y9 and therefore
aa < y, showing that V X = aa in A. Thus >4 is complete. 0
proof of theorem 14. Let / be injective in one of D, D{0, i>, B. Then / is a
distributive lattice, and thus (by Theorem 10.1) it has an extension B9
which is a complete (atomic) Boolean lattice. By Lemma 15, / is a retract
of B\ therefore, by Lemma 16, / is a complete Boolean lattice.
Conversely, if / is a complete Boolean lattice, then / is injective in B by
Lemma 5. Since (Corollary 10.7) every homomorphism <p:L0->L1 of
bounded distributive lattices can be extended to a homomorphism of the
Boolean lattices they generate, / is injective in D and D{0, i> also. 0
In the category S of sets and maps,
(i) every epimorphism has a left inverse that is mono, and
(ii) every monomorphism has a right inverse that is epi.
G. Birkhoff [1967] asks whether (i) or (ii) holds for B or Dfln (the category
of finite distributive lattices and lattice homomorphisms).
Using the results of this section we can easily answer this question. By
Corollary 6, (i) means that in B every homomorphic image can be regarded
as a retract, which is obviously not the case since a complete Boolean

Section 13 Some Categorical Concepts
147
a
A — ►C
y
Figure 13.6
algebra can have noncomplete homomorphic images (see exercise 17);
it can have only complete retracts (Lemma 16). Similarly, (ii) means that
every subalgebra is a retract, which is false for the same reason. Thus in B
neither (i) nor (ii) holds.
In Dfln let a: (£3 ->• ((£2)2 be one-to-one. Therefore, a is epi by Theorem 4.
If j8 is a left inverse of a and /? is mono, then /? is one-to-one from a four-
element set into a three-element set, a contradiction. Thus (i) fails in Dfin
and so does (ii): Any distributive lattice is a sublattice of a Boolean lattice,
but only the Boolean ones are retracts. Thus in Dfin also, neither (i) nor (ii)
holds.
Finally, we consider a categorical property not of maps or algebras but
of whole classes of algebras.
Definition 17 (B. Jonsson [1961] and [1965]). Let K be a class of
algebras. K is said to have the Amalgamation Property if whenever
A,B9C e K with A a subalgebra of B and of C, then there exists an algebra
D e K containing up to isomorphism B and C as subalgebras such that
BC\C^ A.
An equivalent definition is illustrated by Figure 13.6. If j3: A -> B and
y\ A -> C are one-to-one, then there exist D and one-to-one maps
$:B->D and c: C-> D in K such that £5 = yc.
Many important classes of algebras have this property and many others
do not. A simple condition equivalent to the Amalgamation Property is
given in the following result.
Theorem 18. Let K be an equational class of algebras. K has the
Amalgamation Property if whenever A,B9C e K with A a subalgebra of B and of C,
a,bgB9a ^ b9 then there exist an algebra DeK and homomorphisms
<p: B-+ Z>, 0: C-> D such that for every t e A, t<p = t\ft and a<p ^ kp
(see Figure 13.7).

148 DISTRIBUTIVE LATTICES
Figure 13.7
proof. The "only if" part is obvious. To prove the "if" part, again, all
we have to do is to copy the proof of Theorem 5.5; the set S has to be of
cardinality \B\ + \C\ + |F| + N0, and we sha11 take a11 Pairs <G, (A rK>
where GeK,G^ S, p: B-> G,y: C->G are homomorphisms agreeing
on A and where G is generated by Bp u Cy. The details are left to the
reader. %
Corollary 19 (R. S. Pierce [1968]). Let K be an equational class of
algebras in which every algebra can be embedded in an injective algebra.
Then K has the Amalgamation Property.
proof. Let A, B, and C be given as in Figure 13.7, a,b e B, and a ^ b.
Let <p be an embedding of B in an injective algebra D; then a<p ^ b<p. The
restriction ^ of <p to A can be extended to a homomorphism t/tofC into Z),
since D is injective. Thus the condition of Theorem 18 is satisfied. %
By Theorem 14, Theorem 7.19, and Corollary 7.21, the condition of
Corollary 19 holds in D, D{0,i}, and B. Therefore,
Corollary 20. D. D{0f 1}, and B satisfy the Amalgamation Property.
Compare this result with exercises 14 and 16.
To conclude this section, we give an interesting application of the
Amalgamation Property.
Theorem 21 (B. Jonsson [1961] and [1965]). Let K be a category of
algebras satisfying the Amalgamation Property, A9B,C e K with C a free K-

Section 13 Some Categorical Concepts 149
Figure 13.8
product of A and B. (We assume that A and B are subalgebras ofC.) Let A1
be a subalgebra of A, let B1 be a subalgebra ofB, and let C1 be the subalgebra
of C generated by A1 and B^ If a free K-product ofA1 and B1 exists, then C1
is a free K-product ofA1 and Bx (see Figure 13.8).
proof. Let C2 be a free K-product of A1 and Bx. Thus there exists a
homomorphism x of C2 into Cx that is the identity on Al and Bv Since Ax
is a subalgebra of A and of C2, by the Amalgamation Property there is an
algebra D containing A and C2 as subalgebras, A1 ^ A n C2. Similarly,
B1 is a subalgebra of D and B, and thus there exists an algebra E containing
B and D as subalgebras, 2?i ^ B n D (see Figure 13.9). Since C is a free
product of A and By there exists a homomorphism 9^ of C into E that is
the identity on A and 2?. Let 99 be the restriction of 9^ to Cx. Then 99 maps
Cx into C2, <px is the identity on A1 and 2?!, and <px is thus the identity on
Cx. Similarly, x9 is the identity on C2, so 9? is an isomorphism between Q
and C2. 0
Figure 13.9

150
DISTRIBUTIVE LATTICES
Exercises
1. Let K be the category of one- and two-element Boolean algebras, with all
homomorphisms as morphisms. Which maps are epi and which are
mono?
2. Prove that if a and )3 are epi (mono), then so is «)3.
3. An isomorphism is both epi and mono. Is the converse also true?
4. In Definition 7, change "onto" to "epi." Show that in D no lattice
satisfies this definition.
5. Show that a direct product of injective algebras is injective.
6. Show that a free product of projective algebras is projective.
7. Is a retract of an injective algebra injective?
8. Is a retract of a projective algebra projective?
9. Let K be a category of algebras, AyA0yAi g K. Let the algebra A be the
direct product of A0 and Ax. Assume that for any X,YeK there is at least
one homomorphism <p: X —> Y. Show that A is injective iff A0 and Ai
are injective.
10. Let A, B, and C be lattices, let A be a sublattice of B and of C, and let
B n C = A. Define a partial order on B u C as follows: For x,y e B
(for x,ye C), x < y iff x < y in B (in C); for x e B, y g C (and for
x g C, ^ g 5), x < >> iff for some z g A, x < z in B (in C) and z < >> in
C (in B). Show that 5uCisa partial lattice.
11. Prove that the Amalgamation Property holds for the category L of all
lattices.
12. Let K be an equational class of algebras. The Strong Amalgamation
Property holds in K if, whenever A,B,C e K, and A is a subalgebra of B
and of C, there exists an algebra D in K containing B and C as sub-
algebras (up to isomorphism) such that B n C = A. Show that the
Strong Amalgamation Property holds in K iff, in addition to the
condition of Theorem 18, the following condition holds:
If a e B — A and 6eC- A, then there exists an algebra D eK and
homomorphisms <p\ B—> D ip: C-> D such that for all t g A, t<p = t*l*
and acp ^ &</».
13. Show that the Strong Amalgamation Property holds for the class of all
lattices.
14. Show that the Strong Amalgamation Property holds for B.
15. Let A, B, and C be bounded distributive lattices, let A be a {0, 1}-
sublattice of B and of C, and let B n C = A. Show that there exists a
bounded distributive lattice Z) containing B and C as {0, l}-sublattices
such that B n C = /4 iff A is closed under relative complementation in
5 and C.
16. Prove that the Strong Amalgamation Property fails in D{0,i} and D.
17. Let B be the Boolean algebra of all subsets of an infinite set I. Show that,
although B is complete, B/J is not complete if / is the ideal of all finite
subsets of I.
18. Show that Theorem 18 holds for any class closed under direct products.

Further Topics and References
151
19. Let K be a category of algebras. Show that if the Strong Amalgamation
Property holds for K, then every epi is onto.
20. Is the converse of exercise 19 true ?
Further Topics and References
It seems very hard to generalize the theorem on the uniqueness of an
irredundant representation of an element of a finite distributive lattice
(Corollary 7.13). The best generalization in this direction is that of R. P.
Dilworth and P. Crawley [1960] to distributive algebraic lattices in which
every interval has an atom. See the survey article by R. P. Dilworth [1961];
see also S. Kinugawa and J. Hashimoto [1966].
For special classes of lattices, Theorem 7.1 has various stronger forms
that claim the existence of copies of 9?5 and/or 9tt5 that are very large or
very small. For instance, in a bounded relatively complemented non-
modular lattice, 9*5 can be chosen so that the o and / of 9?5 are the 0 and 1
of the lattice. The same is true of 9tt5 in certain complemented modular
lattices (such results follow from J. von Neumann [1936]). If the lattice is
finite and modular, then 9tt5 can be chosen so that a, b, and c cover o and i
covers a, b, and c. If L is finite and nonmodular, then -ft5 can be minimized
by requiring that a >- b.
Using the terminology of Section 16, Theorem 7.19 states that every
distributive lattice is a subdirect product of copies of (£2. Distributive
lattices that are subdirect products of copies of (£n are described in F. W.
Anderson and R. L. Blair [1961].
A more general form of Theorem 8.5 can be found in R. Sikorski [1964];
all such theorems can be derived from a universal algebraic triviality;
see, for example, Theorem 12.2 in G. Gratzer [1968].
Let L be a lattice and let/be an w-ary function on L, that is,/: Ln ->• L.
We say that / has the Congruence Substitution Property if, for every
congruence relation 0 in L and ai9bi eL, 1 < i < n, at = &*(©), 1 < i < n,
imply that f(ai9..., an) = f(bl9..., 6n)(0). On a Boolean algebra B, sl
function has the Congruence Substitution Property iff it is a Boolean
algebraic function, that is, a Boolean polynomial in which elements of B
are substituted for some variables. Functions satisfying the Congruence
Substitution Property on a bounded distributive lattice were described in
G. Gratzer [1964].
Many properties of ideals of a distributive lattice can be generalized to

152
DISTRIBUTIVE LATTICES
certain ideals of a general lattice. One such concept is that of standard ideal
(G. Gratzer [1959], G. Gratzer and E. T. Schmidt [1961]).
The method given in Theorem 9.9 is not the only one used to introduce
ring operations in a generalized Boolean lattice. G. Gratzer and E. T.
Schmidt [1958e] proved that ring operations -f, • can be introduced on
a distributive lattice L such that -f and • satisfy the Congruence
Substitution Property iff L is relatively complemented. Furthermore, +
and are uniquely determined by the zero of the ring, which can be an
arbitrary element of L.
Whether every distributive algebraic lattice is isomorphic to the
congruence lattice of some lattice is one of the longest-standing problems of
lattice theory. The method used in Section 9 for the finite case can be
easily extended to infinite algebraic lattices in which every element is a
finite join of join-irreducible elements. A further extension of this result
is that of E. T. Schmidt [1962] and [1968]. (In reading the two papers, the
reader should disregard the Theorem and Lemmas 9 and 10 of the first
paper.)
Congruence lattices of distributive lattices were not considered in this
text because the problem is trivial by Lemma 10.5: A lattice is isomorphic
to the congruence lattice of a distributive lattice iff it is isomorphic to 1(B),
where B is a generalized Boolean lattice; 1(B), by Theorem 9.13, is
characterized as a distributive algebraic lattice in which the compact
elements form a relatively complemented sublattice.
Algebraic lattices originated in A. Komatu [1943], L. Nachbin [1949],
and J. R. Buchi [1952]. The original definition was presented as follows:
(i) L is complete; (ii) in L every element is the join of join-inaccessible
elements; (iii) L is join-continuous. An element a of L is join-accessible if
there is a nonvoid subset HofL that is directed (for x,y e H there exists an
upper bound z e H), if V H = a, and if a $ H; otherwise, a is join-
inaccessible. L is join-continuous if, for any aeL and directed H ^ L, we
have a /\\/ H =\J (a h h\heH).
Interestingly, it is sufficient to formulate conditions (i) and (iii) of the
previous paragraph for chains only. In other words, a lattice L is complete
iff A C and V C exist for any chain C of L; and a (complete) lattice L is
join-continuous iff a A V C = V (a A c | c e C) for any chain C of L.
These statements are immediate consequences of the following result of
T. Iwamura [1944]. Let H be an infinite directed set. Then H has a
decomposition H = (J (Hy | y < a), where each Hy is directed; for y < 8 <
a we have HY c H6i and for y < a we have \HV\ < H. Extensions of

Further Topics and References
153
lattices preserving (iii) were considered in D. A. Kappos and F.
Papangelou [1966].
The problem of completions of a lattice has been extensively studied.
The standard method is the MacNeille completion (see exercises 10.9-
10.14, which are from H. M. MacNeille [1937]). This method is described in
detail in G. Bruns [1962a]; see also Y. Sampei [1953]. One of the
important shortcomings of this method is that it does not preserve identities,
nor even distributivity; see M. Cotlar [1944] and N. Funayama [1944].
What is preserved for complemented modular lattices is examined in
J. E. McLaughlin [1961].
If we define a completion L of a lattice L as any lattice L containing L
as a sublattice such that all infinite meets and joins that exist in L are
preserved in L, we can ask whether there is any distributive completion of
a distributive lattice. The answer, in general, is in the negative; see P.
Crawley [1962].
For Boolean algebras, set representations preserving all joins and meets,
or joins and meets of certain types, play an important role, especially in
relationship with infinite distributive identities. The larger part of the book
by R. Sikorski [1964] is devoted to this problem. For distributive lattices
the questions become even more complicated—see, for instance, G. Bruns
[1959], [1961], and [1962b]; C. C. Chang and A. Horn [1962]; A. Horn
[1962]; and G. N. Raney [1952], [1953].
Cardinalities of maximal chains in Boolean algebras were'considered by
J. Jakubik [1957] and [1958]; see also exercise 10.28.
Utilizing the sequence Iyi B/Iy described in Theorem 10.24, R. S. Pierce
[a] gave the first deep analysis of the structure of countable Boolean
algebras. Pierce solved many known problems, among others the ones
raised in P. R. Halmos [1963].
The investigations of Section 12 originated with the well-known theorem
that the free Boolean algebras satisfy the Countable Chain Condition (see
I. ReznikofT [1963] and A. Horn [1968]. In fact, this statement is true
whenever a lattice structure is involved—see the beautiful proof in F.
Galvin and B. Jonsson [1961].) The question arose whether this is so
because the free Boolean algebras are free products of copies of the free
Boolean algebra on one generator, which is finite and thus satisfies the
Countable Chain Condition.
R. Balbes [1967] found another interesting property of free distributive
lattices: Every subset S with the property a A b = c A d for all a9b9c,d e S,
a ^ b, c ^ d is finite, and for free Boolean algebras every such set is

154
DISTRIBUTIVE LATTICES
finite or countable. H. Lakser proved that if, for each such set S, we have
|5| < N0 in each lattice, then \S\ < X0 in a free {0, l}-distributive product.
This result has opened up a whole new field for investigation.
The development of the Amalgamation Property is described in B.
Jonsson [1965]; probably B. Jonsson's work more than any other influence
convinced the algebraists of the importance of this property. The
Amalgamation Property for posets and lattices was noted by B. Jonsson [1956];
it was noted for Boolean algebras by A. Daigneault [1959].
As the reader may already have noticed, the conclusion of Theorem
13.21 can be based either on the Amalgamation Property, as suggested by
B. Jonsson, or on the solution to the word problem for free products. (The
solution to the word problem for distributive lattices is given in Corollary
12.6; this general method also works for lattices—see G. Gratzer,
H. Lakser, and C. R. Piatt [1970].) Neither of the two methods includes the
other. For groups the first one works but not the second, whereas for
semigroups it is the other way around.
There have been some further results on projective algebras. The theorem
of R. Balbes describing finite projective distributive lattices has been
extended to the countable case by R. Balbes and A. Horn [1970c]. R.
Engelkind [1965] has shown that a subalgebra of a free Boolean algebra
need not be projective.
The problem of how to prove the distributivity of a sublattice by
imposing relations on the generators is discussed in many papers. For
general lattices and three-generated sublattices, this problem was solved by
O. Ore [1940]. It was observed that when the lattice is modular, the
elements a, b, and c generate a distributive sublattice iff a A (b V c) =
(a A b) v (a A c) (as is evident from Figure 5.7). Far-reaching
generalizations of this result can be found in B. Jonsson [1955]; see also R. Musti and
E. Buttafuoco [1956] and R. Balbes [1969].
The interdependence of the various "finiteness conditions" for
distributive lattices was investigated in S. P. Avann [1964].
Boolean algebras with only the trivial automorphism were constructed
by M. Katetov [1951], B. Jonsson [1951], and S. Rieger [1951]; Katetov's
example has 2Ro elements; the others are very large.
The method used in establishing Lemma 10.22 was employed by B.
Jonsson [1956] to construct "universal" lattices, that is, lattices containing
a large class of lattices as sublattices (see also B. Jonsson [1960] and M. D.
Morley and R. L. Vaught [1962]).
For background material in categories see B. Mitchell [1965].

Problems
155
R. Dedekind found the distributive identity by investigating ideals of
number fields. Rings with a distributive lattice of ideals have been
investigated by E. Noether [1927], L. Fuchs [1949] (who named such rings
arithmetical rings), I. S. Cohen [1950], and Ch. U. Jensen [1963]. Equa-
tional classes of rings with distributive ideal lattices were considered in G.
Michler and R. Wille [1970] and in H. Werner and R. Wille [1970]. E. A.
Behrens [1960] and [1961] considered rings in which one-sided ideals
form a distributive lattice. Rings with a distributive lattice of subrings were
classified in P. A. Freidman [1967].
For groups the corresponding problem is much simpler: The subgroup
lattice of a group G is distributive iff G is locally cyclic (see O. Ore [1937]
and [1938]).
The distributivity of congruence lattices of lattices implies a number of
important consequences. B. Jonsson [1967] discovered that many of these
results hold for arbitrary universal algebras with distributive congruence
lattices. His results have found applications that go far beyond lattice
theory—they have already been applied to lattice-ordered algebras,
closure algebras, nonassociative lattices, cylindric algebras, monadic
algebras, lattices with pseudocomplementation, and primal algebras.
The foregoing examples show the central role played by distributive
lattices in applications of the lattice concept.
PROBLEMS
28. Find short one-identity axioms characterizing Boolean algebras.
29. Is there a self-dual minimal set of identities defining Boolean algebras ?
30. Characterize the automorphism groups of Boolean algebras.9
31. For a Boolean algebra B and cardinal number m > |2?|, can a Boolean
algebra C be found such that | C | = m and the automorphism groups of
B and C are isomorphic ?10
32. Does the endomorphism semigroup determine a Boolean algebra up to
isomorphism?11
9 B. J6nsson has proved (unpublished result) that the finite automorphism groups
of Boolean algebras are exactly the finite symmetric groups.
10 B. J6nsson suggests starting the investigation of this problem with the one-
element automorphism group.
11 This problem was solved in the affirmative by B. J6nsson.

156
DISTRIBUTIVE LATTICES
33. Characterize the poset &*(L) of prime ideals of a distributive lattice.
34. Characterize ^(L) under the additional assumption that L has a 0,
a 1, or both. If L has 1, then every chain in ^(L) has a supremum; if L has
0, then every chain in 0*(L) has an infimum. Are these the only additional
conditions ?
35. For which classes K of finite lattices can we claim that if L e K and L is
nonmodular, then L contains a " minimal" 9£5 ?
36. Characterize the congruence lattices of lattices. (Conjecture: distributive
algebraic lattices).
37. Characterize the congruence lattices of sectionally complemented lattices.
(Conjecture: distributive algebraic lattices).
38. Determine /D(&, n).
39. Is a distributive lattice, in which every ideal is the intersection of maximal
ideals, and dually, necessarily relatively complemented (see A. Monteiro
[1947])?
40. Investigate the cardinalities of maximal chains in complete Boolean
algebras.
41. What can be proved about the cardinalities of maximal chains in a
complete and atomic Boolean algebra without the Generalized Continuum
Hypothesis ?
42. Compare the number and form of the terms in a normal V-representa-
tion and normal A-representation of an element in a free {0, l}-distribu-
tive product.
43. For a bounded distributive lattice L, let F(L) and Fn(L) denote the lattice
of all functions and all /i-ary functions on L, respectively, with the
Congruence Substitution Property. To what extent do F(L) and Fn(L)
determine the structure of L ?
44. Characterize F(L) and Fn(L).
45. Describe the functions with the Congruence Substitution Property on
unbounded distributive lattices.
46. Study problems 43 and 44 in the unbounded case.
47. For a lattice L, let w(L) denote the smallest cardinal number such that
\S\ < w(L) whenever S satisfies the following property: a,&,c,*/e,S, a i^ b,
and c ^ d imply that a A b = c A d. For cardinals m and n (n > 1),
let C(m, n) denote the condition that whenever L,, i'e/, are
distributive lattices and L is the free distributive product, then w(Li) < m
for all i'e/ and |/| < n imply that w(L) < m. Describe the relation
G0n, n).
48. For cardinals m, n (n > 1), define R(m, n) to hold if the free product of
n lattices each satisfying the m chain condition satisfies the m chain
condition. Describe R(m, n). (R(Hl9 n) should always hold. Substitute
G. Gratzer, H. Lakser, and C. R. Piatt [1970] for Theorem 12.5).
49. Investigate R(m, n) for other classes of lattices (modular lattices, for
example).
50. Is it possible to define formally the free {0, l}-distributive product via the
description given in Theorems 12.5 and 12.9?

Problems
157
51. Does the Amalgamation Property hold for the class of all modular
lattices?12
52. Are there 2**o equational classes of lattices for which the Amalgamation
Property holds?
In January 1971, B. J6nsson announced a negative solution to this problem.

CHAPTER
DISTRIBUTIVE LATTICES WITH
PSEUDOCOMPLEMENTATION
14. Introduction and Stone Algebras
In this chapter we shall deal exclusively with pseudocomplemented
distributive lattices. There are two concepts that we should be able to
distinguish: a lattice, <L; a, V>, in which every element has a pseudo-
complement, and an algebra, <L; A, V, *, 0, 1>, where <L; A, V, 0, 1>
is a bounded lattice and where, for every aeL, the element a* is a pseudo-
complement of a. We shall call the former a pseudocomplemented lattice
and the latter a lattice with pseudocomplementation (as an operation)—the
same kind of distinction that we make between Boolean lattices and
Boolean algebras. Thus, in the sense of the exercises following Section 12,
a pseudocomplemented lattice is an algebra of type <2, 2>, whereas a lattice
with pseudocomplementation is an algebra of type <2, 2, 1,0, 0>. To see
the difference in viewpoint, consider the finite distributive lattice of Figure
14.1. As a distributive lattice it has twenty-five sublattices and eight
congruences; as a lattice with pseudocomplementation it has three sub-
algebras and five congruences.

160
DISTRIBUTIVE LATTICES WITH PSEUDOCOMPLEMENTATION
o
Figure 14.1
Thus, for a lattice with pseudocomplementation L, a subalgebra Lx is
a {0, l}-sublattice of L closed under * (that is, a eLx implies that a* el^).
A homomorphism <p is a {0, l}-homomorphism that also satisfies (x<p)* =
x*<p. If there is any danger of confusion, we call such a homomorphism a
*-homomorphism. Similarly, a congruence relation 0 will have the
Substitution Property also for *: a = b(Q) implies that a* = 6*(0).
A wide class of examples is provided by
Theorem 1. Any complete lattice that satisfies the Join Infinite Distributive
Identity (JID) is a pseudocomplemented distributive lattice.
proof. Let L be such a lattice. For aeL, set
a* = V (* I x e L, a A x = 0).
Then, by (JID),
aAa* = aA\/(x\aAx = 0) = \/(aAx\aAx = 0) = \/0 = 0.
Furthermore, if a A x = 0, then x < a* by the definition of a*; thus a* is
indeed the pseudocomplement of a. %
Corollary 2. Every distributive algebraic lattice is pseudocomplemented.
proof. Let L be a distributive algebraic lattice. By Theorem 9.13 and
Lemma 11.1 we can assume that L = I(S), where S is a distributive join-
semilattice with 0. Let IJjeI{S) for jej. Then / A Ik S / A V
(7, |./e/) for any A; e /, and thus
V(/A/,|/e/)£/A V(/y|/e/).

Section 14 Introduction and Stone Algebras
161
To prove the reverse inclusion, let ael A V (//17 e «0» that is, a el,
a e V CO 17 eA The latter implies that a < tx V • • • V tn, where tx e
Ih,...,tneIJnJl9...JneJ. Thus flG/yi v • • • V 7,n and so, using the
distributivity of L, we obtain
ael a (Ih v ••• v/J = (/A/*) v ••• v (/A 7,n) £
\Z(lAl,\jeJ),
completing the proof of the (JID). The statement now follows from
Theorem 1. £
Thus, the lattice of all congruence relations of an arbitrary lattice and
the lattice of all ideals of a distributive (semi-) lattice with zero are examples
of pseudocomplemented distributive lattices. Note that for IeI(K),
I* = {x\xeK9x A i = Oforall/e/}.
Also, any finite distributive lattice is pseudocomplemented. Therefore, our
investigations include all finite distributive lattices.
A model for distributive lattices with pseudocomplementation is the
class of Boolean algebras, and the purpose of much of the research is to
see how far they deviate from Boolean algebras. This purpose will be stated
more precisely in the following paragraphs.
The first class of distributive lattices with pseudocomplementation,
other than the class of Boolean algebras, to be examined in detail was the
class of Stone algebras. A distributive lattice with pseudocomplementation
L is called a Stone algebra iff it satisfies the Stone identity:
a* v <*** = 1.
The corresponding pseudocomplemented lattice is called a Stone lattice.
To understand the meaning of this identity, define the skeleton of L:
S(L) = {a*\aeL}.
The elements of S(L) are called skeletal L is dense if S(L) = {0, 1}. By
Theorem 6.4, (S(L); A, v, *, 0, 1> is a Boolean algebra. For a Stone
algebra L, S(L) is a subalgebra of L:
Theorem 3. For a distributive lattice with pseudocomplementation L, the
following conditions are equivalent:
(i) L is a Stone algebra.

162 DISTRIBUTIVE LATTICES WITH PSEUDOCOMPLEMENTATION
(ii) (a A b)* = a* V b* for a,b e L.
(iii) a,b e S(L) implies that a V be 5(L).
(iv) S{L) is a subalgebra ofL.
proof. The proofs that (ii) implies (iii), that (iii) implies (iv), and that (iv)
implies (i) are trivial. Now let L be a Stone algebra; we show that a* v b*
is the pseudocomplement of a A b. Indeed, (a A b) A (a* V b*) =
(a A b A a*) V (a A b A b*) = 0 V 0 = 0. If (a A b) A x = 0, then
(h x) A a = 0, and so 6 A x < a*. Meeting both sides by a** yields
b A x A a** < a* A a** = 0; that is, x A a** A b = 0, implying that
x A a** < b*. By the Stone Identity, a* V a** = 1, and thus x = x A 1
= X A (a* V a**) = (x A a*) V (x A a**) < a* V 6*. •
This is already enough to yield the structure theorem for finite Stone
algebras (G. Gratzer and E. T. Schmidt [1957b]):
Corollary 4. A finite distributive lattice is a Stone lattice iff it is the
direct product of finite distributive dense lattices, that is, finite distributive
lattices with only one atom.
PROOF. By Theorem 3, a Stone lattice L has a complemented element
a $ {0, 1} iff S(L) t£ {0, 1}; thus the decomposition of Theorem 7.6 can be
repeated until each factor Lt satisfies S(Li) = {0, 1}. In a direct product, *
is formed componentwise; therefore, all the L( are Stone lattices. For a
finite lattice K with S(K) = {0, 1}, the condition that K has one atom is
equivalent to K being a Stone lattice. %
Another significant subset of a Stone algebra is the dense set
D(L) = {a | a* = 0}.
The elements of D(L) are called dense.
We can easily check that D(L) is a dual ideal of L and that 1 e D(L);
thus D(L) is a distributive lattice with 1. Since a V a* e D(L) for every
flel,we can interpret the identity
a = a** A (a V a*)
to mean that every aEL can be represented in the form a = b A c,
where b e S(L), c e D{L). Such an interpretation correctly suggests that if
we know S(L) and D{L) and the relationships between elements of S(L)

Section 14 Introduction and Stone Algebras
Figure 14.2
and D(L), then we can describe L. The relationship is expressed by the
homomorphism <p(L): S(L) ->^(D(L)) defined by
9<L): a -> {x | x e D(L), x > 0*}.
Theorem 5 (C. C. Chen and G. Gratzer [1969b]). Let L be a Stone
algebra. Then S(L) is a Boolean algebra, D(L) is a distributive lattice with 1,
and (p(L) is a {0, \}-homomorphism of S(L) into @(D(L)). The triple
<5(L), D(L), <p(L)> characterizes L up to isomorphism.
proof. The first statement is easily verified. For a e S(L), set
Fa = {x I x** = a}.
The sets {Fa \ a e S(L)} form a partition of L; for a simple example, see
Figure 14.2. Obviously, F0 = {0} and Fx = D(L). The map x -> x V a*
sends Fa into Fx = D(L); in fact, the map is an isomorphism between Fa
and a<p(L) ^ D(L). Thus xsFa is completely determined by a and
x V a* g a<p(L)—that is, by a pair (a, z>, where a e 5(L), z e a<p(L)—and
every such pair determines one and only one element of L. To complete our
proof we have to show how the partial ordering on L can be determined by
such pairs.

164 DISTRIBUTIVE LATTICES WITH PSEUDOCOMPLEMENTATION
Let xGFa and y g Fb. Then x < y implies that *** < y**, that is,
a < b. Since x < y iff
a v x < a V y and x v a* < y V a*
and since the first of these two conditions is trivial, we obtain:
x < y iff a < b and x V a* < y V a*.
Identifying x with <x v a*, a> and 7 with (y v 6*, 6>, we see that the
preceding conditions are stated in terms of the components of the ordered
pairs, except that y V a* will have to be expressed by the triple.
Because <p(L) is a {0, l}-homomorphism and a** is the complement of a*,
we conclude that a**<p(L) and a*<p(L) are complementary dual ideals of
D(L). Therefore, by Theorem 7.6, for any z g D(L), [z) is the direct
product of [z V a*) and [z V a**). Thus, every z can be written in a
unique fashion in the form z = z(a*) A z(a**), where z{a*) g a<p(L)
and z(a**) g a*<p(L). Let ypa denote the element (y<p(L))(a*) and observe
that pa is expressed in terms of the triple. Finally, y V a* = y v 6* v a*
= (}><p(L)) V a* = y/)a. Thus for m g a<p(L) and 1; e £<p(L), we have
<w, fl> < <*>, b} iffa < 6 and u < vpa. %
This theorem shows that for Stone algebras, the behavior of the
skeleton and the dense set is decisive. This conclusion leads us to
formulate the goal of research for Stone algebras and for all distributive
lattices with pseudocomplementation as follows:
A problem for distributive lattices with pseudocomplementation is
considered solved if it can be reduced to two problems: one for Boolean
algebras and one for distributive lattices with 1. We shall see examples in
which this program works and others in which it does not.
Exercises
1. Show that every bounded chain is a pseudocomplemented distributive
lattice.
2. Let L be a lattice with 1. Adjoin a new zero 0 to L: Lx = L u {0}, 0 < x
for all xgL. Show that L± is a pseudocomplemented lattice.
3. Call a lattice with 0 dense if 0 is meet-irreducible. Show that every bounded
dense lattice K is pseudocomplemented and that every such lattice can be
constructed by the method of exercise 2 with L = D(K).

Section 14 Introduction and Stone Algebras
165
4. Find an example of a complete distributive lattice L that is not pseudo-
complemented.
5. Prove that if L is a complete Stone lattice, then so is /(L). (Hint: /* =
(«], where a = A (x* \ x e /).)
6. Show that a distributive pseudocomplemented lattice is a Stone lattice
iff (a v b)** = a** v b** for a9b gL.
7. Find a small set of identities characterizing Stone algebras.
8. Let L be a Stone algebra. Show that S(L) is a retract of L.
9. Let L be a Stone algebra, a,6 e SXL), and a < b. Prove that
x -> {x V a*) A 6 embeds Fa into Fb.
10. Let B be a Boolean lattice. Define B™ c 52 by <fl> £> G #2] iff a < ^
Verify that 2?[2] is a sublattice of B2 but not a subalgebra of B2. Show
that B™ is a Stone lattice.
Exercises 11-23 are from C. C. Chen and G. Gratzer [1969b]. Exercises 24
and 25 are from C. C. Chen and G. Gratzer [1969c].
Let B be a Boolean algebra, let D be a distributive lattice with 1, and let <p be
a {0, l}-homomorphism of B into ^(D). Set L = {<*, a> | a e B9 x e a<p} and
define <x, a> < O, &> iff a < b and x < ypa, where [^pa) = a? A [y).
11. Verify the following formulas:
(i) If a e B and </ e £>, then </pa = d iff d = a<p.
(ii) </pa >^/forfle5,i/ei).
(iii) dpa A </pa' = d for aeB,de D (where a' is the complement of a
\nB).
(iv) PaPb = PaAbfora^eC.
12. Prove that:
(i) dpa A dpb = i/povb for a9b e B9de D.
(ii) dpaSb = </pa v dpb for a,6 e B,de D.
13. Show that L is a poset under the given partial ordering.
14. Let <x, a>, <>>, &> gL. Verify that <x, a> A (y, b> = (xpb A ypay a A &>.
♦15. Show that <x, a> v (y9 b} = <(xpb> A y) v (x A ypa>\a v &>.
16. Let <x, a>, <>>, &>, and <z, c> el,;
^ = «x, a> A <^, £>» v <z, c>,
and
,8 = «x, a> V <z, c» A «>>, £>> V <z, c».
Compute ,4; show that B = <</, (a v c) A (6 V c)>, where d = d0 V
dx v d2 V d3i and </0 = *p&ac- A ypaAc A zid1 = xpbAC' A J7>aAC A
z, </2 = xpbvc A ^PaAC' A z, and d3 = xpbvc A ^pbvc A zpa'vb'.
17. Show that </0 ^ ^1 and d0 > d2; therefore, d = d0 v d3.
18. Show that L is distributive.
19. Show that L is a Stone lattice.
20. Identify aeB with <1, &> and deD with <</, 1>. Verify that then
S(L) = 5, D(L) = £>, and <p(£) = <p. In other words, we have proved the
Construction Theorem of C. C. Chen and G. Gratzer [1969b]: Given a

166 DISTRIBUTIVE LATTICES WITH PSEUDOCOMPLEMENTATION
Boolean algebra B9 a distributive lattice D with 1, and a {0, \}~homo-
morphism <p:B—>@(D), there exists a Stone algebra L whose triple is
<B, D, 9>>.
Describe isomorphisms and homomorphisms of Stone algebras in terms
of triples.
Describe subalgebras of Stone algebras in terms of triples.
For a given Boolean algebra B with more than one element and
distributive lattice D with 1, construct a Stone algebra with S(L) ^ B9
D(L) ^ D. (S(L) and D(L) are independent.)
Show that a Stone algebra L is complete if S(L) and D(L) are complete.
Characterize the completeness of Stone algebras in terms of triples.
Let L be a distributive lattice. Prove that L is a Stone lattice iff a unary
operation * can be defined on L that satisfies the following implication:
aAb<c<aVb implies that c A b* < a < c** V b*
(K. M. Koh).
15. Identities and Congruences
Since one of our objectives in this chapter is to learn how to compute with
*, we might as well start by collecting some important rules of the
arithmetic of the operations *, A , V .
Theorem 1. Let L be a pseudocomplemented distributive lattice, S(L) =
{a* \aeL}, and D(L) = {a \ a* = 0}. Then for a,beL:
(i) a A a* = 0.
(ii) a < b implies that a* > b*.
(iii) a < a**.
(iv) a* = a***.
(v) (a V b)* = a* A b*.
(vi) (a A 6)** = a** A 6**.
(vii) a A b = 0 iff a** A b** = 0.
(viii) a A (a A b)* = a A b*.
(ix) 0* = 1 andl* = 0.
(x) aeS(L)iffa = a**.
(xi) a9b g S(L) implies that a A be S(L).
(xii) supsa) {a, b} = (av 6)** = (a* A 6*)*.
(xiii) 0,1 g S(L), 1 g D(L\ and S(L) n D(L) = {1}.
(xiv) ajb g D(L) implies that a A be D(L).
(xv) a g D(L) and a < b imply that b e D(L).

Section 15 Identities and Congruences
167
(xvi) a V a*e D(L).
(xvii) x -> *** is a meet-homomorphism ofL onto S(L).
proof, (i), (ii), (iii), and (iv) were proved in Section 6. To show (v),
observe that {a V b) A (a* A b*) = (a A a* A b*) V (b A a* A 6*) =
0 v 0 = 0. Now (a V b) A x = 0 implies that (a A x) V (jb A x) = 0;
therefore, a A x = 0 and 6 A x == 0, and so x < a*,x < b*9 thus
x < a* A b*. Note that x** is the smallest element of S(L) containing x.
By Formula (5) of Section 6, a** A b** e 5(L), and it is obviously the
smallest element of S(L) containing a A b9 hence (vi); (vii) is an immediate
consequence of (vi) and (iii); (viii)-(xv) and (xvii) are either trivial or
known from Section 6; (xvi) follows directly from (v). 0
The Stone identity has a nice generalization to an identity in n variables,
which is attributed to K. B. Lee [1970] :
(Ln) (*x A • • • A Xn)* V (X? A • • • A Xn)* V • • • V (*! A • • ■ A **)* = 1.
Note that (Lx) is x* V *** = 1, the Stone identity. For n — 2 and n = 3
we obtain
(La) to A x2)* V (x? A x2)* V (*! A 4)* = 1,
(L3) (*! A X2 A X3)* V (*? A X2 A X3)* V (*! A xt A X3)* V
(*! A X2 A X*)* = 1.
The larger n is, the harder (Ln) is to work with. The following lemma
(G. Gratzer and H. Lakser [1969b]) is sometimes useful:
Lemma 2. Let L be a distributive lattice with pseudocomplementation.
Then the following conditions are equivalent:
(i) L satisfies the identity (Ln).
(ii) L satisfies the implication:
x{ A Xj = 0, / 7^ h Uj = 0,..., w, imply that
X* V X? V ••• V X* = 1.
(iii) L satisfies the implication:
Xt A x, = 0, i ^ j9 Uj = 0,..., n, x09..., xn g S(L)
imply that x? v x* V • • • V x* = 1.

168 DISTRIBUTIVE LATTICES WITH PSEUDOCOMPLEMENTATION
remark. For n = 1, (ii) is: x0 A x1 = 0 implies that x* V x* = 1.
For n = 2, (ii) is: x0 A *i = x0 A x2 = xx A xa = 0 *mPlv t^iat
xj V x* V x*. = 1.
PROOF.
(i) implies (ii). Let x{ A x, = 0 for i ^ y, /j = 0,..., n. Then for / ^ j\
Xi < xf; therefore, x0 < x* A A x*, Xi < X** A x* A • • • A x*,
..., xn < x? A x* A • • • A x**. Thus by applying (Ln) to xf,..., x* we
obtain
X^ V xf V • • • V X* > (xf A • • • A X*)* V (xf* A • • • A X*)* V • • •
V(*?A-..A***)* = 1. ft
(ii) implies (iii), by specialization, ft
(iii) implies (i). Set }>o = *i A • • • A xn, y1 = x* A • • • A xn,..., yn =
x1 A • • • A x*. Then ;{ a ^ = 0 for / ^ j\ and so, by Theorem l(vii),
we have yf* A y** = 0, for / ^ y, and >>* *, • • •, 7* * e #(£)• We can thus
apply (iii) to the elements y$*,..., y** to obtain .y*** v .y *** V • • • V y***
= 1. This reduces to (Ln) by Theorem l(iv). 0
As an immediate application of Lemma 2 we can check whether (Ln)
holds in certain examples. Let (£n denote the 2n-element Boolean lattice
with a new 1 (see (£0> ®i, ®2> and (£3 in Figure 15.1). More formally, (£n is a
bounded lattice with 0 and 1 and an element e such that [0, e] is a 2n-
element Boolean lattice, e -< 1, and (£n = [0, e] u {1}. It is easily seen that
(£n is pseudocomplemented; in fact, 1* =0,0* = 1, and for every xe
[0, e], x^0, the pseudocomplement of x in (£n is the relative complement
of* in [0, e].
Lemma 3. Let m be a nonnegative integer and let n be a positive integer.
Then (Ln) holds in (£m iffm < n.
proof. Note that S((£m) £ [0, e]9 which is a 2m-element Boolean lattice
having m atoms pl9..., pm. Thus if m < n, then (since S(($,m) does not have
n + 1 pairwise disjoint nonzero elements) Lemma 2(iii) holds in (£m
trivially, and so by Lemma 2, (Ln) holds in (£m. If n < m, then pl9...,
/?n+1 satisfy/?, A /?, = 0 for i # j, pK e 5(6:m), but/?? V • • • V p*+1 < e < 1.
Thus Lemma 2(iii), and therefore (Ln) fails in (£m. 0
Now we turn our attention to congruence relations. The intimate
relation between congruences and prime ideals that we observed in

Section 15 Identities and Congruences
169
o
0
d
0
distributive lattices prevails for distributive lattices with pseudocomple-
mentation. A new feature is the important role played by minimal prime
ideals. A prime ideal P is called minimal if there is no prime ideal Q with
Q<^P.
Lemma 4. Let L be a lattice with 0. Then every prime ideal contains a
minimal prime ideal
proof. Let P be a prime ideal of L and let X denote the set of all prime
ideals Q contained in P. Then X is not void since P g $. If C is a chain in X
and Q = C\(X\ Xe C), then Q is nonvoid because 0g Q and Q is an
ideal; in fact, Q is prime. Indeed, if a A be Q for some afieL, then
a A bg X for all XgC; since X is prime, either aG X or bG X. Thus
either Q = (~)(X\ aG X) or Q = 0(X\ bG X), proving that a or b e £?.
Therefore, we can apply to #" the dual form of Zorn's Lemma (which is
also an equivalent form of the Axiom of Choice) to conclude the existence
of a minimal member of 9£. %
The following characterization of minimal prime ideals is quite useful.
Lemma 5. Let Lbea pseudocomplemented distributive lattice and let Pbea
prime ideal ofL. Then the following four conditions are equivalent:
(i) P is minimal.
(ii) x g P implies that jc* £ P.
(iii) x g P implies that x** gP.
(iv) P n D(L) = 0.

170 DISTRIBUTIVE LATTICES WITH PSEUDOCOMPLEMENTATION
PROOF.
(i) implies (ii). Let P be minimal and let (ii) fail, that is, a* eP for some
a e P. Let D = (L - P) V [a). We claim that 0 £ D. Indeed, if 0 e D9 then
0 = q A a for some qeL — P9 which implies that q < a* e P9 &
contradiction. Thus by Theorem 7.15, there exists a prime ideal Q disjoint to D.
Then Q c p since 0 n (£ - -P) = 0> and Qi- P since a <£ g,
contradicting the minimality of P. \
(ii) implies (iii). Indeed, x* A *** = OeP for any xeL; thus if xeP,
then by (ii), x* $ P, implying that x** eP. )
(iii) implies (iv). If aeP r\ D(L) for some aeL, then a** = 1 <£P, a
contradiction to (iii). Thus P n £(L) = 0. )
(iv) implies (i). If P is not minimal, then Q <= P for some prime ideal Q of
L. Let xgP- g. Then x A x* = Oe Q and x£ Q; therefore x* e
Q <= P, which implies that ^V^Gf. By Theorem l(xvi), x V x* e
Z>(L); thus we obtain x V x*e?n £>(L), contradicting (iv.) #
The importance of minimal ideals for Stone algebras was observed in G.
Gratzer and E. T. Schmidt [1957b]; their importance in describing homo-
morphisms onto <B± was illustrated by R. Balbes and G. Gratzer [a].
The next lemma, which (in one direction) is due to K. B. Lee [1970],
generalizes this observation to arbitrary <Bn.
Lemma 6. Let L be a distributive lattice with pseudocomplementation.
The homomorphisms of L onto <Bn are in one-to-one correspondence with
sequences <P, Ql9.. ., Qn} of prime ideals ofL, where Ql9..., Qn are all the
distinct minimal prime ideals contained in P.
proof. If 9? is a homomorphism of L onto <Bnipl9.. .9pn are the atoms of
<Sn, and e = p1 v • • • V pni then we set P = {x \ x e L, x<p < e}9 Qt =
{x | x e L, x<p < />*}, / = 1,..., n. Clearly, P, Ql9..., Qn are prime ideals,
and Ql9..., Qnsatisfy Lemma 5(iv); thus, Ql9..., Qn are minimal prime
ideals contained in P. Let R be a minimal prime ideal of L contained in P
and distinct from all the Qt. Then there exist a^^eR- Ql9.. .,aneR- Qni
and so a = ax V • • • V an satisfies asR — <2t for all / = 1, ..., w.
Therefore, A<p ^ />* for all /. This implies that a<p = e or 1. Thus (a<p)** = 1
and so a**<p ~ 1, implying that a**£P, contradicting a**eP, which
holds by Lemma 5(iii).
Conversely, let P, Ql9..., 2n be given as in the lemma and let us define
the map <p by
__ (I ifa<£P,
a<P~\V(Pi\a$Qi) for aeP.

Section 15 Identities and Congruences
171
We know from Section 9 that <p is a lattice homomorphism. To verify that
(a<p)* = a*<p, first assume that a$P\ in this case a A a* e Qi9 and thus
a* e Qt for / = 1,..., n; that is, a* e Gi n • • • n gn- By definition, then,
a*9? = 0. Also, a<p = 1, and so (#<?)* = 1* = 0 = a*<p. Second, let
a e Qi n • - r\ Qn. If a* e P, then, just as in the first step of Lemma 5, we
conclude that there is a minimal prime ideal Q satisfying a $ Q £ P9 a
contradiction. Therefore, a* $ P and so (a<p)* = 0* = 1 = a*<p. Third,
let a e P — Qt for some /, \ < i < n. Then a<p ^ 0 and 09? e [0, e]; thus
(a<p)* is the complement of a<p in [0, e]. Therefore,
fo»* = (V (A I « # GO)* = V (Pi I «e Qi) = V (Pi I ** # Qt) = «V
Finally, we have to show that 9? is onto. Obviously, 0,1 e L<p. To finish the
proof it suffices to show that />*,..., pi e £<p. If we can choose a e Qt such
that a £ Qj for ally 7^ /, then a<p = pf as desired. Otherwise, for some Qi9
say Qu we have Gi c Q2 u • • u Qn. Choose a minimal set I c
{2,...,«} such that Gi c (J (Qt | / e /). Then for each / e I there is an
flf e Ci - (J (2; 17 e / - {/}). Let a = V fat I * e /)• Then a e Gi, and so
a g G; for some j e I; therefore, a{ e Qj for all / e 7, contradicting at £ Qj
fori*/ •
A description of all congruence relations of a distributive lattice with
pseudocomplementation has been given by H. Lakser [a]. His result,
which is the next theorem, is an interesting application of the principle
stated at the end of Section 14.
Let L be a distributive lattice with pseudocomplementation. For a
congruence relation 0 of L, let 0S and 0D denote the restrictions of 0 to
S(L) and D(L\ respectively. Obviously, 0D is a congruence relation of
D(L). In S(L) the operations are x A y, x v y = (x* A y*)*9 and *;
therefore, 0S is clearly a congruence relation of S(L). Thus <0S, 0D> e
C(S(L)) x C(D(L)). An arbitrary pair <0, T> e C(S(L)) x C(D(L)) will
be called a congruence pair if a e S(L), u e D(L)9 u > a, and a = 1(0)
imply that u = 1(T).
Theorem 7. Let L be a distributive lattice with pseudocomplementation.
Then every congruence relation 0 of L determines a congruence pair
<0s, ©d>- Conversely, every congruence pair (0^ 02> uniquely determines
a congruence relation 0 on L with 0S = 0X and 0D = 02 by the following
rule:
x^y(®) #(0 ** = **(ei)
and(ii) x v u = y v u(®2)for all u e D(L).

172 DISTRIBUTIVE LATTICES WITH PSEUDOCOMPLEMENTATION
proof. The first statement is obvious. Let 0 be a congruence of
L, xyeL, x = y(®). By Theorem l(i) and (iii), x = x** A (x V x*)9
y = y** a (y v y*)9 and x** = >>**(0s), x V x* = y v y*(®D)', thus
0S and 0D do indeed determine 0.
Let <0i, 02> be a congruence pair and let 0 be defined by (i) and (ii).
0 is obviously an equivalence relation. Theorem l(v) implies the
Substitution Property for V ; for A it follows from Theorem 15.1(iv) and (vi).
To show the Substitution Property for *, let x = y(®). Then by (i),
x* = j;*(01), and thus x** = }>**(®i), which is (i) for x* and y*. Since
x* = .y*(0i) and S(L) is Boolean, by Corollary 9.4 and Theorem 9.7 there
is an a e S(L) such that a = 1(00 and x* A a = y* A fl(0i). Thus for
any u e D(L) we obtain u V a = 1(02) by the definition of the congruence
pair, and so
x* v w = (x* v u) A (a V u) = (x* A a) v u = (y* A a) v u
= (y* V u) A (a v u) = y* V m(02),
proving (ii) for x* and >>*. Therefore, 0 is a congruence relation.
For x9y e S(L), x = y(S) iff x* = ^*(0O (since (ii) is trivial), and so
x = y(@s) iff x = X^i), that is, 0$ = ®x. For x,y e Z>(L), (i) is trivial,
and thus x = .y(0) iff, for all u e D(L), we have x V w = y V m(02),
which is equivalent tox = }>(02), and so 02 = 0D. #
We shall apply this description of congruences to establish an important
property of pseudocomplemented distributive lattices, namely, the
analogue of Theorem 9.6. To facilitate this we prove (a part of) the so-
called Second Isomorphism Theorem of Algebra (see, for example, G.
Gratzer [1968], Theorem 11.4). The word "algebra" in this lemma could
be replaced by "lattice" or "lattice with pseudocomplementation."
Lemma 8. Let L be an algebra and let 0 be a congruence relation of L.
For any congruence <E> of L such that O > 0, define the relation O/0 on
LI® by
M0 ee [j,]0(O/0) iffx^ym
Then 0/0 is a congruence ofL/@. Conversely, every congruence To/1/0
can be (uniquely) represented in the form T = O/0/or some congruence
O > 0.

Section 15 Identities and Congruences
173
proof. We have to prove that 0/ 0 (i) is well-defined, (ii) is an equivalence
relation, and (iii) has the Substitution Property. To represent T, define O
by
x = y(<!>) iff M6 s \y\eQS).
Again, we have to verify that 0 is a congruence. <E>/0 = T follows from
the definition of O. The trivial details are left to the reader. £
Definition 9. A class K of algebras is said to have the Congruence
Extension Property if for A,BeK with A a subalgebra of B and 0 a
congruence of A, there exists a congruence O on B such that <bA = 0,
that is, <E> restricted to A is 0.
Using this terminology, Theorem 9.6 states that the class of distributive
lattices D has the Congruence Extension Property.
Theorem 10 (G. Gratzer and H. Lakser [a]). The class of all distributive
lattices with pseudocomplementation enjoys the Congruence Extension
Property.
proof. Let L and Kbe distributive lattices with pseudocomplementation,
let L be a subalgebra of K, and let 0 be a congruence of L given by
the congruence pair <0x, 02>. It is clear from Theorem 7 that we need
only show the existence of a congruence pair (Oj, 02> of K such that
(<E>i)sa> = ©i and (<E>2)D(L) = 02.
Let JL = [l]0i and put JK = [JL)9 the dual ideal generated by JL in
S(K). Then Ox can be defined as the congruence of S(K) associated with
JKi that is, [l]Oi = JK. Set / = {/1 / e D(K\ i > u for some u e JL}. Then /
is a dual ideal of D(K); in fact, / = [JK) n D(K). By the definition of
congruence pair, we have to find a congruence <E>2 on D(K) such that
(02)D(t) = 02 and [1]<E>2 ^ /. Note that /has the following property:
If uel,ve D(L), and v < w, then there exists a v1 e D(L),
v1 < u such that v1 e [1]02.
Indeed, us I means that u > x for some x e/L, and thus v1 = v V x
will do the trick.
Summarizing, to complete the proof it suffices to prove the following
statement:

174
DISTRIBUTIVE LATTICES WITH PSEUDOCOMPLEMENTATION
Let A and B be distributive lattices with \, A a. {l}-sublattice of B9 0 a
congruence of A, and / a dual ideal of B satisfying the condition:
If we/, v e A, and v < w, then v1 < u for some v± e [1]0.
Then there exists a congruence relation O on B satisfying ®A = 0 and
[1]<E> => /.
To prove this statement, consider 0[/] defined by the dual of Corollary
9.4. If a9b e A and a = 6(0[/]), then a A b = (a V b) A i for some i e/.
Thus by our assumption on /, there is an ix e [1]0 such that ix < i.
Therefore, a A b = (a V b) A i > (a V b) A ix = (a V b) A I = a V 6(0),
and so a = 6(0). Having shown that (®[I])A < 0, we can form A/®[I]A9
B/®[I], and ©[Z]^/©. By Theorem 9.6 there exists a congruence T on
5/0[/] such that T restricted to A/S[I]A is ©[Z]^/©. By Lemma 8 there is
a unique congruence <E> on £ such that O/0[/] = T and O > 0[/].
Obviously, O satisfies the requirements. 0
Exercises
1. List the rules in Theorem 1 that hold for all pseudocomplemented
lattices.
2. Let L be a distributive lattice in which every prime ideal contains a
minimal prime ideal. Prove that L has 0.
3. Let A and B be dual ideals of a pseudocomplemented distributive
lattice L. Verify that A n S(L) and B n S(L) are dual ideals of S(L)
and 04 v B) n S(L) = (A n 5(L)) v(5n 5(L)).
4. For a pseudocomplemented distributive lattice L, define the relation R
by: x s >>CK) iff ** = y*> Show that it is a congruence on L and
L/R - 5(L).
5. Show that in a Stone algebra every prime ideal contains exactly one
minimal prime ideal.
6. Prove that a prime ideal P of a Stone algebra L is minimal iff P =
(Pr>S(L)]L.
7. Let 0 be a principal congruence relation of a distributive lattice L with
pseudocomplementation. Show that ®D(d need not be principal.
8. Conclude from Lemma 8 that [0) of C(L) is isomorphic to C(L/0).
9. Does Theorem 10 imply the Congruence Extension Property for D?
"10. Find a proof of Theorem 10 that does not make use of the Congruence
Extension Property for D (G. Gratzer and H. Lakser [a]).
11. Show that a distributive lattice with pseudocomplementation is a Stone
algebra iff every prime ideal contains exactly one minimal prime ideal
(G. Gratzer and E. T. Schmidt [1957b]).

Section 16 Representation Theorems
175
*12. Prove that a poset Q is isomorphic to the poset of all prime ideals of a
Stone algebra iff (i) every element of Q contains exactly one minimal
element, and (ii) for every minimal element m of Q, the poset {x \ x > my
x e Q] is isomorphic to the poset of all prime ideals of some distributive
lattice with 1 (C. C. Chen and G. Gratzer [1969c]).
13. A lattice L is a relative Stone lattice if every interval of L is a Stone
lattice. Show that the Stone lattice L is a relative Stone lattice iff D(L) is a
relative Stone lattice (C. C. Chen and G. Gratzer [1969c]).
14. Find a complete Boolean lattice B such that ©x is a subalgebra of 1(B).
Exercises 15 and 16 are from B. Banaschewski [1970].
15. Let K be an equational class of algebras in which every algebra can be
embedded in an injective algebra. Show that K has the Congruence
Extension Property.
16. Let K be the same as in exercise 15. Show that K has the Amalgamation
Property.
17. Find a direct proof of the Congruence Extension Property for Stone
algebras.
18. For a class K of algebras, let H(K), S(K), and I(K) denote the classes of
homomorphic images of algebras in K, subalgebras of algebras in K, and
algebras isomorphic to algebras in K, respectively. Show that if K has the
Congruence Extension Property, then HS(K) = ISH(K).
19. Show that the congruence pairs form a sublattice of C(S(L)) x
C(D(L)) (H. Lakser [a]).
20. Let L be a distributive lattice with 0 and 1. For an ideal / of L, we set
/* = {x | x A i = 0 for all i e /}. Let M be a prime ideal of L. Show that
M is a minimal prime ideal iff x e M implies that (x]* $ M (T. P.
Speed).
21. Let Li and L2 be Stone algebras and let <p be a {0, l}-homomorphism of
Lx into L2. Show that <p is an algebra homomorphism iff D(L^)cp c
D(L2).
22. Let Li and L2 be distributive lattices with pseudocomplementation and
let <p be a {0, l}-homomorphism of Lx into L2. Prove that <p is an algebra
homomorphism iff S(£i)<p c S(L2) and Z)(Li)<p c D(L2) (H. Lakser).
16. Representation Theorems
The BirkhofF-Stone Theorem (Theorem 7.19), which claims that every
distributive lattice can be represented by sets, is the model for the
representation theorems we want to prove in this section. All such theorems are
based on a universal algebraic result, the Subdirect Representation
Theorem of G. Birkhoff [1944]. In certain special cases its use can be
avoided—we gave a direct proof for distributive lattices and we could give

176 DISTRIBUTIVE LATTICES WITH PSEUDOCOMPLEMENTATION
a direct proof for Stone algebras. However, avoidance of it in this section
is neither possible nor desirable. So we begin by developing universal
algebraic tools to deal with such problems. As usual, for our purposes,
"algebra" could be read "lattice" or "lattice with pseudocomplementa-
tion."
Definition 1. An algebra A is called subdirectly irreducible if there exist
elements u,v e A such that u ^ v and u = v(®)for all congruences 0 > a>.
In other words, C(A) = {w} u [0(w, v)). An equivalent form is:
Corollary 2. The algebra A is subdirectly irreducible iff o> =
/\(@( | i e I) (®t e C(A) for i e /) implies that 0, = w for some i e /.
example 3. A distributive lattice L is subdirectly irreducible iff \L\ = 2.
proof. If \L\ = 1, then L is not subdirectly irreducible by definition.
If \L\ = 2, then obviously L is subdirectly irreducible. Let \L\ > 2. Then
there exist a,b,c eL9a < b < c. We claim that 0(a, b) A 0(6, c) = o>,
which by Corollary 2 shows that L is not subdirectly irreducible. Let
x = y(®(a,b) A 0(6, c)). By Theorem 9.3 this implies that x V b = y V b
and x A b = y A b; thus x = y by Corollary 7.3. 0
example 4. 932 is the only subdirectly irreducible Boolean algebra.
proof. Let B be Boolean. The statement is obvious for \B\ < 2. If
\B\ > 2, then B has a direct product representation, B = B1 x 2?2, |2*i|,
|2?2| ^ 2; thus B cannot be subdirectly irreducible. 0
Equational classes of universal algebras can be introduced by defining
polynomials and identities, just as in the case of lattices. However, in the
next theorem the reader can avoid the use of this terminology by
substituting for "equational class" the phrase "class closed under the
formation of subalgebras, homomorphic images, and direct products."
(This, incidentally, is an equivalent formulation by G. Birkhoff [1935];
such an equivalence will be established for the classes of algebras we
consider. For the general case see exercises 6-12 and G. Gratzer [1968],
Theorem 26.3.)

Section 16 Representation Theorems
177
Theorem 5 (G. Birkhoff [1944]). Let K be an equational class of algebras.
Every algebra A in K can be embedded in a direct product of subdirectly
irreducible algebras in K.
proof. For a ,b eA9a^b, let ff denote the set of all congruences 0 of A
satisfying a ^ 6(0). Since o> e &, 2£ is not empty. Let C be a chain in O.
Then 0 = (J (O | O e C) is a congruence, a ^ 6(0), and thus every chain
in 3C has an upper bound. By Zorn's Lemma, % has a maximal element
T(a, 6). We claim that AP¥(a, b) is subdirectly irreducible; in fact, u =
[aY¥(a, b) and v = [6]T(a, 6) satisfy the condition of Definition 1.
Indeed, if 0 is any congruence of Ap¥(a, 6), 0 ^ o>, then by Lemma 15.8,
0 = 0/T(a, b). Since 0 # a>, we obtain O > T(a, 6), and so a = 6(0).
Thus w = i>(0), as claimed.
Let B = n (A/Y(a, b) \a,beA,a^b); then £ is a direct product of
subdirectly irreducible algebras. We embed A into B by <p:x-*fX9 where
/* takes on the value [x]T(a, b) in the algebras AfT(a, b). Clearly, <p is a
homomorphism. To show that 9? is one-to-one, assume that/* = fy. Then
jc 5 >^CF(a, b)) for all a,6 e A, a ^ b. Therefore, x = y(/\ C^fo *) I fl^ e
A9a ^ 6), and so x = >>(o>), that is, x — y. 0
We got a little bit more than claimed. If we pick x e A/Wfa b\ then
x = [yW(a, b) for some y e A. Thus there is an element in the
representation of A whose component in A/Y(a, b) is x; such a representation is
called subdirect.
Corollary 6. In an equational class K, every algebra has a subdirect
representation by subdirectly irreducible algebras in K.
Observe how strong Theorem 5 is. If combined with Example 3 it
yields Theorem 7.19; when we combine it with Example 4 we obtain
Corollary 7.21.
The main result of this section is the determination of subdirectly
irreducible distributive lattices with pseudocomplementation. For a
Boolean algebra B with smallest element 0 and unit e, set 6 = B u {1} and
let x < 1 for all x e B. Note that if B = (932)n, then £ is the algebra <Sn
defined in Section 15. Again, for x e £,
CI if x = 0
x* = lx' \fxeB9x^O
[O if jc - 1

178 DISTRIBUTIVE LATTICES WITH PSEUDOCOMPLEMENTATION
Theorem 7 (H. Lakser [a]). A distributive lattice with pseudocomplemen-
tation L is subdirectly irreducible iffL is isomorphic to 6 for some Boolean
algebra B.
remark. For finite L this result was proved independently by K. B.
Lee [1970].
proof. In S let a = e, let b = 1, and let 0 be any congruence other than
o>. Then there exist x9y e A x < y, such that x = y(S). If y = 1, then
x V e == y V e(0),thatis,a = 6(0). Ify ^ 1, theny < e. Setz = x' A y.
Then z = 0(0), and thus z* = O*(0); that is, z' = 1(0), which again
implies that a = 6(0).
Conversely, let L be a subdirectly irreducible distributive lattice with
pseudocomplementation. Then \L\ > 2 by Definition 1. Let a9beL,
a < b be the elements establishing that L is subdirectly irreducible and let
<0i, 02> be the congruence pair describing 0(a, b).
If D(L) = {1}, then L is Boolean. (By Theorem 15.1(xvi) for all xeL,
x v x* g D(L) = {1}; that is, x V x* = 1, and so x* is the complement of
x.) Thus \L\ = 2 by Example 4, and so L = B, where B is a one-element
Boolean algebra.
Therefore, we can assume that |£>(£)| > 1- Let O be a nontrivial
congruence on D(L). Then <a>, 0> is a congruence pair that represents
a nontrivial congruence, and thus we must have 0X < a> and 02 < $. We
conclude that G1 = a> and that 02 is contained in every nontrivial
congruence of D(L). Therefore, D(L) is a subdirectly irreducible distributive
lattice, and so by Example 3, \D(L)\ = 2; let D(L) = {e, 1}. We conclude
that a = e, b = 1.
Let xeL, x ^ 1, and x £ e. Then by Theorem 15.1(xvi), x v x* e
{e, 1}. Since x £ e, we conclude that x v x* = 1, and so x e S(L). Let T
be the congruence on S(L) determined by [x). If ye S(L), u e D(L),
y < m, and y = 1(T), then x < y9 and x < ue D{L\ so we can conclude
that u = 1. Thus <T, a>> is a congruence pair that determines a nontrivial
congruence that does not collapse e and 1, a contradiction. Therefore, for
xGLwe obtain x < e for all x ^ 1. Since for x < e we have x* < e,
we conclude that x v x* = e. For x < e, define x' = x* if x ^ 0 and
x' = e if x = 0. Then B = <(e]; a, V,', 0, e) is a Boolean algebra
and L = A 0
Our next task is to describe all equational classes of distributive lattices
with pseudocomplementation. We start with a few examples: B_x is the

Section 16 Representation Theorems
179
class of one-element lattices with pseudocomplementation; let B0 denote
the class of Boolean algebras, and for n > 1, let Bn be the class of all
distributive lattices with pseudocomplementation satisfying the identity
(Ln) (defined in Section 15). It follows from exercise 6.23 that there is a set
of identities 2 (in terms of polynomials built up by A, V, and *) such that
2 characterizes distributive lattices with pseudocomplementation (P.
Ribenboim [1949]). Let Bw be the class determined by 2. Thus Bn for
n > 1 is defined by 2 u {Ln}. B_x is defined by x = y9 and B0 is defined by
2 u {L0}, where
(L0) x V x* = 1
defines Boolean algebras.
It is obvious from Lemma 15.2 that (Ln) implies (Ln + 1) for n > 1; this
implication is also obvious for n = 0, 1. Lemma 15.3 shows that Bn ^
Brt+1 for n > 1; it is obvious for n = 0, 1. Thus
B-lcB0c.cJlc Bn+1 c.cBw
is a strictly increasing sequence of equational classes of distributive
lattices with pseudocomplementation.
Theorem 8 (K. B. Lee [1970]). The Bn (-1 < n < cS) are the only
equational classes of distributive lattices with pseudocomplementation.
Moreover, ifn>0 and A e Bn, then A is a subdirect product of copies of
<30,...,<3n.
proof (H. Lakser [a]). Let K be a class of distributive lattices with pseudo-
complementation closed under the formation of subalgebras, homomor-
phic images, and direct products, and let S denote the class of sub-
directly irreducible algebras in K. Let Sn denote the subdirectly irreducible
algebras in Bn. Combining Lemma 15.3 and Theorem 7, we obtain (up to
isomorphism) for n < o>
Sn = {<30,...,©n},
and Sw is the class of all subdirectly irreducible algebras.
If <&n g S, then S„cS since <B0i ..., ©n_x are subalgebras of <&n. We
have to distinguish two cases.
Case 1. There is a largest integer N such that <BN e S. We claim that in this
case K = B^.

180
DISTRIBUTIVE LATTICES WITH PSEUDOCOMPLEMENTATION
Case 2. There is no largest integer N with <5N e S. Then we claim that
K = BW.
proof of case 1. By assumption, S^cS and <BN+1 $ S. If A e S - S^,
then A = £ for some Boolean lattice 2? with \B\ > 2N; thus @*+i is a
subalgebra of A,<5N+1 e S, a contradiction. Therefore, S = S^ and so, by
Theorem 5, K = BN. )
proof of case 2. In this case 6 e S for any finite Boolean lattice B. To
show that 8 = 8^ and therefore that K = Ba, it suffices to show that if B
is any Boolean lattice, then £eS. Indeed, let 3C denote the family of finite
subalgebras of B\ for Bx e&, let £x denote Bx u {1}, where 1 is the unit
element of A Then Sx is a subalgebra of A Set fc = {S1\B1g #}. Observe
that each A e & is finite; thus £ s S, & is directed, and (J G* I ^ e T) =
A Since these imply that ^eK (see exercises 8-12), we conclude that
SeS. #
The representation theorem of Section 7 stated that every Boolean
algebra can be embedded in some P(X). Verifying a conjecture of O. Frink
[1962], G. Gratzer [1963] proved that every Stone algebra can be
embedded in some I(P(X)), viewed as a distributive lattice with pseudo-
complementation. This cannot be improved upon because, according to
exercise 14.5,I(P(X)) is a Stone algebra. It was conjectured in G. Gratzer
[1963] that by replacing P(X) by a (noncomplete) atomic Boolean algebra,
the previous result extends to all distributive lattices with pseudocomple-
mentation. This idea was proved in H. Lakser [a], utilizing Theorem 7:
Theorem 9. Let L be a distributive lattice with pseudocomplementation.
Then there is an atomic Boolean algebra B such that L can be embedded into
1(B),
proof. Let L be subdirectly irreducible. Then L = S0 for some
Boolean lattice B0. By Corollary 7.21 there is an embedding q>: B0 -> P(X).
We thus obtain an embedding #: L -> P(X) by setting ay = a<p for aeB0
and 1$ = 1. Choose an infinite set Y and let B be the Boolean algebra
of all finite and cofinite (that is, complements of finite) subsets of X x Y.
Obviously, B is atomic. For A ^ X define
Af = {Z\ZeB9ZcA x 7,Zfinite};

Section 16 Representation Theorems
181
Then 0 maps P(X) into 1(B). Observe that 00 = 0 and
JT0 = {Z|Ze£,Z finite}.
The formulas (In 7)0 = 10 A 70 and (Xkj 7)0 = Z0 v 70 are
immediate. Thus 0 is a lattice-embedding of P(X) into 1(B). Let A^ X;
then
040)* = {Z|Ze£,Zn JF= 0forall JFe,40}
= {Z\ZeB9Z^(X-A) x Y}.
Thus if A = 0, then 040)* = £. If ^4 ^ 0, then the complement of
(X - A) x Yin X x 7is infinite (7is infinite); therefore, Z^(X- A) x Y
with ZgB cannot be cofinite, so such a Z must be finite. Consequently,
040*) - {Z | Z s (X - ,4) x 7, Zfinite} = (X - ,4)0.
Thus 0 is an embedding of the algebra P(X) into 1(B), and <p0 embeds L
into /(£).
Since the existence of an embedding into some 1(B) is preserved under
the formation of subalgebras (which is obvious) and direct products (see
exercise 16), a reference to Theorems 5 and 7 concludes the proof. 0
Exercises
1. Show that every simple algebra A (that is, |C04)| = 2) is subdirectly
irreducible.
2. Show that for every cardinal m > 4, there is a simple lattice of cardinality
m.
3. Verify the statement of Example 3 without any reference to Section 9.
*4. Show that Theorem 5 is equivalent to the Axiom of Choice (G. Gratzer,
Notices Amer. Math. Soc. 14 (1967), 133).
5. Let B be a Boolean algebra. Show that every subalgebra of a £ is either
Boolean or of the form 6l9 where Bx is a subalgebra of B.
In exercises 6-12, let K be a class of algebras closed under the formation
of subalgebras, homomorphic images, and direct products.
6. Let A be an algebra and let sf be a family of subalgebras of A such that
for Bes/ we have BeK9 (J (X\ Xej*) = A, and for X,Yes/ there

182 DISTRIBUTIVE LATTICES WITH PSEUDOCOMPLEMENTATION
is a Zest such that X c Z and Y ^ Z. Define B c nWl A'e^)
by fe B iff there exists an X e j^ such that for all K,Z e st with Jc K
and X c Z9 we have /(Y) = /(Z). Show that B is a subalgebra of
Y\(X\Xesf\ and therefore 5e K.
7. Define a relation 0 on the algebra B of exercise 6: / = g(®) iff there
exists an Xes/ such that for all Ye*/ with K2l,we have f(Y) =
£(K). Show that 0 is a congruence relation on B and that A £ 5/0.
Conclude that ^eK.
8. Let FK(a>) denote the free algebra over K freely generated by *0, *i, • • •,
*„, Let 2 denote the set of all identities p = q such that
P(x0,xly...) = q(x09xl9...).
Show that the identities in 2 hold in FK(a>).
9. For an ordinal a let FK(«) denote the free algebra over K freely generated
by jc0> ..., JCy, ..., y < a. Show that 2 holds in FK(a).
10. Show that every A e K is a homomorphic image of some FK(«). Conclude
that 2 is satisfied in every A e K.
11. Show that every algebra v4 satisfying 2 is the homomorphic image of
some FK(«).
12. Combine exercises 8-11 to show that K is an equational class of algebras.
(This is G. BirkhofTs characterization theorem for equational classes;
see G. BirkhofT [1935].)
13. Show that a distributive lattice with pseudocomplementation L belongs
to B„ (1 < n < co) iff every prime ideal of L contains at most n distinct
prime ideals (K. B. Lee [1970]).
14. A congruence relation 0 is completely meet-irreducible if 0 =
A (0f | / e /) implies that 0 = 0i for some / e I. Show that Theorem 5
is equivalent to the statement that for any algebra A, any congruence is
the complete meet of completely meet-irreducible congruences.
15. Combine exercises 13 and 14 to show the Congruence Extension Property
for Bn (1 < n < o>).
16. Let the Li9ieJ, be distributive lattices with 0. Define the mapping
9>: n (/(« I / e /) — I(U (U | / e /)) as follows: If fe U (/(« I i e J),
then/(/) is an ideal of L, for each is J; set/9 = U(f(i) | ieJ). Show
that 9? embeds the distributive lattice with pseudocomplementation
II (/(« I/e/) into/(nail/€/)).
17. Let X be an infinite set. Define <£: <3i -+ I(P(X)) by 0^ = {0}, 1^ =
P(X), ^ = {Z|Zc x, Z finite}. Show that 0 embeds the algebra ©,.
into I(P(X)).
18. Show that an algebra <L; A , v , *, 0, 1> is a Stone algebra iff it can be
embedded in some 1(B) where B is an atomic and complete Boolean
lattice (G. Gratzer [1963]).
19. Show that the free algebra over Bn with m generators is finite, provided
that/i,m < o).

Section 17 Injective and Free Stone Algebras
183
20. Does the statement of exercise 19 hold for Bo, ?
21. Show that a finitely generated subalgebra of a distributive lattice with
pseudocomplementation is finite (K. B. Lee [1970]).
22. Find a direct proof of the statement that every Stone algebra is a sub-
direct product of copies of ©o and ©i (use exercise 15.11).
17. Injective and Free Stone Algebras
Many of the categorical concepts introduced in Section 13 can be
fruitfully studied in the categories Bn for 0 < n < w. To illustrate this we shall
investigate injectives in Bi, proving the Characterization Theorem of R.
Balbes and G. Gratzer [a]. Further developments along this and other
lines are mentioned in the exercises and in the Further Topics and
References at the end of this section.
Let us recall that a Stone algebra / is injective if, whenever A and B are
Stone algebras and B is a subalgebra of A, then any homomorphism p of B
into / can be extended to a homomorphism a of A into /.
We shall need two simple facts about injective algebras.
Lemma 1.
(i) A direct product of injective algebras is injective.
(ii) A retract of an injective algebra is injective.
proof. Let A and B be Stone algebras and let B be a subalgebra of A.
(i) Let Ij,jeJ, be injective algebras and let P be a homomorphism of B
into n(Ij\jeJ). Define ft :£-*/, by xft = (xp)(j)9 that is, p
followed by the jth projection. Since Ij is injective, there exists an
extension a, of ft to A. Define a: A ->Yl (Jj \JeJ) by (xa)(J) = xaf
for j e J. Then a is a homomorphism extending p to A. )
(ii) Let / be an injective algebra, let p be a retraction on /, and let p be a
homomorphism of B into Ip. We can view p as a homomorphism of B
into /; thus p has an extension ax to A. It is easily seen that a = axp is
an extension of p to A and that it maps A into Ip. %
Our results on injective Stone algebras can be summarized in the form of
two theorems, the first of which can be called the Characterization
Theorem, the second of which can be called the Representation Theorem
(both are from R. Balbes and G. Gratzer [a]):

184 DISTRIBUTIVE LATTICES WITH PSEUDOCOMPLEMENTATION
Theorem 2. A Stone algebra I is injective iff it satisfies the following
conditions:
(i) / is complete.
(ii) / has a smallest dense element d.
(iii) Every element a has a dual pseudocomplement a+ (that is,
a V x » 1 iff x > a+) and a+ A a++ = 0 (that is, a+ e
S(I)).
(iv) a* = b* anda+ = b+ imply that a = b.
Let us recall a construction of Stone algebras (exercise 14.10): For a
Boolean algebra B let Bm = {{a, b> \ a < b9 a,b e £}; observe that Pt2] is
a sublattice of B2 and that <a, by* = <6', 6'>; thus <a, &>* V <a, &>** =
<b', b'y v <6, ^> = <1, 1>. Therefore, B™ is a Stone algebra.
Theorem 3. A Stone algebra I is injective iff it can be represented in the
form B0 x B[2\ where B0 and Bx are complete Boolean algebras. In this
representation B0 and Bx are uniquely determined up to isomorphism.
We prepare the proofs of Theorems 2 and 3 in a series of ten steps.
The proofs of the theorems will be easy combinations of the following
statements.
(a) A prime ideal of a Stone algebra contains one and only one minimal
prime ideal. This is an obvious combination of Lemma 15.6 and the case
n = 1 of Theorem 16.8. )
(b) <Si is injective. Let A and B be Stone algebras, let B be a subalgebra
of A, and let j8 be a homomorphism of B into ©i. Set Q = {x \ x e B,
xp = 0} and P = {x \ xfl < e}. By Lemma 15.6, P and Q are prime ideals
and Q is the minimal prime ideal contained in P. Again by Lemma 15.6,
it suffices to find prime ideals Px and Q1 of A, Qx minimal in Pl9 such that
P = P1n B and Q = Ql n B. By Theorem 7.15 there exists a prime ideal
Px of A containing (P]A and disjoint to [B - P)A. Then P1dB = P. Let
Q1 denote the minimal prime ideal of A contained in Px. Then Q1C\ B
is minimal in B by Lemma 15.5, and Qir\B^Px; thus by (a),
Q1C\B=Q. »
(c) <B0 is injective. This is obvious by (b) and by Lemma l(ii). )
(d) A Stone algebra / is injective iff it is a retract of some ©? x ®I-
The "if" part follows from (b), (c), Lemma l(i) and (ii). The "only if"
part follows from Theorem 16.8 and the fact that an injective algebra is a
retract of any extension. )

Section 17 Injective and Free Stone Algebras
185
(e) An injective algebra / satisfies (i)-(iii) of Theorem 2, and
furthermore, D(I) is Boolean. Indeed, ($$ x &i satisfies (i)-(iii) of Theorem 2 and
the dense set is Boolean; these properties are preserved under retraction,
and thus (d) completes the proof. }
(f) An injective Stone algebra / satisfies (iv) of Theorem 2. Let a,b e I,
a ^ b,a* = b*9 and a+ = b + . Take a prime ideal P containing exactly
one of a and b—for example, aeP, b$P. Then a** = b** > b<£P,
and thus a** $ P. By Lemma 15.5, P is not minimal. Let Q be the minimal
prime ideal in P. Using the dual argument (justified by condition (iii) of
Theorem 2 as proved in (e)), there exists a prime ideal R such that R => P
and L — R is a minimal prime dual ideal, that is, R is a maximal prime
ideal. Pick an xeP n D(I) (see Lemma 15.5) and a yeR — P with
x < y. Consider the map <p:a^>Q for ae Q;a^>x for aeP — Q;
a -> y for a e R — P; a -> 1 for a e L — R. It is easily seen (just as in the
proof of Lemma 15.6) that <p is a retraction; therefore, by Lemma l(ii),
the four-element chain is injective, contradicting the last statement of (e).
(The dense set is the three-element chain, which is not Boolean.) )
(g) Let A satisfy (i)-(iv) °f Theorem 2; then A can be represented in the
form B0 x Bl?\ where B0 and B1 are complete Boolean algebras. Indeed,
A £ (a] x (a*]9 where a = d++ (dis given by (ii)). Since by (iii) d+ + has a
complement (namely, d+)9 we obtain a e S(A), hence the possibility of
such a decomposition. The pseudocomplement of b e (A] is b° = a A b*,
so b V b° = b V (a A b*) = (b V b*) A (b V a) > a, since b V b*e D(A),
and so b v b* > d > d+ + = a. On the other hand, b V b° < a,
and thus b V b° = a, verifying that (a] is a Boolean algebra. By (i), (a] is
complete; so is (a*]. Therefore, it suffices to show that (a*] ^ (S((a*]))l2\
For x e (a*], let x° denote the pseudocomplement of x in (a*], that is,
x° = a* A x*. We claim that the dual pseudocomplement x°° of
x in (a*] is x°° = x+ A a*. Indeed, x v x°° = x V (x+ A a*) =
(x v x+) A (x V a*) = 1 A a* = a*;if x V y = a*, then x V >> V a**
= a* V a** = 1, and so y > (x v tf**) + = x+ A (a**)+ = x+ A a* =
x00. The map *-* <x0000, x00> embeds (a*] into (^((tf*]))™ by (iii) and
(iv). The map is onto because if <6, c> e (S((a*]))t2], that is, b,c e S((a*]),
6 ^ c, then x -> <6, c>, where x = (6 v d) A c. Indeed, using d++ = a
and d* = 0, we obtain jc0000 = (x+ a a*)00 = (x+ A a*) + A a* =
(x+ + v a**) a a* = (((6++ v d++) a c++) v a**) A a* =(((*> v a)
A c) v a) A a* = (b v a) a a* = (b a a*) v (a A a*) = b v 0 = b;
similarly, x00 = c. )
(h) Every complete Boolean algebra is injective. Every complete

186 DISTRIBUTIVE LATTICES WITH PSEUDOCOMPLEMENTATION
Boolean algebra is a retract of some power of <50 by Corollary 7.21 and
Theorem 13.14; thus Lemma l(i) and (ii) complete the proof. )
(j) If B is a complete Boolean algebra, then B121 is injective. By Theorem
16.8, B™ can be embedded in some A = &$ x (£?. Thus B ~ S(B™) is a
subalgebra of S(A), and so, by Theorem 13.14, there is a retraction
i/s: S(A) -> S(B™). Then for a e A define a<p = <a+ +0, a**<£>. It is easily
seen that <p is a retraction of >4 onto Bm\ therefore, Bm is injective by
(d). »
(k) If B0 and B1 are complete Boolean algebras, then B0 x B?1 is
injective. This is proved trivially, by (h), (j), and Lemma l(i). )
proof of theorem 2. If 7 is injective, then /satisfies (i)-(iv) of Theorem 2
by (e) and (f). Conversely, if / satisfies (i)-(iv), then by (g), I ^ B0 x Bl?\
where B0 and B1 are complete Boolean algebras, and so / is injective by
(k). •
proof of theorem 3. The first statement has already been proved in
verifying Theorem 2. The uniqueness of the representation follows from
the fact that B0 x Bf} has a smallest dense element d, and B0 is (lattice)
isomorphic to (d+ +] and B1 is (lattice) isomorphic to [d). 0
The second topic to be taken up is free products and free algebras.
Although we prove the existence of free products in all classes Bn, 0 <
n < o>, we shall investigate it in detail only for Stone algebras.
Lemma 4. Let Ai9 i e I, be algebras of more than one element in Bn, where
0 < n < a). Then a free ^-product ofA^iel, exists.
proof. By the proof of Theorem 13.18 or, more explicitly, by exercise
12.9, we have only to prove that if a,b e Ak and a ^ b, then for some
AeBn there exist homomorphisms yx\ AK^ A for all ie I such that
a<Pk =£ b<pk. By Theorems 16.5 and 16.7, there exists a Boolean algebra B
such that there is a homomorphism <p:Ak^>£ satisfying a<p ^ b<p.
Set <pk = <p. By Lemmas 15.4 and 15.6 (case n = 1, P = gi), every algebra
At has a homomorphism <pt into £ (in fact, Am = {0, 1}) so choose this <p,
for i=£k. #
For Stone algebras the situation is surprisingly simple (G. Gratzer and
H. Lakser [1969b]):

Section 17 Injective and Free Stone Algebras
187
0
Figure 17.1
Theorem 5. Let Ati i e /, be Stone lattices. Regarding them as distributive
lattices with 0 and 1, form a free {0, \}-distributive product A of the Au
i g L Then A is a Stone lattice. Furthermore, A as a Stone algebra is a free
{Stone) product ofAi9 i e L
proof. Let L be a distributive lattice with 0 and 1 and let <£%L) be the
Stone space of L. An ideal / of L is represented by an open set r(I) of
Sf(L). For aeL, r((a]*) is the interior of Sf(L) - r(a) (that is, the largest
open set contained in £?(L) - r(a)). Since L is Stone iff (a]* = (b] for
some b that is complemented, this translates into the condition that
Sf{L) — r(a) be clopen, or equivalently:
(S4) The closure of a compact open set is open.
Clearly, (S4), as well as (S1)-(S3) of Section 11, are preserved under
topological product. Thus A is a Stone lattice.
To show that A is a free Stone product, let D be a Stone algebra, and for
each / g / let <p, be a homomorphism of the Stone algebra At into D. Since A
is a free {0, l}-distributive product of the Ai9 iel, there exists a {0, 1}-
homomorphism <p extending all the <p„ / e /. The fact that (x<p)* = x*<p
remains to be proven. We know this to hold for x e At. Let x e A> x =
V A Xae, where xa0 e (J (At \ i e T). Then
(*?)* = ((V A *«*)?)* = (V A *.*)*
= V A (*«/#>)* = V A *&*> = (V A x£tto = x*y. •
The free Stone algebra on one generator -FBl(l) is shown in Figure 17.1.
Observe that FBl(l) = <B0 x <5l9 and so ^(FBl(l)) = ^(©0) u ^(©J,
where the u is disjoint union. Since FBl(n) is the free product of n copies of
FBl(l), &(FMl(n)) is (^(©o) u ^(©!))n. Note that ^(©0) is a singleton, and
©x is the free distributive lattice with 0 and 1. Therefore, (^(©i))* is the
Stone space of jFD{0 1}(k). Thus,

188 DISTRIBUTIVE LATTICES WITH PSEUDOCOMPLEMENTATION
Theorem 6. The following isomorphism holds:
FBl(ri) s n (fl>{0.i>(l*l) I x £ 0. • • •>*»•
This result is due to R. Balbes and A. Horn [1970a]; the present proof is
from G. Gratzer and H. Lakser [1969b].
Exercises
1. Let A be a Stone lattice, let B be a Boolean lattice, and let <p: A —> B be a
{0, l}-homomorphism. Show that ?: x —► <x<p, x**<p> is a homo-
morphism of the Stone algebra /I into the Stone algebra 2?t2].
2. Show that <p —> ^ establishes a one-to-one correspondence between the
{0, l}-homomorphisms of A into B and the algebra homomorphisms of
A into fl[2J.
3. Use exercises 1 and 2 to show that, for a complete Boolean lattice B,
the algebra 2?t2] is injective. (This proof is due to H. Lakser [1970]).
4. For a Boolean lattice B and integer n > 2, define
Partially order 2?[nl componentwise. Show that 2?[nl is a pseudocomple-
mented distributive lattice.
5. Show that Z?[nl as a distributive lattice with pseudocomplementation is
in Bn_i and that if \B\ > 1, then B™ £ B„_2 (G. Gratzer and H. Lakser
[b]).
6. Let K be a category of algebras. Let A and B be algebras in K and let A
be a subalgebra of B. Then 5 is called an essential extension of A if, for
any algebra C e K and homomorphism <p: 5 -» C, if the restriction of <p
to /* is one-to-one, then <p is one-to-one. Assume that K is closed under
the formation of homomorphic images. Show that an essential extension
of a subdirectly irreducible algebra is itself subdirectly irreducible
(see, for example, B. Banaschewski [1970]).
7. An injective essential extension is called an injective hull. Show that
if 2?i and B2 are injective hulls of A, then there is an isomorphism
<p: 2?i —> B2 that is the identity map on A.
8. Let K be a category of algebras closed under the formation of
homomorphic images, A,B e K. Let A be a subalgebra of B. Show that there
exists a congruence relation @ maximal with respect to the property that
®A = o). Show also that B/Q is an essential extension (up to
isomorphism) of A.
9. Let K be an equational class of algebras in which every algebra can be

Section 17 Injective and Free Stone Algebras
189
Figure 17.2
embedded in an injective algebra. Show that every algebra in K has an
injective hull (B. Banaschewski [1970]).
♦10. Determine the injective hulls of Stone algebras (H. Lakser [1970]).
♦11. Verify the Amalgamation Property for Stone algebras.
♦12. Verify the Amalgamation Property for B2.
♦13. Show that the Amalgamation Property holds for Bn iff n < 2 or n = w
(G. Gratzer and H. Lakser [a]).
14. Show that every algebra in Bn can be embedded in an injective algebra
iff/i < 2 (A. Day [1970]).
15. An implicational class K is a class of algebras closed under isomorphism,
subalgebras, and direct products. Find an implicational class K of
distributive lattices with pseudocomplementation such that K^Bn for
any n < o> (G. Gratzer and H. Lakser [c]). (Hint: Take the smallest
implicational class containing the algebra shown in Figure 17.2. Then B2 c
K c B3.)
16. Let A9 B, and C be distributive lattices with pseudocomplementation, let
C be a subalgebra of B9 and let A be a homomorphic image of C. Prove
the existence of a homomorphic image DofB such that A is a subalgebra
of D.
17. Let K be a nontrivial equational class of lattices in which every lattice
can be embedded in an injective lattice. Show that K = D (A. Day [a]).
(Hint: Use exercise 15.15.)
18. Describe FB(B(1). (It has seven elements.)
19. Show that Theorem 5 does not generalize to B2.
20. Show that (S4) is characteristic of the Stone spaces of Stone lattices.
21. Characterize the Stone spaces of L e Bn.

190 DISTRIBUTIVE LATTICES WITH PSEUDOCOMPLEMENTATION
♦22. Show that FBw(/i) £ FB2„(«). Apply this to get a new proof of exercise
16.21 (G. Gratzer and H. Lakser).
Exercises 23-26 are from R. Balbes and G. Gratzer [a].
23. Let A and B be Stone algebras. Show that A x B is projective iff both A
and B are projective.
24. Let A be a projective Stone algebra satisfying S(A) = {0, 1}. Show that
A0 = A - {0, 1} is a sublattice of /* and that ^0 is projective in D.
25. Prove the converse of exercise 24.
26. Characterize the finite projective Stone algebras.
27. Let A be a finite Stone algebra. Why is it not true that A is projective
iff S(A) is projective in B and D(A) is projective in D ?
Further Topics and References
Except for V. Glivenko's early work [1929], the study of pseudocomple-
mented distributive lattices started only in 1956 with a solution to Problem
70 of G. Birkhoff [1948], which gave a characterization of Stone lattices by
minimal prime ideals (G. Gratzer and E. T. Schmidt [1957b]; for a
simplified proof see J. Varlet [1966]). The idea of a triple was conceived by
the author in 1961 as a method of proving O. Frink's conjecture on the
representation of Stone lattices (see exercises 16.18). His attempt to prove
the conjecture failed, however, and as a result triples were not utilized until
1968 (C. C. Chen and G. Gratzer [1969b] and [1969c]). In the sixties the
structure of Stone lattices was the subject of extensive research. The
following list—which is far from complete—may prove of interest to the
reader: G. Bruns [1965] (simplifying the Representation Theorem of
G. Gratzer [1963]—an even simpler proof is in G. Gratzer [1969]), R. Balbes
and G. Gratzer [a] (characterizing finite projective Stone algebras), R. Balbes
and A. Horn [1970a], O. Frink [1962], T. Katrinak [1967], H. Lakser
[1970], T. Speed [1969], and J. Varlet [1963], [1966].
Originally, Chapter 3 of the present text gave a survey of this
development. However, the discovery of the classes Bn (K. B. Lee [1970]) and the
subsequent realization that many of the known facts about Stone lattices
generalize to some or all the Bn made it necessary to revise this chapter.
H. Lakser [a] generalized the Representation Theorem of G. Gratzer [1963];
the description of injectives for Stone algebras is generalized to B2 in G.
Gratzer and H. Lakser [b]. There are no non-Boolean injectives in Bn for
n > 2 (A. Day [1970]), but the description generalizes to give all absolute
retracts for n > 2 (which is the same as injective if every algebra can be

Further Topics and References
191
embedded in an injective). We are, of course, only at the very beginning of
this development.
For a class of algebras K, the concept of the Amalgamation Class ofK, in
notation Amal (K), can be introduced: For A e K we have A e Amal (K)
iff, for all B,CgK, with A a subalgebra of B and C, the amalgamation of
A, B, and Ccan be effected in K in the sense of Definition 13.17. Of course,
the Amalgamation Property means that Amal (K) = K. Since the
Amalgamation Property fails for Bn, 2 < n < a>, Amal (Bn) is investigated in G.
Gratzer and H. Lakser [a] to determine the extent to which Bn fails to
satisfy the Amalgamation Property.
Relatively pseudocomplemented distributive lattices (often called
Heyting algebras) arise from nonclassical logic and were first investigated
by T. Skolem about 1920. For a detailed development, see H. B. Curry
[1963], which contains all the important rules of computation with a * b.
Unfortunately, the triple method is not applicable here. Recall that the
effectiveness of the triple method is due to the fact that the class of Boolean
algebras and the class of distributive lattices with unit are both classes
with structures simpler than that of the class of distributive lattices with
pseudocomplementation. Unfortunately, if L is relatively
pseudocomplemented, so is D(L), and consequently there is no reduction. Using
equivalence classes of ordered pairs, <a, b}, a e D(L), b e S(L), W. C.
Nemitz [1965] developed a Construction Theorem for relatively
pseudocomplemented distributive lattices. It should also be mentioned that
injectives and projectives were investigated in the work of R. Balbes and A.
Horn [1970b] and A. Day [1970]; see also A. Horn [1969a] and [1969b].
Pseudocomplemented distributive semilattices were investigated by
T. Katrinak [1970]; unfortunately, the Construction Theorem uses
equivalence classes of ordered pairs {a, by, a e D(L), b e S(L).
In connection with nonclassical logic, many algebras emerged that are
distributive lattices endowed with some additional structure. The best
known of these, the Post algebras, happen to be Stone lattices. In fact, a
Post algebra is a free {0, l}-distributive product of a finite chain and a
Boolean algebra, in which the elements of the chain are regarded as
nullary operations; see E. L. Post [1921], P. C. Rosenbloom [1942], G.
Epstein [1960], T. Traczyk [1963] and [1967], C. C. Chang and A. Horn
[1961], R. Balbes and Ph. Dwinger [a] and [b], and M. Mandelker [1970].
For some other types of distributive lattices with additional operations,
see N. D. Belnap and J. H. Spencer [1966], J. M. Dunn and N. D. Belnap
[1968], H. Rasiowa and R. Sikorski [1963], and J. Varlet [1968].

192 DISTRIBUTIVE LATTICES WITH PSEUDOCOMPLEMENTATION
PROBLEMS
53. Describe the projective Stone algebras.
54. Describe the triples associated with free Stone algebras.
55. Find a direct (less-computational) proof of the Construction Theorem
for Stone Algebras.
56. Determine the free algebras FBk(«).
57. Let B be a Boolean lattice, let D be a distributive lattice with unit, and
let A be a sublattice of C(B) x C(D). Under what conditions does there
exist a distributive lattice with pseudocomplementation L such that
S(L) = B, D(L) = D, and A consists of all congruence pairs of L?
58. Characterize free Byproducts (n > 1).
59. Describe Amal (Bn) for n > 2.
60. Determine the projectives in Bn, n > 1.
61. Let n > 0. Determine all injective structures in the sense of J. M.
Maranda [1964] in the category Bn.
62. For every identity / for distributive lattices with pseudocomplementation,
there exists a first-order sentence $(/) such that / holds for L iff $(/)
holds for ^(L). (In K. B. Lee [1970], the sentence for (Ln) is given as
follows: "Every element contains at most n minimal elements." For
n = 0 use Theorem 7.22.) Is there a natural class of first-order sentences
properly containing all identities for which the same statement can be
made?
63. Show that there are 2K<> implicational classes of distributive lattices with
pseudocomplementation.
64. Investigate the lattice of implicational classes of distributive lattices with
pseudocomplementation.
65. Show that there are 2Ro equational classes of lattices with pseudo-
complementati on.
66. For a distributive lattice L, define the number n(L) as follows: Let n(L)
be the smallest integer n such that I(L) e B„ if L has a zero and /0(L) e Bn
if L does not have a zero. Classify and investigate distributive lattices
according to the value of n(L). (The case n = 1 was considered by
M. Mandelker [1970].)
67. Let K be an equational class of algebras satisfying HS(Ki) = ISH(Ki)
for all Kx c K. Does K satisfy the Congruence Extension Property?
Does the assumption that the congruence lattices of algebras in K are
distributive make any difference (see exercise 15.18)?

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1963. The mathematics of metamathematics. Monog. Mat. Vol. XLI. Warsaw:
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1951. The theory of lattices. Cambridge, England: Forster and Jagg.
I. Reznikoff
1963. Chaines de formules. C. R. Acad. Sci. Paris 256:5021-5023.
P. Ribenboim
1949. Characterization of the sup-complement in a distributive lattice with last
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1958. To the axiomatics of modular lattices (in Slovak). Acta Fac. Rerum Natur.
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L. Rieger
1949. A note on topological representations of distributive lattices. Casopis P€st.
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1951. Some remarks on automorphisms of Boolean algebras. Fund. Math. 38:209-
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H. L. Rolf
1958. The free lattice generated by a set of chains. Pacific J. Math. 8:585-
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P. C. Rosenbloom
1942. Post algebras I. Postulates and general theory. Amer. J. Math. 64:167-
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G. C. Rota
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S. Rudeanu
1963. Axioms for lattices and Boolean algebras (in Roumanian). Bucharest:
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D. Sachs
1962. The lattice of subalgebras of a Boolean algebra. Canad. J. Math. 14:451-
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Y. Sampei
1953. On lattice completions and closure operations. Comment. Math. Univ. St.
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1962. Uber die Kongruenzverbande der Verbande. Publ. Math. Debrecen 9:243-
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1968. Zur Charakterisierung der Kongruenzverbande der Verbande. Mat. Casopis
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L. N. Sevrin
1964. Projectivities of semi-lattices (in Russian). Dokl. Akad. Nauk SSSR. 154:
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204
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R. Sikorski
1964. Boolean algebras. 2nd ed. Ergebnisse der Mathematik und ihrer Grenz-
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F. M. Sioson
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A. K. Steiner
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1936. The theory of representations for Boolean algebras. Trans. Amer. Math.
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INDEX
Abian, S., 64
Absorption identity, 5
Aigner, M., 11
Alexander, J. W., 126
Algebra
Boolean, 58
generalized Boolean, 78
Hey ting, 191
partial, 54
Post, 191
simple, 181
Stone, 161
See also Universal algebra, 62
Algebraic function, 86
Boolean, 86
Algebraic lattice, 93
Altwegg, ML, 62
Amalgamation class, 191
Amalgamation Property, 147
Strong, 150
Anderson, F. W., 151
Anderson, L. W., 65
Areskin, G. Ja., 100, 138
Ascending Chain Condition, 15
Associativity, 5
Atom, 59
dual, 60
Atomic Boolean polynomial, 82
Augmented ideal lattice, 22
Automorphism, 64
Avann, S. P., 154
Balbes, R., 61, 65, 144, 153, 154, 170, 1
190, 191
Banaschewski, B., 175, 188
Base, 125
Basis, 56
Behrens, E. A., 155
Belnap, N. D., 141, 191
Betweenness relation, 62
Binary bracketing, 17
Binary operation, 4, 62
Binary relation, 2
transitive extension of, 10
BirkhofF, G., vii, viii, 3, 53, 64, 65, 75,146,
175, 176, 177, 182, 190
Blair, R. L., 151
Block, 10
Boole, G., vii
Boolean algebra, 58
generalized, 78
Boolean algebraic function, 86
Boolean lattice, 58
generalized, 91
7?-generated by a chain, 109
7?-generated by a distributive lattice, 115
Boolean polynomial, 81
atomic, 82
Boolean ring, 92
Boolean space, 123
generalized, 127
Booleanization, 124
Bound
greatest lower, 3
least upper, 2
lower, 3
upper, 2
Bounded poset, 56
Bracketing
binary, 17
right, 17
Bracketing function, 17
Brown, A. B., 64
Bruns, G., 153, 190

208
Index
Buchi, J. R., 115, 152
Buttafuoco, E., 154
Byrne, L., 63
Campbell, A. D., 53
Chain, 2
length of, 73
maximal, 79
strongly maximal, 114
Chain Condition
Ascending, 15
Descending, 60
m-, 159
Chang, C. C, 153, 191
Chen, C C, 67, 163, 165, 175, 190
Church, R., 80
Class
amalgamation, 191
implicational, 189
Clopen set, 123
Closed set, 125
Closure, 125
Cohen, I. S., 155
Commutative diagram, 39
Commutativity, 5
Compact element, 93
Compact set, 126
Compact space, 126
locally, 127
Compactly generated lattice, 93
Comparability relation, 10
Comparable elements, 2
Complement, 57
pseudo-, 58
relative, 57
Complemented lattice, 58
relatively, 58
sectionally, 100
Complete Infinite Distributive Identity,
116
Complete lattice, 29
conditionally, 31
Completely meet-irreducible, 182
Completion, 153
MacNeille, 116, 153
Complexity, 66
Conditionally complete lattice, 31
Congruence class, 24
Congruence extension property, 173
Congruence lattice, 25
Congruence pair, 171
Congruence relation, 24, 67, 96, 160
principal, 87
Congruence relations, permuting, 79
Congruence Substitution Property, 151
Continuous map, 126
Convex, 20
Convex sublattice generated by H, 21
Cotlar, M., 153
Covering relation, 12
Crapo, H., 3
Crawley, P., 151, 153
Curry, H. B., 191
Curzio, M., 63
Daigneault, A., 154
Davis, A. C, 64
Day, A., 189, 190, 191
Dean, R. A., 65, 128
Dedekind, R., vii, 46, 80, 154
Demarr, R., 64
De Morgan's identities, 57
Dense element, 162
Dense lattice, 161, 164
Dense set, 162
Descending Chain Condition, 60
Devide, V., 64
Diagram, 13, 39
commutative, 39
optimal, 13
planar, 13
Diamond, A. H., 62, 66
Dilworth, R. P., 63, 67, 95, 151
Direct power, 28
Direct product, 27, 28
Discrete topological space, 135
Distributive identity, 35, 36
complete infinite, 116
Distributive inequality, 35
Distributive join-semilattice, 117
Distributive lattice 7?-generated by a chain,
109
Dual, 3
Dual atom, 60
Dual ideal, 23, 50, 119
generated by //, 23
prime, 23
principal, 23
proper, 23
Dual pseudocomplement, 184
Duality Principle, 3, 7
Dunn, J. M., 191
Dwinger, Ph., 191
Embedding, 19, 51
Endomorphism, 64, 143
{0,1}-, 64
Engelking, R., 154
Epi, 140
Epimorphism, 141
Epstein, G., 191
Equational class of lattices, 38
Equivalence relation, 24
Essential extension, 188

Index
209
Field of sets, 76
Filippov, N. D., 63
Filter, 23
prime, 23
principal, 23
ultra, 23
Finitely presented lattice, 66
Free distributive product, 128
{0,1}-, 129
Free product
distributive, 128
K-, 128
{0,l}-distributive, 129
Free K-product, 128
Free lattice
generated by P, 40
on m generators, 40
over K generated by 21, 52
over K generated by P, 39
Free {0,1 {-distributive product, 129
Freidman, P. A., 155
Friedman, H., 17
Frink, O., 59, 65, 180, 190
Frucht, R., 64
Fuchs, L., 155
Fully ordered set, 2
Funayama, N., 48, 51, 93, 107, 109, 153
Function
algebraic, 86
Boolean algebraic, 86
Galvin, F., 153
Generalized Boolean algebra, 78
Generalized Boolean lattice, 91
Generalized Boolean space, 127
Generating set, 20
Ghouila-Houri, A., 11
Gilbert, N., 80
Gillman, L., 65
Gilmore, P. C, 11
Glivenko, V., vii, 58, 107, 190
Gluhov, M. M., 56
Graph, 63
Greatest lower bound, 3
Hajnal, A., 139
Half-open interval, 20
Halmos, P. R., 64, 153
Hansel, G., 80
Harrison, M. A., 64
Hashimoto, F., 77, 79, 91, 100, 102, 10<
151
Hausdorff space, 126
Havas, G., 63, 67
Hereditary, 52, 72
Herrmann, C, 56
Hey ting algebra, 191
Higgs, D., 30
Hoffman, A. J., 11
Homeomorphism, 126
Homomorphic image, 26
maximal, 56
Homomorphism, 51, 55, 58, 160
join-, 19
meet-, 19
{0}-, 57
(0,1)-, 56
Homomorphism Theorem, The, 26
Horn, A., 61, 65,153,154,188,190,191
Huang, S., 17
Hull, injective, 188
Huntington, E. V., vii, 62
Ideal, 20, 50
dual, 23,50, 119
dual principal, 23
generated by //, 21
maximal prime, 79
minimal prime, 79, 169
normal, 116
prime, 21, 119
prime dual, 23
principal, 21
proper, 21
proper dual, 23
standard, 152
Ideal kernel, 31
Ideal lattice, 22
augmented, 22
Idempotency, 5
Identity, 33
De Morgan's, 57
distributive, 35, 36
modular, 36
Stone, 161
Implicational class, 184
Incomparable elements, 2
Inequality, 33
distributive, 35
modular, 35
Infimum, 3
Injective, 145
Injective hull, 188
Injective Stone algebra, 183
Injective structure, 192
Interior, 187
Interval, 20
half-open, 20
open, 20
prime. 111
Intrinsic topology, 65
Irreducible, subdirectly, 176
Irredundant representation, 73
Isomorphism, 17, 18, 51
Isotone map, 19
Iwamura, T., 152

210
Index
Jakubik, J., 63, 153
Jensen, Ch. U., 155
Jerison, M., 65
Join, 4
Join-accessible element, 152
Join-continuous lattice, 152
Join-homomorphism, 19
Join-inaccessible element, 152
Join-irreducible, 60
Join- (v-) reduct, 133
Join- (V-) representation, 133
normal, 134
Join-semilattice, 8
distributive, 117
modular, 127
Join table, 11
Jonsson, B., 132, 147, 148, 153, 154, 155,
157
Kalman, J. A., 10
Kaplansky, I., 63, 65
Kappos, D. A., 153
Katetov, ML, 154
Katrindk, T., 190, 191
Kernel, ideal, 31
Kinugawa, S., 151
Kleitman, D., 80
Kogalovskii, S. R., 64
Koh, K. M., 166
Kolibiar, M., 62, 63
Komatu, A., 152
Lakser, H., 61, 65, 95, 99, 101, 131, 138,
154,156,167, 171,173, 174, 175, 178,
179,180,186,188,189,190,191
Lattice, 3, 4, 6
algebraic, 93
augmented ideal, 22
Boolean, 58
compactly generated, 93
complemented, 58
complete, 29
conditionally complete, 31
congruence, 25
dense, 161, 164
distributive, 36
finitely presented, 66
generalized Boolean, 91
ideal, 22
join-continuous, 152
modular, 36
partial, 48
partial weak, 49
pseudocomplemented, 58, 159
quotient, 26
/^-generated Boolean, 115
relative Stone, 175
relatively complemented, 57
sectionally complemented, 100
Stone, 161
topological, 65
Lattice completion, 153
Lattice polynomial, 32
Least upper bound, 2
Lee, K. B., 167, 170, 178, 179, 182, 183,
190, 192
Length of a chain, 73
Linearly ordered set, 2
Locally compact space, 127
Lower bound, 3
greatest, 3
Lowig, H. F. J., 63
m-chain condition, 139
McKenzie, R. N., 10, 38, 62, 63, 64, 66
McKinsey, J. C. C, 62, 66
McLaughlin, J. E., 153
MacNeille, H. M., 102, 116, 153
MacNeille completion, 116
Mandelker, ML, 191, 192
Map, 139
Maranda, J. M., 192
Maximal chain, 79
strongly, 114
Maximal element, 15
Maximal homomorphic image, 56
Maximal prime ideal, 79
Meet, 4
Meet-homomorphism, 19
Meet-irreducible, 60
completely, 182
Meet-polynomial, 80
Meet-(A-) reduct, 133
Meet- (A-) representation, 133
normal, 134
Meet-semi lattice, 8
relatively pseudocomplemented, 61
Meet table, ii
Menger, K., vii
Michler, G., 155
Minimal element, 15
Minimal prime ideal, 79, 169
Mitchell, B., 154
Modular identity, 36
Modular inequality, 35
Modular join-semilattice, 127
Mono, 140
Monoid, 64
Monomorphism, 140
Monotone map, 19
Monteiro, A., 64, 156
Morley, M. D., 154
Morphism, 139
Mostowski, M., 109, 115
Musti, R., 154

Index
211
w-ary operation, 61
Nachbin, L., 76, 152
Nakayama, T., 93
Nemitz, W. C, 191
Nerode, A., 129
Neumann, J. von, vii, 107, 151
Noether, E., 155
Normal form, 133
Normal ideal, 116
Normal join- (V-) representation, 134
Normal meet- (A-) representation, 134
Nullary operation, 62
Object, 139
projective, 143
Oehmke, R. H., 65, 128
Open interval, 20
Open map, 126
Open set, 125
Operation
binary, 4, 62
H-ary, 61
nullary, 62
partial, 48
unary, 62
Optimal diagram, 13
Ore, O., vii, 154, 155
Padmanabhan, R., xi, xii, 10, 38, 62, 63
Papangelou, F., 153
Papert, D., 65, 128
Partial algebra, 54
Partial lattice, 48
weak, 49
Partial operation, 48
Partial ordering relation, 2
Partially ordered set, 2
Peirce, C. S., vii
Pelczar, A., 64
Permuting congruence relations, 79
Permutti, R., 65
Pierce, R. S., 148, 153
Planar diagram, 13
Piatt, C. R., 154, 156
Polynomial, 32
atomic Boolean, 82
binary, 32
Boolean, 81
meet-, 80
rank of, 32
unary, 32
Polynomial function, 31
Poset, 2
bounded, 56
Post, E. L., 191
Post algebra, 191
Potts, D. H., 63
Prime dual ideal, 23
Prime filter, 23
Prime ideal, 21, 119
maximal, 79
minimal, 79, 169
Prime interval, 111
Principal congruence relation, 87
Principal dual ideal, 23
Principal filter, 23
Principal ideal, 21
Product, direct, 27, 28
Product topology, 126
Projective object, 143
Proper dual ideal, 23
Proper ideal, 21
Proper sublattice, 79
Pseudocomplement, 58
dual, 184
relative, 61
Pseudocomplemented lattice, 58, 159
Pseudocomplemented semilattice, 58
Quasi-ordering relation, 10
Quotient lattice, 26
/^-generated Boolean lattice, 115
^-generated sublattice, 102
/^-generation, 109
Raney, G. N., 153
Rank of countable Boolean algebra, 112
Rank of polynomial, 32
Rasiowa, H., 191
Reduced choice function, 134
Reduced set, 133
Redundant representation, 73
Regular cardinal, 139
Relation
betweenness, 62
binary, 2
congruence, 24, 67, 96, 160
equivalence, 24
partial ordering, 2
principal congruence, 87
quasi-ordering, 10
Relative complement, 57
Relative pseudocomplement, 61
Relative Stone lattice, 175
Relative sublattice, 38
Relatively complemented lattice, 57
Relatively pseudocomplemented meet-
semilattice, 61
Rennie, B. C, 65
Representation
irredundant, 73
redundant, 73
subdirect, 177
Retract, 143
Retraction, 143
Reznikoff, I., 153

212
Index
Ribenboim, P., 64, 179
Riecan, J., 62
Rieger, L., 76, 117, 154
Right bracketing, 17
Ring, Boolean, 92
Ring of sets, 73
Rolf, H. L., 17
Rosenbloom, P. C, 191
Rota, G. C, 3
Rudeanu, S., 63
Sachs, D., 63
Sampei, Y., 153
Schmidt, E. T., 24, 58, 59, 64, 65, 67, 79,
87,91,95,99,100, 101, 102,115,116,
117,152,162, 170, 174, 190
Schroder, E., vii
Second Isomorphism Theorem, 172
Sectionally complemented lattice, 100
Semiassociative identity, 18
Semilattice, 9
distributive join-, 117
join-, 8
meet-, 8
modular join-, 127
pseudocomplemented, 58
Sevrin, L. N., 63
Sholander, M., 62
Sichler, J., 64, 66
Sikorski, R., 151, 153, 191
Simple algebra, 181
Singular cardinal, 139
Sioson, F. M., 63
Skeletal element, 161
Skeleton, 161
Skolem, T., 191
Sorkin, Ju. I., 17, 62
Speed, T. P., 175, 190
Spencer, J. H., 191
Standard ideal, 152
♦-homomorphism, 160
Steiner, A. K., 65
Stone, M. H., 74, 75, 76, 92, 117, 123
Stone algebra, 161
injective, 183
Stone identity, 161
Stone lattice, 161
relative, 175
Stone space, 119
Strong Amalgamation Property, 150
Strongly maximal chain, 114
Subalgebra, 58,160
generated by //, 138
Subbase, 125
Subdirect representation, 177
Subdirectly irreducible, 176
Sublattice, 20
generated by /f, 20
proper, 79
/^-generated, 102
relative, 48
(0,1}-, 56
Substitution Property, 24
Supremum, 2
Symmetric difference, 84
To-space, 126
Ti-space, 126
T2-space, 126
Takeuchi, K., 79
Tamari, D., 17
Tarski, A., 62, 63, 64, 109, 115, 116
Tihonov's Theorem, 126
Topological lattice, 65
Topological space, 125
discrete, 135
Topology, intrinsic, 65
Totally disconnected space, 123, 126
Traczyk, T., 191
Transitive extension of binary relation, 10
Triple, 163
Trivial equational class, 38
Type of algebra, 137
Ultra filter, 23
Unary operation, 62
Unit, 56
Universal albebra, 62
type of, 137
Upper bound, 2
Vaidyanathaswamy, R., 65
Varlet, J., 65, 190, 191
Vaught, R. L., 154
Wallace, A. D., 65
Wallman, H., 65
Ward, M., 63, 67
Weak partial lattice, 49
Werner, H., 155
Whitman, P. M., 64
Wille, R., 65, 155
Wolk, B., 67, 144
Wolk, E. S., 64
Zacher, G., 65
Zero, 56
(0)-homomorphism, 57
{0,l}-endomorphism, 56
(0,1)-homomorphism, 56
{0,1}-sublattice, 56
Zorn's Lemma, 74

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radium chloride; determination of atomic weight of radium; plus electric, photographic,
luminous, heat, color effects of radioactivity. ii+94pp. 5% x81 0-486-42550-9
CHEMICAL MAGIC, Leonard A. Ford. Second Edition, Revised by E. Winston
Grundmeier. Over 100 unusual stunts demonstrating cold fire, dust explosions,
much more. Text explains scientific principles and stresses safety precautions.
128pp. 5% x 81/;. 0-486-67628-5
THE DEVELOPMENT OF MODERN CHEMISTRY, Aaron J. Ihde. Authoritative
history of chemistry from ancient Greek theory to 20th-century innovation. Covers
major chemists and their discoveries. 209 illustrations. 14 tables. Bibliographies.
Indices. Appendices. 851pp. 5% x 8%. 0-486-64235-6
CATALYSIS IN CHEMISTRY AND ENZYMOLOGY, William P. Jencks.
Exceptionally clear coverage of mechanisms for catalysis, forces in aqueous solution,
carbonyl- and acyl-group reactions, practical kinetics, more. 864pp. 5% x 8%.
0-486-65460-5
ELEMENTS OF CHEMISTRY, Antoine Lavoisier. Monumental classic by founder
of modern chemistry in remarkable reprint of rare 1790 Kerr translation. A must for
every student of chemistry or the history of science. 539pp. 5% x 8'/>. 0-486-64624-6
THE HISTORICAL BACKGROUND OF CHEMISTRY, Henry M. Leicester.
Evolution of ideas, not individual biography. Concentrates on formulation of a
coherent set of chemical laws. 260pp. 5% x 8%. 0-486-61053-5
A SHORT HISTORY OF CHEMISTRY, J. R. Partington. Classic exposition
explores origins of chemistry, alchemy, early medical chemistry, nature of
atmosphere, theory of valency, laws and structure of atomic theory, much more. 428pp.
5*x8fc. (Available in U.S. only.) 0-486-65977-1
GENERAL CHEMISTRY, Linus Pauling. Revised 3rd edition of classic first-year
text by Nobel laureate. Atomic and molecular structure, quantum mechanics,
statistical mechanics, thermodynamics correlated with descriptive chemistry. Problems.
992pp. 5% x 8%. 0-486-65622-5
FROM ALCHEMY TO CHEMISTRY, John Read. Broad, humanistic treatment
focuses on great figures of chemistry and ideas that revolutionized the science. 50
illustrations. 240pp. 5% x 8fc. 0-486-28690-8

CATALOG OF DOVER BOOKS
Engineering
DE RE METALLICA, Georgius Agricola. The famous Hoover translation of greatest
treatise on technological chemistry, engineering, geology, mining of early modern
times (1556). All 289 original woodcuts. 638pp. 6% x 11. 0-486-60006-8
FUNDAMENTALS OF ASTRODYNAMICS, Roger Bate et al. Modern approach
developed by U.S. Air Force Academy. Designed as a first course. Problems,
exercises. Numerous illustrations. 455pp. 5% x SlL 0-486-60061-0
DYNAMICS OF FLUIDS IN POROUS MEDIA, Jacob Bear. For advanced
students of ground water hydrology, soil mechanics and physics, drainage and irrigation
engineering and more. 335 illustrations. Exercises, with answers. 784pp. 6% x 974.
0-486-65675-6
THEORY OF VISCOELASTICITY (Second Edition), Richard M. Christensen.
Complete consistent description of the linear theory of the viscoelastic behavior of
materials. Problem-solving techniques discussed. 1982 edition. 29 figures. xiv+364pp.
6K x 9%. 0-486-42880-X
MECHANICS, J. P. Den Hartog. A classic introductory text or refresher. Hundreds
of applications and design problems illuminate fundamentals of trusses, loaded
beams and cables, etc. 334 answered problems. 462pp. 5% x 8%. 0-486-60754-2
MECHANICAL VIBRATIONS, J. P. Den Hartog. Classic textbook offers lucid
explanations and illustrative models, applying theories of vibrations to a variety of
practical industrial engineering problems. Numerous figures. 233 problems,
solutions. Appendix. Index. Preface. 436pp. 5% x 8%. 0-486-64785-4
STRENGTH OF MATERIALS, J. P. Den Hartog. Full, clear treatment of basic
material (tension, torsion, bending, etc.) plus advanced material on engineering
methods, applications. 350 answered problems. 323pp. 5% x 8%. 0-486-60755-0
A HISTORY OF MECHANICS, Rene Dugas. Monumental study of mechanical
principles from antiquity to quantum mechanics. Contributions of ancient Greeks,
Galileo, Leonardo, Kepler, Lagrange, many others. 671pp. 5% x 8%. 0-486-65632-2
STABILITY THEORY AND ITS APPLICATIONS TO STRUCTURAL
MECHANICS, Clive L. Dym. Self-contained text focuses on Koiter postbuckling
analyses, with mathematical notions of stability of motion. Basing minimum energy
principles for static stability upon dynamic concepts of stability of motion, it develops
asymptotic buckling and postbuckling analyses from potential energy considerations,
with applications to columns, plates, and arches. 1974 ed. 208pp. 5% x 8%.
0-486-42541-X
METAL FATIGUE, N. E. Frost, K. J. Marsh, and L. P. Pook. Definitive, clearly
written, and well-illustrated volume addresses all aspects of the subject, from the
historical development of understanding metal fatigue to vital concepts of the cyclic stress
that causes a crack to grow. Includes 7 appendixes. 544pp. 5% x SlL 0-486-40927-9

CATALOG OF DOVER BOOKS
ROCKETS, Robert Goddard. Two of the most significant publications in the history
of rocketry and jet propulsion: "A Method of Reaching Extreme Altitudes" (1919) and
"Liquid Propellant Rocket Development" (1936). 128pp. 5% x 81/;. 0-486-42537-1
STATISTICAL MECHANICS: PRINCIPLES AND APPLICATIONS, Terrell L.
Hill. Standard text covers fundamentals of statistical mechanics, applications to
fluctuation theory, imperfect gases, distribution functions, more. 448pp. 5% x 8%.
0-486-65390-0
ENGINEERING AND TECHNOLOGY 1650-1750: ILLUSTRATIONS AND
TEXTS FROM ORIGINAL SOURCES, Martin Jensen. Highly readable text with
more than 200 contemporary drawings and detailed engravings of engineering
projects dealing with surveying, leveling, materials, hand tools, lifting equipment,
transport and erection, piling, bailing, water supply, hydraulic engineering, and more.
Among the specific projects outlined-transporting a 50-ton stone to the Louvre,
erecting an obelisk, building timber locks, and dredging canals. 207pp. 8% x 11%.
0-486-42232-1
THE VARIATIONAL PRINCIPLES OF MECHANICS, Cornelius Lanczos.
Graduate level coverage of calculus of variations, equations of motion, relativistic
mechanics, more. First inexpensive paperbound edition of classic treatise. Index.
Bibliography. 418pp. 5% x 8%. 0-486-65067-7
PROTECTION OF ELECTRONIC CIRCUITS FROM OVERVOLTAGES,
Ronald B. Standler. Five-part treatment presents practical rules and strategies for
circuits designed to protect electronic systems from damage by transient overvoltages.
1989 ed. xxiv+434pp. 6K x M. 0-486-42552-5
ROTARY WING AERODYNAMICS, W. Z. Stepniewski. Clear, concise text
covers aerodynamic phenomena of the rotor and offers guidelines for helicopter
performance evaluation. Originally prepared for NASA. 537 figures. 640pp. 6ll x 9lL
0-486-64647-5
INTRODUCTION TO SPACE DYNAMICS, William Tyrrell Thomson.
Comprehensive, classic introduction to space-flight engineering for advanced
undergraduate and graduate students. Includes vector algebra, kinematics, transformation of
coordinates. Bibliography. Index. 352pp. 5% x 8%. 0-486-65113-4
HISTORY OF STRENGTH OF MATERIALS, Stephen P. Timoshenko. Excellent
historical survey of the strength of materials with many references to the theories of
elasticity and structure. 245 figures. 452pp. 5% x 8%. 0-486-61187-6
ANALYTICAL FRACTURE MECHANICS, David J. Unger. Self-contained text
supplements standard fracture mechanics texts by focusing on analytical methods for
determining crack-tip stress and strain fields. 336pp. 6!6 x 9ft. 0-486-41737-9
STATISTICAL MECHANICS OF ELASTICITY, J. H. Weiner. Advanced, self-
contained treatment illustrates general principles and elastic behavior of solids. Part
1, based on classical mechanics, studies thermoelastic behavior of crystalline and
polymeric solids. Part 2, based on quantum mechanics, focuses on interatomic force
laws, behavior of solids, and thermally activated processes. For students of physics
and chemistry and for polymer physicists. 1983 ed. 96 figures. 496pp. 5% x 8%.
0-486-42260-7

CATALOG OF DOVER BOOKS
Mathematics
FUNCTIONAL ANALYSIS (Second Corrected Edition), George Bachman and
Lawrence Narici. Excellent treatment of subject geared toward students with
background in linear algebra, advanced calculus, physics and engineering. Text covers
introduction to inner-product spaces, normed, metric spaces, and topological spaces;
complete orthonormal sets, the Hahn-Banach Theorem and its consequences, and
many other related subjects. 1966 ed. 544pp. 6% x 9tt. 0-486-40251-7
ASYMPTOTIC EXPANSIONS OF INTEGRALS, Norman Bleistein & Richard A.
Handelsman. Best introduction to important field with applications in a variety of
scientific disciplines. New preface. Problems. Diagrams. Tables. Bibliography. Index.
448pp. 5% x 8fc. 0-486-65082-0
VECTOR AND TENSOR ANALYSIS WITH APPLICATIONS, A. I. Borisenko
and I. E. Tarapov. Concise introduction. Worked-out problems, solutions, exercises.
257pp. 5'% x 81 0-486-63833-2
AN INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS, Earl
A. Coddington. A thorough and systematic first course in elementary differential
equations for undergraduates in mathematics and science, with many exercises and
problems (with answers). Index. 304pp. 5% x 8%. 0-486-65942-9
FOURIER SERIES AND ORTHOGONAL FUNCTIONS, Harry F. Davis. An
incisive text combining theory and practical example to introduce Fourier series,
orthogonal functions and applications of the Fourier method to boundary-value
problems. 570 exercises. Answers and notes. 416pp. 5% x 8%. 0-486-65973-9
COMPUTABILITY AND UNSOLVABILITY, Martin Davis. Classic graduate-level
introduction to theory of computability, usually referred to as theory of recurrent
functions. New preface and appendix. 288pp. 5% x SlL 0-486-61471-9
ASYMPTOTIC METHODS IN ANALYSIS, N. G. de Bruijn. An inexpensive,
comprehensive guide to asymptotic methods-the pioneering work that teaches by
explaining worked examples in detail. Index. 224pp. 5% x 8% 0-486-64221-6
APPLIED COMPLEX VARIABLES, John W. Dettman. Step-by-step coverage of
fundamentals of analytic function theory-plus lucid exposition of five important
applications: Potential Theory; Ordinary Differential Equations; Fourier Transforms;
Laplace Transforms; Asymptotic Expansions. 66 figures. Exercises at chapter ends.
512pp. 5% x 8%. 0-486-64670-X
INTRODUCTION TO LINEAR ALGEBRA AND DIFFERENTIAL EQUATIONS,
John W. Dettman. Excellent text covers complex numbers, determinants,
orthonormal bases, Laplace transforms, much more. Exercises with solutions. Undergraduate
level. 416pp. 5% x 8%. 0-486-65191-6
RIEMANN'S ZETA FUNCTION, H. M. Edwards. Superb, high-level study of
landmark 1859 publication entitled "On the Number of Primes Less Than a Given
Magnitude" traces developments in mathematical theory that it inspired. xiv+315pp.
5%x8%. 0-486-41740-9

CATALOG OF DOVER BOOKS
CALCULUS OF VARIATIONS WITH APPLICATIONS, George M. Ewing.
Applications-oriented introduction to variational theory develops insight and
promotes understanding of specialized books, research papers. Suitable for advanced
undergraduate/graduate students as primary, supplementary text. 352pp. 5% x SlL
0-486-64856-7
COMPLEX VARIABLES, Francis J. Flanigan. Unusual approach, delaying complex
algebra till harmonic functions have been analyzed from real variable viewpoint.
Includes problems with answers. 364pp. 5% x 8^. 0-486-61388-7
AN INTRODUCTION TO THE CALCULUS OF VARIATIONS, Charles Fox.
Graduate-level text covers variations of an integral, isoperimetrical problems, least
action, special relativity, approximations, more. References. 279pp. 5% x 8lL
0-486-65499-0
COUNTEREXAMPLES IN ANALYSIS, Bernard R. Gelbaum and John M. H.
Olmsted. These counterexamples deal mostly with the part of analysis known as
"real variables." The first half covers the real number system, and the second half
encompasses higher dimensions. 1962 edition. xxiv+ 198pp. 5% x 8^. 0-486-42875-3
CATASTROPHE THEORY FOR SCIENTISTS AND ENGINEERS, Robert
Gilmore. Advanced-level treatment describes mathematics of theory grounded in
the work of Poincare, R. Thorn, other mathematicians. Also important applications
to problems in mathematics, physics, chemistry and engineering. 1981 edition.
References. 28 tables. 397 black-and-white illustrations, xvii + 666pp. 6% x 9lL
0-486-67539-4
INTRODUCTION TO DIFFERENCE EQUATIONS, Samuel Goldberg.
Exceptionally clear exposition of important discipline with applications to sociology,
psychology, economics. Many illustrative examples; over 250 problems. 260pp. 5%
x 8%. 0-486-65084-7
NUMERICAL METHODS FOR SCIENTISTS AND ENGINEERS, Richard
Hamming. Classic text stresses frequency approach in coverage of algorithms,
polynomial approximation, Fourier approximation, exponential approximation, other
topics. Revised and enlarged 2nd edition. 721pp. 5% x SlL 0-486-65241-6
INTRODUCTION TO NUMERICAL ANALYSIS (2nd Edition), F. B. Hilde-
brand. Classic, fundamental treatment covers computation, approximation,
interpolation, numerical differentiation and integration, other topics. 150 new problems.
669pp. 5% x 8fc. 0-486-65363-3
THREE PEARLS OF NUMBER THEORY, A. Y. Khinchin. Three compeUing
puzzles require proof of a basic law governing the world of numbers. Challenges
concern van der Waerden's theorem, the Landau-Schnirelmann hypothesis and Mann's
theorem, and a solution to Waring's problem. Solutions included. 64pp. 53/» x Slk
0-486-40026-3
THE PHILOSOPHY OF MATHEMATICS: AN INTRODUCTORY ESSAY,
Stephan Korner. Surveys the views of Plato, Aristotle, Leibniz & Kant
concerning propositions and theories of applied and pure mathematics. Introduction. Two
appendices. Index. 198pp. 5% x 8%. 0-486-25048-2

CATALOG OF DOVER BOOKS
INTRODUCTORY REAL ANALYSIS, A.N. Kolmogorov, S. V. Fomin. Translated
by Richard A. Silverman. Self-contained, evenly paced introduction to real and
functional analysis. Some 350 problems. 403pp. 5% x 81 0-486-61226-0
APPLIED ANALYSIS, Cornelius Lanczos. Classic work on analysis and design of
finite processes for approximating solution of analytical problems. Algebraic
equations, matrices, harmonic analysis, quadrature methods, much more. 559pp. 5% x 8%.
0-486-65656-X
AN INTRODUCTION TO ALGEBRAIC STRUCTURES, Joseph Landin. Superb
self-contained text covers "abstract algebra": sets and numbers, theory of groups,
theory of rings, much more. Numerous well-chosen examples, exercises. 247pp. 5%
x 81/;. 0-486-65940-2
QUALITATIVE THEORY OF DIFFERENTIAL EQUATIONS, V. V. Nemytskii
and V.V. Stepanov. Classic graduate-level text by two prominent Soviet
mathematicians covers classical differential equations as well as topological dynamics and
ergodic theory. Bibliographies. 523pp. 5% x SlL 0-486-65954-2
THEORY OF MATRICES, Sam Perlis. Outstanding text covering rank, nonsingu-
larity and inverses in connection with the development of canonical matrices under
the relation of equivalence, and without the intervention of determinants. Includes
exercises. 237pp. 5% x 8%. 0-486-66810-X
INTRODUCTION TO ANALYSIS, Maxwell Rosenlicht. Unusually clear,
accessible coverage of set theory, real number system, metric spaces, continuous functions,
Riemann integration, multiple integrals, more. Wide range of problems.
Undergraduate level. Bibliography. 254pp. 5% x 8fc. 0-486-65038-3
MODERN NONLINEAR EQUATIONS, Thomas L. Saaty. Emphasizes practical
solution of problems; covers seven types of equations. "... a welcome contribution
to the existing literature...."-Math Reviews. 490pp. 5% x SlL 0-486-64232-1
MATRICES AND LINEAR ALGEBRA, Hans Schneider and George Phillip
Barker. Basic textbook covers theory of matrices and its applications to systems of
linear equations and related topics such as determinants, eigenvalues and differential
equations. Numerous exercises. 432pp. 5% x 8%. 0-486-66014-1
LINEAR ALGEBRA, Georgi E. Shilov. Determinants, linear spaces, matrix
algebras, similar topics. For advanced undergraduates, graduates. Silverman translation.
387pp. 5% x 8%. 0-486-63518-X
ELEMENTS OF REAL ANALYSIS, David A. Sprecher. Classic text covers
fundamental concepts, real number system, point sets, functions of a real variable, Fourier
series, much more. Over 500 exercises. 352pp. 5% x SlL 0-486-65385-4
SET THEORY AND LOGIC, Robert R. Stoll. Lucid introduction to unified theory
of mathematical concepts. Set theory and logic seen as tools for conceptual
understanding of real number system. 496pp. 5% x 874. 0-486-63829-4

CATALOG OF DOVER BOOKS
TENSOR CALCULUS, J.L. Synge and A. Schild. Widely used introductory text
covers spaces and tensors, basic operations in Riemannian space, non-Riemannian
spaces, etc. 324pp. 5% x 874. 0-486-63612-7
ORDINARY DIFFERENTIAL EQUATIONS, Morris Tenenbaum and Harry
Pollard. Exhaustive survey of ordinary differential equations for undergraduates
in mathematics, engineering, science. Thorough analysis of theorems. Diagrams.
Bibliography. Index. 818pp. 5% x 81 0-486-64940-7
INTEGRAL EQUATIONS, F. G. Tricomi. Authoritative, well-written treatment
of extremely useful mathematical tool with wide applications. Volterra Equations,
Fredholm Equations, much more. Advanced undergraduate to graduate level.
Exercises. Bibliography. 238pp. 5% x 81 0-486-64828-1
FOURIER SERIES, Georgi P. Tolstov. Translated by Richard A. Silverman. A
valuable addition to the literature on the subject, moving clearly from subject to subject
and theorem to theorem. 107 problems, answers. 336pp. 5% x 8%. 0-486-63317-9
INTRODUCTION TO MATHEMATICAL THINKING, Friedrich Waismann.
Examinations of arithmetic, geometry, and theory of integers; rational and natural
numbers; complete induction; limit and point of accumulation; remarkable curves;
complex and hypercomplex numbers, more. 1959 ed. 27 figures. xii+260pp. 5% x 8%.
0-486-63317-9
POPULAR LECTURES ON MATHEMATICAL LOGIC, Hao Wang. Noted
logician's lucid treatment of historical developments, set theory, model theory, recursion
theory and constructivism, proof theory, more. 3 appendixes. Bibliography. 1981
edition, ix + 283pp. 5% x 81 0-486-67632-3
CALCULUS OF VARIATIONS, Robert Weinstock. Basic introduction covering
isoperimetric problems, theory of elasticity, quantum mechanics, electrostatics, etc.
Exercises throughout. 326pp. 5% x 81 0-486-63069-2
THE CONTINUUM: A CRITICAL EXAMINATION OF THE FOUNDATION
OF ANALYSIS, Hermann Weyl. Classic of 20th-century foundational research deals
with the conceptual problem posed by the continuum. 156pp. 5% x 81
0-486-67982-9
CHALLENGING MATHEMATICAL PROBLEMS WITH ELEMENTARY
SOLUTIONS, A. M. Yaglom and I. M. Yaglom. Over 170 challenging problems
on probability theory, combinatorial analysis, points and lines, topology, convex
polygons, many other topics. Solutions. Total of 445pp. 5% x 8%. Two-vol. set.
Vol. I: 0-486-65536-9 Vol. II: 0-486-65537-7
INTRODUCTION TO PARTIAL DIFFERENTIAL EQUATIONS WITH
APPLICATIONS, E. C. Zachmanoglou and Dale W. Thoe. Essentials of partial
differential equations applied to common problems in engineering and the physical
sciences. Problems and answers. 416pp. 5% x 81 0-486-65251-3
THE THEORY OF GROUPS, Hans J. Zassenhaus. Well-written graduate-level text
acquaints reader with group-theoretic methods and demonstrates their usefulness in
mathematics. Axioms, the calculus of complexes, homomorphic mapping, />-group
theory, more. 276pp. 5% x 81 0-486-40922-8

CATALOG OF DOVER BOOKS
HYDRODYNAMIC AND HYDROMAGNETIC STABILITY, S. Chandrasekhar.
Lucid examination of the Rayleigh-Benard problem; clear coverage of the theory of
instabilities causing convection. 704pp. 5% x 8%. 0-486-64071-X
INVESTIGATIONS ON THE THEORY OF THE BROWNIAN MOVEMENT,
Albert Einstein. Five papers (1905-8) investigating dynamics of Brownian motion
and evolving elementary theory. Notes by R. Furth. 122pp. 5% x M. 0-486-60304-0
THE PHYSICS OF WAVES, William C. Elmore and Mark A. Heald. Unique
overview of classical wave theory. Acoustics, optics, electromagnetic radiation, more.
Ideal as classroom text or for self-study. Problems. 477pp. 5% x 8%. 0-486-64926-1
GRAVITY, George Gamow. Distinguished physicist and teacher takes reader-
friendly look at three scientists whose work unlocked many of the mysteries behind
the laws of physics: Galileo, Newton, and Einstein. Most of the book focuses on
Newton's ideas, with a concluding chapter on post-Einsteinian speculations
concerning the relationship between gravity and other physical phenomena. 160pp. 5% x 8%.
0-486-42563-0
PHYSICAL PRINCIPLES OF THE QUANTUM THEORY, Werner Heisenberg.
Nobel Laureate discusses quantum theory, uncertainty, wave mechanics, work of
Dirac, Schroedinger, Compton, Wilson, Einstein, etc. 184pp. 5% x 8V2.0-486-60113-7
ATOMIC SPECTRA AND ATOMIC STRUCTURE, Gerhard Herzberg. One
of best introductions; especially for specialist in other fields. Treatment is physical
rather than mathematical. 80 illustrations. 257pp. 5% x SlL 0-486-60115-3
AN INTRODUCTION TO STATISTICAL THERMODYNAMICS, Terrell L.
Hill. Excellent basic text offers wide-ranging coverage of quantum statistical
mechanics, systems of interacting molecules, quantum statistics, more. 523pp. 5% x SlL
0-486-65242-4
THEORETICAL PHYSICS, Georg Joos, with Ira M. Freeman. Classic overview
covers essential math, mechanics, electromagnetic theory, thermodynamics,
quantum mechanics, nuclear physics, other topics. First paperback edition, xxiii + 885pp.
5% x 8'/i. 0-486-65227-0
PROBLEMS AND SOLUTIONS IN QUANTUM CHEMISTRY AND PHYSICS,
Charles S. Johnson, Jr. and Lee G. Pedersen. Unusually varied problems, detailed
solutions in coverage of quantum mechanics, wave mechanics, angular momentum,
molecular spectroscopy, more. 280 problems plus 139 supplementary exercises.
430pp. 6K x 9%. 0-486-65236-X
THEORETICAL SOLID STATE PHYSICS, Vol. 1: Perfect Lattices in Equilibrium;
Vol. II: Non-Equilibrium and Disorder, William Jones and Norman H. March.
Monumental reference work covers fundamental theory of equilibrium properties of
perfect crystalline solids, non-equilibrium properties, defects and disordered systems.
Appendices. Problems. Preface. Diagrams. Index. Bibliography. Total of 1,301pp. 5%
x 8'/i. Two volumes. Vol. I: 0-486-65015-4 Vol. II: 0-486-65016-2
WHAT IS RELATIVITY? L. D. Landau and G. B. Rumer. Written by a Nobel Prize
physicist and his distinguished colleague, this compelling book explains the special
theory of relativity to readers with no scientific background, using such familiar
objects as trains, rulers, and clocks. 1960 ed. vi+72pp. 5% x SlL 0-486-42806-0

CATALOG OF DOVER BOOKS
A TREATISE ON ELECTRICITY AND MAGNETISM, James Clerk Maxwell.
Important foundation work of modern physics. Brings to final form Maxwell's theory
of electromagnetism and rigorously derives his general equations of field theory.
1,084pp. 5% x 8%. Two-vol. set. Vol. I: 0-486-60636-8 Vol. II: 0-486-60637-6
QUANTUM MECHANICS: PRINCIPLES AND FORMALISM, Roy McWeeny.
Graduate student-oriented volume develops subject as fundamental discipline,
opening with review of origins of Schrodinger's equations and vector spaces. Focusing on
main principles of quantum mechanics and their immediate consequences, it
concludes with final generalizations covering alternative "languages" or representations.
1972 ed. 15 figures. xi+155pp. 5% x 8%. 0-486-42829-X
INTRODUCTION TO QUANTUM MECHANICS With Applications to
Chemistry, Linus Pauling & E. Bright Wilson, Jr. Classic undergraduate text by
Nobel Prize winner applies quantum mechanics to chemical and physical problems.
Numerous tables and figures enhance the text. Chapter bibliographies. Appendices.
Index. 468pp. 5% x 8%. 0-486-64871-0
METHODS OF THERMODYNAMICS, Howard Reiss. Outstanding text focuses
on physical technique of thermodynamics, typical problem areas of understanding,
and significance and use of thermodynamic potential. 1965 edition. 238pp. 5% x 8!6.
0-486-69445-3
THE ELECTROMAGNETIC FIELD, Albert Shadowitz. Comprehensive
undergraduate text covers basics of electric and magnetic fields, builds up to
electromagnetic theory. Also related topics, including relativity. Over 900 problems. 768pp.
55k x 8%. 0-486-65660-8
GREAT EXPERIMENTS IN PHYSICS: FIRSTHAND ACCOUNTS FROM
GALILEO TO EINSTEIN, Morris H. Shamos (ed.). 25 crucial discoveries: Newton's
laws of motion, Chad wick's study of the neutron, Hertz on electromagnetic waves,
more. Original accounts clearly annotated. 370pp. 5% x SlL 0-486-25346-5
EINSTEIN'S LEGACY, Julian Schwinger. A Nobel Laureate relates fascinating story
of Einstein and development of relativity theory in well-illustrated, nontechnical
volume. Subjects include meaning of time, paradoxes of space travel, gravity and its
effect on light, non-Euclidean geometry and curving of space-time, impact of radio
astronomy and space-age discoveries, and more. 189 b/w illustrations. xiv+250pp.
8% x 9%. 0-486-41974-6
STATISTICAL PHYSICS, Gregory H. Wannier. Classic text combines
thermodynamics, statistical mechanics and kinetic theory in one unified presentation of thermal
physics. Problems with solutions. Bibliography. 532pp. 5% x 81/;. 0-486-65401-X
Paperbound unless otherwise indicated. Available at your book dealer, online at
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