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The Congruences of a Finite Lattice, A Proof-by-Picture Approach

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George Grätzer The Congruences of a Finite Lattice
The new edition of this self-contained exposition by one of the leading experts in lattice
theory, George Grätzer, presents the major results from the past ten years on congruence
lattices of nite lattices featuring the author's signature "Proof-by-Picture" method.
Key features:
Insightful discussion of techniques to construct "nice" fi nite lattices with given
congruence lattices and "nice" congruence-preserving extensions
Contains complete proofs, an extensive bibliography and index, and about
150 illustrations
This new edition includes two new parts on Planar Semimodular Lattices and
The Order of Principle Congruences
The book is appropriate for a one-semester graduate course in lattice theory, yet is also
designed as a practical reference for researchers studying lattices.
Reviews of the fi rst edition:
There exist a lot of interesting results in this area of lattice theor y, and some of them are
presented in this bo ok. [This] monograph…is an exc eptional work in lattice theor y, like all
the contributions by this author. … The way this book is written makes it extremely inter-
esting for the speci alists in the fi eld but also for the st udents in lattice the ory. Moreover, the
author provides a s eries of companion lectur es which help the reader to approach the Proof-
by-Picture sections. —Cosmin Pelea, Studia Universitatis Babes-Bolyai Mathematica,
Vol. LII (1), 2007
The book is self-contain ed, with many detailed proofs p resented that can be followed ste p-
by-step. [I]n addition to giving the full formal details of the proofs, the author chooses a
somehow more pedagogical way that he calls Proof-by-Picture, somehow related to the
combinatorial (as opposed to algebraic) nature of many of the presented results. I believe
that this book is a m uch-needed tool for any mathematic ian wishing a gentle introduct ion
to the fi eld of congruences representations of fi nite lattices, with emphasis on the more
'geometric' aspects. Mathematical Reviews
George Grätzer
The Congruences of a Finite Lattice
A Proof-by-Picture Approach, 2nd Edition
The Congruences
of a Finite Lattice
George Grätzer
9 783319 387963
ISBN 978-3-319-38796-3
2nd Ed.
A Proof-by-Picture Approach
Second Edition
The Congruences of
a Finite Lattice
Second Edition
George Gr¨atzer
Birkh¨auser Verlag, New York
@ 2016 Birkh¨auser Verlag
To L´aszl´o Fuchs,
my thesis advisor, my teacher,
who taught me to set the bar high;
and to my coauthors,
who helped me raise the bar.
Short Contents
Table of Notation xvii
Picture Gallery xxi
Preface to the Second Edition xxiii
Introduction xxv
I A Brief Introduction to Lattices 1
1 Basic Concepts 3
2 Special Concepts 19
3 Congruences 35
4 Planar Semimodular Lattices 47
II Some Special Techniques 55
5 Chopped Lattices 57
6 Boolean Triples 67
7 Cubic Extensions 81
III Congruence Lattices of Finite Lattices 87
8 The Dilworth Theorem 89
ix
xSHORT CONTENTS
9 Minimal Representations 101
10 Semimodular Lattices 113
11 Rectangular Lattices 123
12 Modular Lattices 127
13 Uniform Lattices 141
IV Congruence Lattices and Lattice Extensions 155
14 Sectionally Complemented Lattices 157
15 Semimodular Lattices 165
16 Isoform Lattices 173
17 The Congruence Lattice and the Automorphism Group 189
18 Magic Wands 201
V Congruence Lattices of Two Related Lattices 223
19 Sublattices 225
20 Ideals 237
21 Tensor Extensions 253
VI The Ordered Set of Principal Congruences 273
22 Representation Theorems 275
23 Isotone Maps 285
VII Congruence Structure 291
24 Prime Intervals and Congruences 293
25 Some Applications of the Swing Lemma 317
Bibliography 325
Index 339
Contents
Table of Notation xvii
Picture Gallery xxi
Preface to the Second Edition xxiii
Introduction xxv
I A Brief Introduction to Lattices 1
1 Basic Concepts 3
1.1 Ordering 3
1.1.1 Ordered sets 3
1.1.2 Diagrams 5
1.1.3 Constructions of ordered sets 5
1.1.4 Partitions 7
1.2 Lattices and semilattices 9
1.2.1 Lattices 9
1.2.2 Semilattices and closure systems 10
1.3 Some algebraic concepts 12
1.3.1 Homomorphisms 12
1.3.2 Sublattices 13
1.3.3 Congruences 15
2 Special Concepts 19
2.1 Elements and lattices 19
2.2 Direct and subdirect products 20
2.3 Terms and identities 23
2.4 Gluing 28
xi
xii Contents
2.5 Modular and distributive lattices 30
2.5.1 The characterization theorems 30
2.5.2 Finite distributive lattices 31
2.5.3 Finite modular lattices 32
3 Congruences 35
3.1 Congruence spreading 35
3.2 Finite lattices and prime intervals 38
3.3 Congruence-preserving extensions and variants 41
4 Planar Semimodular Lattices 47
4.1 Basic concepts 47
4.2 SPS lattices 48
4.3 Forks 49
4.4 Rectangular lattices 51
II Some Special Techniques 55
5 Chopped Lattices 57
5.1 Basic definitions 57
5.2 Compatible vectors of elements 59
5.3 Compatible vectors of congruences 60
5.4 From the chopped lattice to the ideal lattice 62
5.5 Sectional complementation 63
6 Boolean Triples 67
6.1 The general construction 67
6.2 The congruence-preserving extension property 70
6.3 The distributive case 72
6.4 Two interesting intervals 73
7 Cubic Extensions 81
7.1 The construction 81
7.2 The basic property 83
III Congruence Lattices of Finite Lattices 87
8 The Dilworth Theorem 89
8.1 The representation theorem 89
8.2 Proof-by-Picture 90
8.3 Computing 92
8.4 Sectionally complemented lattices 93
8.5 Discussion 95
Contents xiii
9 Minimal Representations 101
9.1 The results 101
9.2 Proof-by-Picture for minimal construction 102
9.3 The formal construction 104
9.4 Proof-by-Picture for minimality 105
9.5 Computing minimality 107
9.6 Discussion 108
10 Semimodular Lattices 113
10.1 The representation theorem 113
10.2 Proof-by-Picture 114
10.3 Construction and proof 115
10.4 Discussion 122
11 Rectangular Lattices 123
11.1 Results 123
11.2 Proof-by-Picture 124
11.3 Discussion 125
12 Modular Lattices 127
12.1 The representation theorem 127
12.2 Proof-by-Picture 128
12.3 Construction and proof 131
12.4 Discussion 137
13 Uniform Lattices 141
13.1 The representation theorem 141
13.2 Proof-by-Picture 141
13.3 The lattice N(A, B) 144
13.4 Formal proof 149
13.5 Discussion 150
IV Congruence Lattices and Lattice Extensions 155
14 Sectionally Complemented Lattices 157
14.1 The extension theorem 157
14.2 Proof-by-Picture 158
14.3 Simple extensions 160
14.4 Formal proof 162
14.5 Discussion 164
xiv Contents
15 Semimodular Lattices 165
15.1 The extension theorem 165
15.2 Proof-by-Picture 165
15.3 The conduit 168
15.4 The construction 170
15.5 Formal proof 171
15.6 Discussion 171
16 Isoform Lattices 173
16.1 The result 173
16.2 Proof-by-Picture 174
16.3 Formal construction 177
16.4 The congruences 183
16.5 The isoform property 184
16.6 Discussion 185
17 The Congruence Lattice and the Automorphism Group 189
17.1 Results 189
17.2 Proof-by-Picture 190
17.3 The representation theorems 194
17.4 Formal proofs 195
17.4.1 An automorphism-preserving simple extension 195
17.4.2 A congruence-preserving rigid extension 197
17.4.3 Proof of the independence theorems 197
17.5 Discussion 199
18 Magic Wands 201
18.1 Constructing congruence lattices 201
18.1.1 Bijective maps 201
18.1.2 Surjective maps 202
18.2 Proof-by-Picture for bijective maps 203
18.3 Verification for bijective maps 206
18.4 2/3-boolean triples 209
18.5 Proof-by-Picture for surjective maps 215
18.6 Verification for surjective maps 217
18.7 Discussion 218
V Congruence Lattices of Two Related Lattices 223
19 Sublattices 225
19.1 The results 225
19.2 Proof-by-Picture 226
19.3 Multi-coloring 229
Contents xv
19.4 Formal proof 230
19.5 Discussion 231
20 Ideals 237
20.1 The results 237
20.2 Proof-by-Picture for the main result 238
20.3 The main result 240
20.3.1 A formal proof 240
20.3.2 The final step 248
20.4 Proof-by-Picture for planar lattices 249
20.5 Discussion 251
21 Tensor Extensions 253
21.1 The problem 253
21.2 Three unary functions 254
21.3 Defining tensor extensions 256
21.4 Computing 258
21.4.1 Some special elements 258
21.4.2 An embedding 260
21.4.3 Distributive lattices 261
21.5 Congruences 262
21.5.1 Congruence spreading 262
21.5.2 Some structural observations 265
21.5.3 Lifting congruences 267
21.5.4 The main lemma 269
21.6 The congruence isomorphism 270
21.7 Discussion 271
VI The Ordered Set of Principal Congruences 273
22 Representation Theorems 275
22.1 Representing the ordered set of principal congruences 275
22.2 Proof-by-Picture 275
22.3 An independence theorem 277
22.4 Discussion 280
23 Isotone Maps 285
23.1 Two isotone maps 285
23.1.1 Sublattices 285
23.1.2 Bounded homomorphisms 286
23.2 Sublattices, sketching the proof 287
23.3 Isotone surjective maps 287
23.4 Proving the Representation Theorem 288
xvi Contents
23.5 Discussion 289
VII Congruence Structure 291
24 Prime Intervals and Congruences 293
24.1 Introduction 293
24.2 The Prime-projectivity Lemma 294
24.3 The Swing Lemma 296
24.4 Some consequences of the Swing Lemma 306
24.5 Fork congruences 309
24.6 Discussion 315
25 Some Applications of the Swing Lemma 317
25.1 The Trajectory Theorem for SPS Lattices 317
25.2 The Two-cover Theorem 319
25.3 A counterexample 320
25.4 Discussion 323
Bibliography 325
Index 339
Glossary of Notation
Symbol Explanation Page
0Iand 1Ithe zero and unit of the interval I14
Atom(U) set of atoms of the ideal U94
Aut Lautomorphism group of L12
Bnboolean lattice with natoms 4
Cl(L),Cll(L),Cul (L) left boundary chains of a planar lattice 47
Cnn-element chain 4
con(a, b) smallest congruence under which ab16
con(c) principal congruence for a color c41
con(H) smallest congruence collapsing H16
con(p) principal congruence generated by a prime interval p39
Con Lcongruence lattice of L15, 58
ConJLordered set of join-irreducible congruences of L39
ConMLordered set of meet-irreducible congruences of L81
Cr(L),Clr(L),Cur (L) right boundary chains of a planar lattice L47
Cube Kcubic extension of K81
Dclass (variety) of distributive lattices 24
Diag(K) diagonal embedding of Kinto Cube K81
Down Pordered set of down sets of the ordered set P5, 9, 242
ext : Con KCon Lfor KL, extension map: α7→ conL(α) 42
fil(a) filter generated by the element a15
fil(H) filter generated by the set H15
FreeD(3) free distributive lattice on three generators 25
FreeK(H) free lattice generated by Hin a variety K27
FreeM(3) free modular lattice on three generators 27
Frucht CFrucht lattice of a graph C190
hom{∨,0}(X, Y ) the set of {∨,0}-homomorphisms of Xinto Y261
id(a) ideal generated by the element a14
id(H) ideal generated by the set H14
Id Lideal lattice of L14, 59
(Id) condition to define ideals 14, 59
Isoform class of isoform lattices 153
xviii Table of Notation
Symbol Explanation Page
J(D) ordered set of join-irreducible elements of D19
J(γ) J(γ): J(E)J(D), the “inverse” of γ:DE31
J(a) set of join-irreducible elements below a19
ker(γ) congruence kernel of γ17
Lclass (variety) of all lattices 25
Lbottom, Ltop bottom and top of a rectangular lattice L53
lc(L) left corner of a rectangular lattice L53
Lleft, Lright left and right of a rectangular lattice L53
Mclass (variety) of modular lattices 25
Max maximal elements of an ordered set 59
mcr(n) minimal congruence representation function 97
mcr(n, V)mcr for a class V97
M(D) ordered set of meet-irreducible elements of D32
M3five-element modular nondistributive lattice xxi, 11, 30
M3[L] ordered set of boolean triples of L68
M3[L, a] interval of M3[L] 73
M3[L, a, b] interval of M3[L] 75
M3[a, b] ordered set of boolean triples of the interval [a, b] 68
M3[α] restriction of α3to M3[L] 70
M3[α, a] restriction of α3to M3[L, a] 74
M3[α, a, b] restriction of α3to M3[L, a, b] xxi, 77
N5five-element nonmodular lattice xxi, 11, 20, 30
N5,5seven-element nonmodular lattice 102
N6=N(p, q) six-element nonmodular lattice xxi, 20, 90
N6[L] 2/3-boolean triple construction 209
N(A, B) a lattice construction 144
O(f) Landau big Onotation xxxii
Part Apartition lattice of A7, 9
Pow Xpower set lattice of X5
Pow+Xordered set of nonempty subsets of X229
Princ Lordered set of principal congruences L38
rc(L) right corner of a rectangular lattice 52
Prime(L) set of prime intervals of L39
re : Con LCon Krestriction map: α7→ αeK41
SecComp class of sectionally complemented lattices 97
SemiMod class of semimodular lattices 122
Simp Ksimple extension of K81
(SP) join-substitution property 15, 58
(SP) meet-substitution property 15, 58
sub(H) sublattice generated by H13
S7seven-element semimodular lattice xxi, 34
S8eight-element semimodular lattice 34
Swing,
x
p
x
q,pswings to q296
Tclass (variety) of trivial lattices 25
Uniform class of uniform lattices 153
Traj Lset of all trajectories of L48
Table of Notation xix
Symbol Explanation Page
Relations and
Congruences
A2set of ordered pairs of A3
%, τ, π , . . . binary relations
α,β, . . . congruences
0zero of Part Aand Con L8
1unit of Part Aand Con L8
ab(mod π)aand bin the same block of π7
a % b a and bin relation %3
ab(mod α)aand bin relation α3
a/π block containing a7, 15
H/π blocks represented by H7
αβproduct of αand β21
αr
βreflexive product of αand β29
αeKrestriction of αto the sublattice K15
L/αquotient lattice 16
β/αquotient congruence 17
πiprojection map: L1× · · · × LnLi20
α×βdirect product of congruences 21
Ordered sets
, < ordering 3
, > ordering, inverse notation 3
KL K a sublattice of L13
Qordering of Prestricted to a subset Q4
akb a incomparable with b3
ab a is covered by b5
ba b covers a5
0 zero, least element of an ordered set 4
1 unit, largest element of an ordered set 4
abjoin operation 9
WHleast upper bound of H3
abmeet operation 9
VHgreatest lower bound of H4
Pddual of the ordered set (lattice) P4, 10
[a, b] interval 14
Hdown set generated by H5
adown set generated by {a}5
P
=Qordered set (lattice) Pisomorphic to Q4, 12
xx Table of Notation
Symbol Explanation Page
Constructions
P×Qdirect product of Pand Q5, 20
P+Qsum of Pand Q6
PuQglued sum of Pand Q17
A[B] tensor extension of Aby B256
ABtensor product of Aand B253
U~Vmodular lattice construction 131
Prime intervals
p,q, . . . prime intervals
con(p) principal congruence generated by p39
Prime(L) set of prime intervals of L39
Princ Lthe ordered set of principal congruences 38
Perspectivities
[a, b][c, d] [a, b] perspective to [c, d] 32
[a, b]up
[c, d] [a, b] up-perspective to [c, d] 32
[a, b]dn
[c, d] [a, b] down-perspective to [c, d] 32
[a, b][c, d] [a, b] projective to [c, d] 32
[a, b][c, d] [a, b] congruence-perspective onto [c, d] 36
[a, b]up
[c, d] [a, b] up congruence-perspective onto [c, d] 35
[a, b]dn
[c, d] [a, b] down congruence-perspective onto [c, d] 35
[a, b][c, d] [a, b] congruence-projective onto [c, d] 35
pp
q p prime-perspective to q294
pp-up
q p prime-perspective up to q294
pp-dn
q p prime-perspective down to q294
pp
=q p prime-projective to q294
p
x
q p swings to q296
pin
x
qinternal swing 296
pex
x
qexternal swing 296
Miscellaneous
xclosure of x12
empty set 4
Picture Gallery
Preface to the Second Edition
A few years after the publication of the first edition of this book, I submitted
a paper on congruence lattices of finite lattices to a journal. The referee
expressed surprise that I have more to say on this topic, after all, I published
a whole book on the subject.
This Second Edition, will attest to the fact the we, indeed, have a lot more
to say on this subject. In fact, we have hardly started.
The new topics include
The minimal representation theorem for rectangular lattices, a combina-
torial result, see Chapter 11; this is joint work with E. Knapp.
A new field investigating the ordered set
Princ K
of principal congruences
of a lattice
K
, see Part VI. I started this field; it was continued by
G. Cz´edli and myself.
Congruences and prime intervals, a field concerned with the congruence
structure of finite lattices, especially of SPS lattices, see Part VII. For
SPS lattices this field was started by G. Cz´edli; the results are due to
him and myself.
Part I has been extended to also provide an introduction to the new topics.
LTF refers to the book Lattice Theory: Foundation, [
62
], and LTS1 to the
book Lattice Theory: Special Topics and Applications, Volume 1 [152].
We use now the notation as in LTF, in particular, the functional notation
for maps and bold Greek letters for congruences. This introduced
0
incorrect
formulas and diagrams. I think I have been able to catch 99% of the errors.
However, if you find a map ϕψ difficult to figure out, just change it to ψϕ.
I would like to thank all those who sent me corrections, especially, G. Cz´edli,
K. Kaarli, H. Lakser, R. Padmanabhan, and F. Wehrung.
Note
All the unpublished papers in the references can be found in the
ResearchGate Website,
researchgate.net
; since
Gratzer
will not find my
publications, use the link
https://www.researchgate.net/profile/George_Graetzer
Introduction
The topic of this book, congruences of finite lattices, naturally splits into three
fields of research:
congruence lattices of finite lattices;
the ordered set of principal congruences of finite lattices;
the congruence structure of finite lattices.
Congruence lattices of finite lattices
This is the major topic of this book, it covers over 70 years of research and
almost 200 papers.
The congruences of a finite lattice Lform a lattice, called the congruence
lattice of
L
, denoted by
Con L
. According to a 1942 result of N. Funayama
and T. Nakayama [
53
], the lattice
Con L
is a finite distributive lattice. The
converse is a result of R.P. Dilworth from 1944 (see [11]):
Dilworth Theorem.
Every finite distributive lattice
D
can be represented
as the congruence lattice of a finite lattice L.
This result was not published until 1962, see G. Gr¨atzer and E. T. Schmidt
[
117
]. In the almost 60 years since the discovery of this result, a large number
of papers have been published, strengthening and generalizing the Dilworth
Theorem. These papers form two distinct fields:
(i)
Representation theorems of finite distributive lattices as congruence
lattices of lattices with special properties.
(ii)
The Congruence Lattice Problem (CLP): Can congruence lattices of
lattices be characterized as distributive algebraic lattices?
xxvi Introduction
A nontrivial finite distributive lattice
D
is determined by the ordered set
J
(
D
) of join-irreducible elements. So a representation of
D
as the congruence
lattice of a finite lattice
L
is really a representation of a finite ordered set
P
(=
J
(
D
)), as the ordered set,
ConJL
, of join-irreducible congruences of a
finite lattice
L
. A join-irreducible congruence of a nontrivial finite lattice
L
is
exactly the same as a congruence of the form
con
(
a, b
), where
ab
in
L
; that
is, the smallest congruence collapsing a prime interval. Therefore, it is enough
to concentrate on such congruences, and make sure that they are ordered as
required by P.
The infinite case is much different. There are really only two general
positive results:
1. The ideal lattice of a distributive lattice with zero is the congruence
lattice of a lattice—see E. T. Schmidt [177] (also P. Pudl´ak [168]).
2. Any distributive algebraic lattice with at most
1
compact elements
is the congruence lattice of a lattice—A. P. Huhn [
159
] and [
160
] (see also
H. Dobbertin [45]).
The big breakthrough for negative results came in 1999 in F. Wehrung
[190], based on the results of F. Wehrung [189].
This book deals with the finite case. For a detailed review of the infinite
case, see F. Wehrung [192]–[194], Chapter 7–9 of the book LTS1.
The two types of representation theorems
The basic representation theorems for the finite case are all of the same general
type. We represent a finite distributive lattice
D
as the congruence lattice of
a “nice” finite lattice
L
. For instance, in the 1962 paper (G. Gr¨atzer and E. T.
Schmidt [
117
]), we already proved that the finite lattice
L
for the Dilworth
Theorem can be constructed as a sectionally complemented lattice.
To understand the second, more sophisticated, type of representation
theorem, we need the concept of a congruence-preserving extension.
Let
L
be a lattice, and let
K
be a sublattice of
L
. In general, there is
not much connection between the congruence lattice of
L
and the congruence
lattice of
K
. If they happen to be naturally isomorphic, we call
L
acongru-
ence-preserving extension of K. (More formally, see Section 3.3.)
For sectionally complemented lattices, the congruence-preserving extension
theorem was published in a 1999 paper, G. Gr¨atzer and E. T. Schmidt [
129
]:
Every finite lattice
K
has a finite, sectionally complemented, congruence-pre-
serving extension
L
. It is difficult, reading this for the first time, to appreciate
how much stronger this theorem is than the straight representation theorem.
While the 1962 theorem provides, for a finite distributive lattice
D
, a finite
sectionally complemented lattice
L
whose congruence lattice is isomorphic
to
D
, the 1999 theorem starts with an arbitrary finite lattice
K
, and builds a
sectionally complemented lattice
L
extending it with the “same” congruence
structure.
Introduction xxvii
The ordered set of principal congruences
of a finite lattice
A large part of this book investigates the congruence lattice,
Con L
, of a finite
lattice
L
. But
Con L
is not the only interesting congruence construct we can
associate with a finite lattice
L
. A new one is
Princ L
, the ordered set of
principal congruences of L. We discuss this topic in Part VI.
It turns out that only a tiny consequence of finiteness (being bounded) is
of importance for this topic. So we will phrase the results for bounded lattices.
We start discussing this new field in Chapter 22, characterizing
Princ K
of
a bounded lattice
K
as a bounded ordered set, see G. Gr¨atzer [
65
]. Utilizing
this result, we prove in Section 22.3 that for a bounded lattice
1L
, the two
related structures Princ Land Aut Lare independent, see G. Cz´edli [28].
If
K
and
L
are bounded lattices and
ϕ
is a bounded homomorphism
2
of
K
into
L
, then there is a natural bounded isotone map of
Princ K
into
Princ L
.
The main result of Chapter 23 is the converse: every bounded isotone map
of
Princ K
into
Princ L
can be so represented, see G. Cz´edli [
26
], [
27
] and
G. Gr¨atzer [75], [77].
Congruence structure of finite lattices
The spreading of a congruence from a prime interval to another prime interval
involves intervals of arbitrary size. Can we describe such a spreading with
prime intervals only?
We can indeed, by introducing the concept of prime-projectivity, see Sec-
tion 24.2, and obtaining Prime-Projectivity Lemma, see G. Gr¨atzer [69].
Then, in Section 24.3, we develop much sharper forms of this result for SPS
(slim, planar, semimodular) lattices. The main result is the Swing Lemma,
G. Gr¨atzer [
70
], from which we derive many of the known results of G. Cz´edli
and myself concerning congruences of SPS lattices.
Proof-by-Picture
In 1960, trying with E. T. Schmidt to prove the Dilworth Theorem (unpublished
at the time) we came up with the construction—more or less—as presented
in Section 8.2. In 1960, we did not anticipate the 1968 result of G. Gr¨atzer
and H. Lakser [
89
], establishing that the construction of a chopped lattice
solves the problem. So we translated the chopped lattice construction to a
closure space, as in Section 8.4, proved that the closed sets form a sectionally
complemented lattice
L
, and based on that, we verified that the congruence
lattice of Lrepresents the given finite distributive lattice.
1A lattice is bounded, if it has 0 and 1, see Section 1.1.1.
2A homomorphism is bounded, if it preserves 0 and 1, see Section 1.3.1.
xxviii Introduction
When we submitted the paper [
117
] for publication, it had a three-page
section explaining the chopped lattice construction and its translation to
a closure space. The referee was strict: “You cannot have a three-page
explanation for a two-page proof.” I believe that in the 50 plus years since the
publication of that article, few readers have developed an understanding of
the idea behind the published proof.
The referee’s dictum is quite in keeping with mathematical tradition and
practice. When mathematicians discuss new results, they explain the construc-
tions and the ideas with examples; when these same results are published,
the motivation and the examples are largely gone. We publish definitions,
constructions, and formal proofs (and conjectures, Paul Erd˝os would have
added).
Tradition has it, when Gauss proved one of his famous results, he was not
ready to publicize it because the proof gave away too much as to how the
theorem was discovered. “I have had my results for a long time: but I do not
yet know how I am to arrive at them”, Gauss is quoted in A. Arber [3].
I try to break with this tradition in this book. In many chapters, after
stating the main result, I include a section: Proof-by-Picture. This is a
misnomer. A Proof-by-Picture is not a proof. The Pythagorean Theorem has
many well known Proofs-by-Picture—sometimes called “Visual Proofs”; these
are really proofs. My Proof-by-Picture is an attempt to convey the idea of the
proof. I trust that if the idea is properly understood, the reader should be
able to provide the formal proof, or should at least have less trouble reading it.
Think of a Proof-by-Picture as a lecture to an informed audience, concluding
with “the formal details now you can provide.”
Outline
In the last paragraph, I call an audience “informed” if they are familiar with
the basic concepts and techniques of lattice theory. Part I provides this. I am
very selective as to what to include. There are no proofs in this part—with a
few exceptions; they are easy enough for the reader to work them out on his
own. For proofs, lots of exercises, and a more detailed exposition, I refer the
reader to my book LTF.
Most of the research in this book deals with representation theorems;
lattices with certain properties are constructed with prescribed congruence
structures. The constructions are ad hoc. Nevertheless, there are three basic
techniques to prove representation theorems:
chopped lattices, used in almost every chapter in Parts III–V;
boolean triples, used in Chapters 12, 14, and 18, and generalized in
Chapter 21; also used in some papers that did not make it in this book,
for instance, G. Gr¨atzer and E. T. Schmidt [131];
Introduction xxix
cubic extensions, used in most chapters of Part IV.
These are presented in Part II with proofs.
Actually, there are two more basic techniques. Multi-coloring (and its
recent variant, quasi-coloring, see many recent papers of G. Cz´edli) is used in
several relevant papers; however, it appears in the book only in Chapter 19,
so we introduce it there. Pruning is utilized in Chapters 13 and 16—it would
seem to qualify for Part II; however, there are only concrete uses of pruning,
there is no general theory to discuss in Part II.
Part III contains the representation theorems of congruence lattices of finite
lattices, requiring only chopped lattices from Part II. I cover the following
topics:
The Dilworth Theorem and the representation theorem for sectionally
complemented lattices in Chapter 8 (G. Gr¨atzer and E. T. Schmidt [
117
],
P. Crawley and R. P. Dilworth [13]; see also [11]).
Minimal representations in Chapter 9; that is, for a given
|J
(
D
)
|
, we
minimize the size of
L
representing the finite distributive lattice
D
(G. Gr¨atzer, H. Lakser, and E. T. Schmidt [
102
], G. Gr¨atzer, Rival, and
N. Zaguia [114]).
The semimodular representation theorem in Chapter 10 (G. Gr¨atzer,
H. Lakser, and E. T. Schmidt [105]).
Another minimal representations in Chapter 11 for rectangular lattices
(G. Gr¨atzer and E. Knapp [87]).
The representation theorem for modular lattices in Chapter 12 (E. T.
Schmidt [
174
] and G. Gr¨atzer and E. T. Schmidt [
134
]); we are forced to
represent with a countable lattice
L
, since the congruence lattice of a
finite modular lattice is always boolean.
The representation theorem for uniform lattices (that is, lattices in
which any two congruence classes of a congruence are of the same size)
in Chapter 13 (G. Gr¨atzer, E. T. Schmidt, and K. Thomsen [141]).
Part IV is mostly about congruence-preserving extension. I present the
congruence-preserving extension theorem for
sectionally complemented lattices in Chapter 14 (G. Gr¨atzer and E. T.
Schmidt [129]);
semimodular lattices in Chapter 15 (G. Gr¨atzer and E. T. Schmidt [
132
]);
isoform lattices (that is, lattices in which any two congruence classes of
a congruence are isomorphic) in Chapter 16 (G. Gr¨atzer, R. W. Quack-
enbush, and E. T. Schmidt [113]).
xxx Introduction
These three constructions are based on cubic extensions, introduced in Part II.
In Chapter 17, I present the congruence-preserving extension version of the
Baranski˘ı-Urquhart Theorem (V. A. Baranski˘ı [
4
], [
5
] and A. Urquhart [
188
])
on the independence of the congruence lattice and the automorphism group of
a finite lattice (see G. Gr¨atzer and E. T. Schmidt [126]).
Finally, in Chapter 18, I discuss two congruence “destroying” extensions,
which we call “magic wands” (G. Gr¨atzer and E. T. Schmidt [
135
], G. Gr¨atzer,
M. Greenberg, and E. T. Schmidt [82]).
What happens if we consider the congruence lattices of two related lattices,
such as a lattice and a sublattice? I take up three variants of this question
in Part V.
Let
L
be a finite lattice, and let
K
be a sublattice of
L
. As we discuss it
in Section 3.3, there is a map
ext
from
Con K
into
Con L
: For a congruence
relation
α
of
K
, let the image
ext α
be the congruence relation
conL
(
α
) of
L
generated by α. The map ext is a {0}-separating join-homomorphism.
Chapter 19 proves the converse, a 1974 result of A. P. Huhn [
158
] and a
stronger form due to G. Gr¨atzer, H. Lakser, and E. T. Schmidt [103].
We deal with ideals in Chapter 20. Let
K
be an ideal of a lattice
L
. Then
the restriction map
re : Con LCon K
(which assigns to a congruence
α
of
L
, the restriction
αeK
of
α
to
K
) is a bounded homomorphism. We prove
the corresponding representation theorem for finite lattices—G. Gr¨atzer and
H. Lakser [90].
We also prove two variants. The first is by G. Gr¨atzer and H. Lakser [
97
]
stating that this result also holds for sectionally complemented lattices. The
second is by G. Gr¨atzer and H. Lakser [
95
] stating that this result also holds
for planar lattices.
The final chapter is a first contribution to the following class of problems.
Let
~
be a construction for finite lattices (that is, if
D
and
E
are finite lattices,
then so is
D~E
). Find a construction
}
of finite distributive lattices (that
is, if
K
and
L
are finite distributive lattices, then so is
K}L
) satisfying
Con(K~L)
=Con K}Con L.
If the lattice construction is the direct product, the answer is obvious since
Con(K×L)
=Con K×Con L.
In Chapter 21, we take up the construction defined as the distributive
lattice of all isotone maps from J(E) to D.
In G. Gr¨atzer and M. Greenberg [
78
], we introduced another construction:
the tensor extension,
A
[
B
], for nontrivial finite lattices
A
and
B
. In Chapter 21,
we prove that Con(A[B])
=(Con A)[Con B].
In 2013, I raised the question whether one can associate with a finite
lattice
L
a structure of some of its congruences? We could take the ordered
set of the principal congruences generated by prime intervals, but this is just
ConJL
, which is “equivalent” with
Con L
(see Section 2.5.2 for an explanation).
We proposed to consider the ordered set
Princ K
of principal congruences of a
lattice K. In Part VI, we describe the first few results of this field.
Introduction xxxi
It turns out that the class of bounded (not necessarily finite) lattices is the
natural setting for this topic.
The highlights include
The Representation Theorem, characterizing
Princ K
of a bounded lattice
Kas a bounded ordered set, see Chapter 22 (G. Gr¨atzer [65]).
For a bounded lattice
L
, the two related structures
Princ L
and
Aut L
are
independent, see Section 22.3 (G. Cz´edli [28], see also G. Gr¨atzer [76]).
If
K
and
L
are bounded lattices and
γ
is a bounded homomorphism
of
K
into
L
, then there is a natural bounded isotone map of
Princ K
into
Princ L
. The main result of Chapter 23 is the converse, every
bounded isotone map of
Princ K
into
Princ L
can be so represented (G.
Cz´edli [26], [27]) and G. Gr¨atzer [75]).
Finally, Part VII deals with another new topic: the congruence structure
of finite lattices, focusing on prime intervals and congruences.
The two main results are
The Prime-Projectivity Lemma, which verifies that, indeed, we can
describe the spreading of a congruence from a prime interval to another
prime interval involving only prime intervals, see G. Gr¨atzer [69].
The Swing Lemma, a very strong form of the Prime-Projectivity Lemma,
for SPS (slim, planar, and semimodular) lattice, see G. Gr¨atzer [70].
Each chapter in Parts III–VII concludes with an extensive discussion section,
giving the background for the topic, further results, and open problems. This
book lists more than 80 open problems, hoping to convince the reader that,
indeed, we have hardly started. There are almost 200 references and a detailed
index.
This book is, as much as possible, visually oriented. I cannot stress too
much the use of diagrams as a major research tool in lattice theory. I did not
include in the book the list of figures because there is not much use to it; it
would list about 150 figures.
Notation and terminology
Lattice-theoretic terminology and notation evolved from the three editions of
G. Birkhoff’s Lattice Theory, [
10
], by way of my books, [
54
]–[
57
], [
59
], [
60
],
LTF, and R. N. McKenzie, G. F. McNulty, and W.F. Taylor [
165
], changing
quite a bit in the process.
Birkhoff’s notation for the congruence lattice and ideal lattice of a lattice
changed from Θ(
L
) and
I
(
L
) to
Con L
and
Id L
, respectively. The advent
of L
A
T
E
X promoted the use of operators for lattice constructions. I try to
xxxii Introduction
be consistent: I use an operator when a new structure is constructed; so I
use
Con L, Id L, Aut L
, and so on, without parentheses, unless required for
readability, for instance,
J
(
D
) and
Con
(
Id L
). I use functional notation when
sets are constructed, as in
Atom
(
L
) and
J
(
a
). “Generated by” uses the same
letters as the corresponding lattice construction, but starting with a lower
case letter:
Con L
is the congruence lattice of
L
and
con
(
H
) is the congruence
generated by
H
, while
Id L
is the ideal lattice of
L
and
id
(
H
) is the ideal
generated by H.
New concepts introduced in more recent research papers exhibit the usual
richness in notation and terminology. I use this opportunity, with the wisdom
of hindsight, to make their use more consistent. The reader will often find
different notation and terminology when reading the original papers. The
detailed Table of Notation and Index may help.
In combinatorial results, I will use Landau’s big
O
notation: for the
functions
f
and
g
, we write
f
=
O
(
g
) to mean that
|f| ≤ C|g|
for a suitable
constant C. Natural numbers start at 1.
Toronto, Ontario George Gr¨atzer
Summer, 2016
Homepage: http://www.maths.umanitoba.ca/homepages/gratzer.html/
Part I
A Brief Introduction to
Lattices
1
Chapter
1
Basic Concepts
In this chapter we introduce the most basic order theoretic concepts: ordered
sets, lattices, diagrams, and the most basic algebraic concepts: sublattices,
congruences, products.
1.1. Ordering
1.1.1 Ordered sets
Abinary relation
%
on a nonempty set
A
is a subset of
A2
, that is, a set
of ordered pairs (
a, b
), with
a, b A
. For (
a, b
)
%
, we will write
a%b
or
ab(mod %).
A binary relation
on a set
P
is called an ordering if it is reflexive (
aa
for all
aP
), antisymmetric (
ab
and
ba
imply that
a
=
b
for all
a, b P
), and transitive (
ab
and
bc
imply that
ac
for all
a, b, c P
).
An ordered set (P, ) consists of a nonempty set Pand an ordering .
a < b
means that
ab
and
a6
=
b
. We also use the “inverse” relations:
ab
defined as
ba
and
a>b
for
b < a
. If more than one ordering is being
considered, we write
P
for the ordering of (
P,
); on the other hand if the
ordering is understood, we will say that
P
(rather than (
P,
)) is an ordered
set. An ordered set Pis trivial if Phas only one element.
The elements
a
and
b
of the ordered set
P
are comparable if
ab
or
ba
.
Otherwise, aand bare incomparable, in notation, akb.
Let
HP
and
aP
. Then
a
is an upper bound of
H
iff
ha
for all
hH
. An upper bound
a
of
H
is the least upper bound of
H
iff
ab
for
any upper bound
b
of
H
; in this case, we will write
a
=
WH
. If
a
=
WH
exists,
3
41. Basic Concepts
then it is unique. By definition,
W
exist (
is the empty set) iff
P
has a
smallest element, zero, denoted by 0. The concepts of lower bound and greatest
lower bound are similarly defined; the latter is denoted by
VH
. Note that
V
exists iff
P
has a largest element, unit, denoted by 1. A bounded ordered
set is one that has both 0 and 1. We often denote the 0 and 1 of
P
by 0
P
and 1
P
. The notation
WH
and
VH
will also be used for families of elements.
The adverb “similarly” (in “similarly defined”) in the previous paragraph
can be given concrete meaning. Let (
P,
) be an ordered set. Then (
P,
) is
also an ordered set, called the dual of (
P,
). The dual of the ordered set
P
will be denoted by
Pd
. Now if Φ is a “statement” about ordered sets, and if
in Φ we replace all occurrences of by , then we get the dual of Φ.
Duality Principle for Ordered Sets.
If a statement Φis true for all
ordered sets, then its dual is also true for all ordered sets.
For
a, b P
, if
a
is an upper bound of
{b}
, then
a
is an upper bound of
b
.
If for all
a, b P
, the set
{a, b}
has an upper bound, then the ordered set
P
is
directed.
Achain (linearly ordered set,totally ordered set) is an ordered set with no
incomparable elements. An antichain is one in which akbfor all a6=b.
Let (
P,
) be an ordered set and let
Q
be a nonempty subset of
P
. Then
there is a natural ordering
Q
on
Q
induced by
: for
a, b Q
, let
aQb
iff
ab
; we call (
Q, Q
) (or simply, (
Q,
), or even simpler,
Q
) an ordered
subset (or suborder) of (P, ).
Achain
C
in an ordered set
P
is a nonempty subset, which, as a suborder,
is a chain. An antichain
C
in an ordered set
P
is a nonempty subset which,
as a suborder, is an antichain.
The length of a finite chain
C
,
length C
, is
|C| −
1. An ordered set
P
is
said to be of length
n
(in formula,
length P
=
n
), where
n
is a natural number
iff there is a chain in Pof length nand all chains in Pare of length n.
The ordered sets
P
and
Q
are isomorphic (in formula,
P
=Q
) and the
map ψ:PQis an isomorphism iff ψis one-to-one and onto and
abin Piff ψa ψb in Q.
Let Cndenote the set {0, . . . , n 1}ordered by
0<1<2<··· < n 1.
Then
Cn
is an
n
-element chain. Observe that
length Cn
=
n
1. If
C
=
{x0, . . . , xn1}
is an
n
-element chain and
x0< x1<··· < xn1
, then
ψ:i7→ xi
is an isomorphism between
Cn
and
C
. Therefore, the
n
-element
chain is unique up to isomorphism.
Let B
n
denote the set of all subsets of the set
{
0
, . . . , n
1
}
ordered by con-
tainment. Observe that the ordered set B
n
has 2
n
elements and
length Bn=n
.
1.1. Ordering 5
In general, for a set
X
, we denote by
Pow X
the power set of
X
, that is, the
set of all subsets of Xordered by set inclusion.
For an ordered set
P
, call
AP
adown set iff
xA
and
yx
in
P
,
imply that
yA
. For
HP
, there is a smallest down set containing
H
,
namely,
{x|xh, for some hH}
; we use the notation
H
for this set.
If
H
=
{a}
, we write
a
for
{a}
. Let
Down P
denote the set of all down
sets ordered by set inclusion. If
P
is an antichain, then
Down P
=
B
n
, where
n=|P|.
The map
ψ:PQ
is an isotone map (resp., antitone map) of the ordered
set
P
into the ordered set
Q
iff
ab
in
P
implies that
ψa ψb
(resp.,
ψa ψb
) in
Q
. Then
ψP
is a suborder of
Q
. Even if
ψ
is one-to-one, the
ordered sets
P
and
ψP
need not be isomorphic. If both
P
and
Q
are bounded,
then the map
ψ:PQ
is bounded or a bounded map, if it preserves the
bounds, that is,
ψ
0
P
= 0
Q
and
ψ
1
P
= 1
Q
. Most often, we talk about bounded
isotone maps (and bounded homomorphisms, see Section 1.3.1).
1.1.2 Diagrams
In the ordered set
P
, the element
a
is covered by
b
or
b
covers
a
(in formula,
ab
or
ba
) iff
a<b
and
a<x<b
for no
xP
. The binary relation
is
called the covering relation. The covering determines the ordering:
Let
P
be a finite ordered set. Then
ab
iff
a
=
b
or if there exists a finite
sequence of elements x1, x2, . . . , xnsuch that
a=x1x2≺ ··· ≺ xn=b.
Adiagram of an ordered set
P
represents the elements with small circles ;
the circles representing two elements
x, y
are connected by a line segment iff
one covers the other; if
x
is covered by
y
, then the circle representing
x
is
placed lower than the circle representing y.
The diagram of a finite ordered set determines the order up to isomorphism.
In a diagram the intersection of two line segments does not indicate an
element. A diagram is planar if no two line segments intersect. An ordered
set
P
is planar if it has a diagram that is planar. Figure 1.1 shows three
diagrams of the same ordered set
P
. Since the third diagram is planar,
P
is a
planar ordered set.
1.1.3 Constructions of ordered sets
Given the ordered sets
P
and
Q
, we can form the direct product
P×Q
,
consisting of all ordered pairs (
x1, x2
), with
x1P
and
x2Q
, ordered
componentwise, that is, (
x1, x2
)
(
y1, y2
) iff
x1y1
and
x2y2
. If
P
=
Q
,
then we write
P2
for
P×Q
. Similarly, we use the notation
Pn
for
Pn1×P
61. Basic Concepts
for
n >
2. Figure 1.2 shows a diagram of C
2×P
, where
P
is the ordered set
with diagrams in Figure 1.1.
Figure 1.1: Three diagrams of an ordered set.
Figure 1.2: A diagram of C2×P.
Figure 1.3: Diagrams of C2+P,P+C2, and PuC2.
Another often used construction is the (ordinal) sum
P
+
Q
of
P
and
Q
,
defined on the (disjoint) union PQand ordered as follows:
xyiff
xPyfor x, y P;
xQyfor x, y Q;
xP, y Q.
Figure 1.3 shows diagrams of C
2
+
P
and
P
+C
2
, where
P
is the ordered
1.1. Ordering 7
set with diagrams in Figure 1.1. In both diagrams, the elements of C
2
are
black-filled. Figure 1.3 also shows the diagram of PuC2.
A variant construction is the glued sum,
PuQ
, applied to an ordered set
P
with largest element 1
P
and an ordered set
Q
with smallest element 0
Q
; then
PuQ
is
P
+
Q
in which 1
P
and 0
Q
are identified (that is, 1
P
= 0
Q
in
PuQ
).
1.1.4 Partitions
We now give a nontrivial example of an ordered set. A partition of a nonempty
set
A
is a set
π
of nonempty pairwise disjoint subsets of
A
whose union is
A
.
The members of
π
are called the blocks of
π
. The block containing
aA
will
be denoted by
a/π
. A singleton as a block is called trivial. If the elements
a
and
b
of
A
belong to the same block, we write
ab
(
mod π
) or
a π b
or
a/π =b/π. In general, for HA,
H/π ={a/π |aH},
a collection of blocks.
An equivalence relation
ε
on the set
A
is a reflexive, symmetric (
aεb
implies that
bεa
, for all
a, b A
), and transitive binary relation. Given a
partition
π
, we can define an equivalence relation
ε
by (
x, y
)
ε
iff
x/π
=
y/π
.
Conversely, if
ε
is an equivalence relation, then
π
=
{a/ε |aA}
is a
partition of
A
. There is a one-to-one correspondence between partitions and
equivalence relations; we will use the two terms interchangeably.
Part Awill denote the set of all partitions of Aordered by
π1π2iff xy(mod π1) implies that xy(mod π2).
We draw a picture of a partition by drawing the boundary lines of the (non-
trivial) blocks. Then
π1π2
iff the boundary lines of
π2
are also boundary
lines of
π1
(but
π1
may have some more boundary lines). Equivalently, the
blocks of π2are unions of blocks of π1; see Figure 1.4.
π2:
and
A
π1π2
π1:
Figure 1.4: Drawing a partition.
81. Basic Concepts
Part A
has a zero and a unit, denoted by
0
and
1
, respectively, defined by
xy(mod 0) iff x=y;
xy(mod 1) for all x, y A.
Figure 1.5 shows the diagrams of
Part A
for
|A| ≤
4. The partitions are
labeled by listing the nontrivial blocks.
{1,2} {1,3} {1,4} {2,3} {2,4} {3,4}
{1,2,4}{1,3,4}
{2,3,4}
{1,2},{3,4}{1,3},{2,4}1}2 3{,4,{,}
{1,2,3}
{1,2,3,4}=
{1,2} {1,3} {2,3}
={1,2}
={1,2,3}
Part {1}
Part {1,2}Part {1,2,3}
Part {1,2,3,4}
0
1
0
1
0
1
Figure 1.5: Part Afor |A| ≤ 4.
Apreordered set is a nonempty set
Q
with a binary relation
that is
reflexive and transitive. Let us define the binary relation
ab
on
Q
as
ab
and
ba
. Then
is an equivalence relation. Define the set
P
as
Q/
, and
on Pdefine the binary relation :
a/≈ ≤ b/iff abin Q.
It is easy to see that the definition of
on
P
is well-defined and that
P
is
an ordered set. We will call
P
the ordered set associated with the preordered
set Q.
1.2. Lattices and semilattices 9
Starting with a binary relation
on the set
Q
, we can define the reflexive-
transitive closure
of
by the formula: for
a, b Q
, let
ab
iff
a
=
b
or
if
a
=
x0x1≺ ··· ≺ xn
=
b
for elements
x1
, . . . ,
xn1Q
. Then
is
a preordering on
Q
. A cycle on
Q
is a sequence
x1
, . . . ,
xnQ
satisfying
x1x2≺ ··· ≺ xnx1
(
n >
1). The preordering
is an ordering iff there
are no cycles.
1.2. Lattices and semilattices
1.2.1 Lattices
We need two basic concepts from Universal Algebra. An (
n
-ary) operation on
a nonempty set
A
is a map from
An
to
A
. For
n
= 2, we call the operation
binary. An algebra is a nonempty set Awith operations defined on A.
An ordered set (
L,
) is a lattice if
W{a, b}
and
V{a, b}
exist for all
a, b L
.
A lattice Lis trivial if it has only one element; otherwise, it is nontrivial.
We will use the notations
ab=_{a, b},
ab=^{a, b},
and call
the join and
the meet. They are both binary operations that
are idempotent (
aa
=
a
and
aa
=
a
), commutative (
ab
=
ba
and
ab
=
ba
), associative ((
ab
)
c
=
a
(
bc
) and (
ab
)
c
=
a
(
bc
)),
and absorptive (
a
(
ab
) =
a
and
a
(
ab
) =
a
). These properties of the
operations are also called the idempotent identities, commutative identities,
associative identities, and absorption identities, respectively. (Identities, in
general, are introduced in Section 2.3.) As always in algebra, associativity
makes it possible to write
a1a2∨ · ·· ∨ an
without using parentheses (and
the same for ).
For instance, for
A, B Pow X
, we have
AB
=
AB
and
AB
=
AB
.
So Pow Xis a lattice.
For
α,βPart A
, if we regard
α
and
β
as equivalence relations, then the
meet formula is trivial:
αβ
=
αβ,
but the formula for joins is a bit more
complicated:
xy
(
mod αβ
)iff there is a sequence
x
=
z0, z1, . . . , zn
=
y
of elements
of Asuch that zizi+1 (mod α)or zizi+1 (mod β)for 0i<n.
So Part Ais a lattice; it is called the partition lattice on A.
For an ordered set
P
, the order
Down P
is a lattice:
AB
=
AB
and
AB=ABfor A, B Down P.
To treat lattices as algebras, define an algebra (
L, ,
) a lattice iff
L
is a nonempty set,
and
are binary operations on
L
, both
and
are
10 1. Basic Concepts
idempotent, commutative, and associative, and they satisfy the two absorp-
tion identities. A lattice as an algebra and a lattice as an ordered set are
“equivalent” concepts: Let the order
L
= (
L,
) be a lattice. Then the algebra
La
= (
L, ,
) is a lattice. Conversely, let the algebra
L
= (
L, ,
) be a
lattice. Define
ab
iff
ab
=
b.
Then
Lp
= (
L,
) is an ordered set, and
the ordered set
Lp
is a lattice. For an ordered set
L
that is a lattice, we have
Lap =L; for an algebra Lthat is a lattice, we have Lpa =L.
Note that for lattices as algebras, the Duality Principle takes on the
following very simple form.
Duality Principle for Lattices.
Let Φbe a statement about lattices ex-
pressed in terms of
and
. The dual of Φis the statement we get from Φby
interchanging
and
. If Φis true for all lattices, then the dual of Φis also
true for all lattices.
If the operations are understood, we will say that
L
(rather than (
L, ,
))
is a lattice. The dual of the lattice
L
will be denoted by
Ld
; the ordered set
Ldis also a lattice.
A finite lattice
L
is planar if it is planar as an ordered set (see Section 1.1.2).
We have quite a bit of flexibility to construct a planar diagram for an ordered
set, but for a lattice, we are much more constrained because
L
has a zero,
which must be the lowest element and a unit, which must be the highest
element—contrast this with Figure 1.1. All lattices with five or fewer elements
are planar; all but the five chains are shown in the first two rows of Figure 1.6.
The third row of Figure 1.6 provides an example of “good” and “bad”
lattice diagrams; the two diagrams represent the same lattice, C
2
3
. Planar
diagrams are the best. Diagrams in which meets and joins are hard to figure
out are not of much value.
In the last row of Figure 1.6 there are two more diagrams. The one on the
left is not planar; nevertheless, it is very easy to work with: joins and meets
are easy to see (the notation M
3
[C
3
] will be explained in Section 6.1). The one
on the right is not a lattice: the two black-filled elements have no join.
In this book, we deal almost exclusively with finite lattices. Some concepts,
however, are more natural to introduce in a more general context. An ordered
set (
L,
) is a complete lattice if
WX
and
VX
exist for all
XL
. All finite
lattices are complete, of course.
1.2.2 Semilattices and closure systems
Asemilattice (
S,
) is an algebra: a nonempty set
S
with an idempotent,
commutative, and associative binary operation
. A join-semilattice (
S, ,
)
is a structure, where (
S,
) is a semilattice, (
S,
) is an ordered set, and
ab
iff
ab
=
b
. In the ordered set (
S,
), we have
W{a, b}
=
ab
. As conventional,
we write (S, ) for (S, ,) or just Sif the operation is understood.
1.2. Lattices and semilattices 11
i
o
a
b
c
i
o
a b c
B2C1+B2B2+C1
N5M3
C3
2
M3[C3]
Figure 1.6: More diagrams.
12 1. Basic Concepts
Similarly, a meet-semilattice (
S, ,
) is a structure, where (
S,
) is a semi-
lattice, (
S,
) is an ordered set, and
ab
iff
ab
=
a
. In the ordered set
(
S,
), we have
V{a, b}
=
ab
. As conventional, we write (
S,
) for (
S, ,
)
or just Sif the operation is understood.
If (
L, ,
) is a lattice, then (
L,
) is a join-semilattice and (
L,
) is a
meet-semilattice; moreover, the orderings agree. The converse also holds.
Let
L
be a lattice and let
C
be a nonempty subset of
L
with the property
that for every
xL
, there is a smallest element
x
of
C
with
xx
. We call
Caclosure system in L, and xthe closure of xin C.
Obviously,
C
, as an ordered subset of
L
, is a lattice: For
x, y C
, the meet
in Cis the same as the meet in L, and the join is
xCy=xLy.
Let
L
be a complete lattice and let
C
be
V
-closed subset of
L
, that is,
if
XC
, then
VXC
. (Since
V
= 1, such a subset is nonempty and
contains the 1 of L.) Then Cis a closure system in L, and for every xL,
x=^(yC|xy).
1.3. Some algebraic concepts
1.3.1 Homomorphisms
The lattices
L1
= (
L1,,
) and
L2
= (
L2,,
) are isomorphic as algebras
(in symbols,
L1
=L2
), and the map
ϕ:L1L2
is an isomorphism iff
ϕ
is
one-to-one and onto and
ϕ(ab) = ϕa ϕb,(1)
ϕ(ab) = ϕa ϕb(2)
for a, b L1.
A map, in general, and a homomorphism, in particular, is called injective
if it is onto, surjective if it is one-to-one, and bijective if it is one-to-one and
onto.
An isomorphism of a lattice with itself is called an automorphism. The
automorphisms of a lattice
L
form a group
Aut L
under composition. A lattice
L
is rigid if the identity map is the only automorphism of
L
, that is, if
Aut L
is the one-element group.
It is easy to see that two lattices are isomorphic as ordered sets iff they
are isomorphic as algebras.
Let us define a homomorphism of the join-semilattice (
S1,
) into the join-
semilattice (
S2,
) as a map
ϕ:S1S2
satisfying
(1)
; similarly, for meet-semi-
lattices, we require
(2)
. A lattice homomorphism (or simply, homomorphism)
ϕ
of the lattice
L1
into the lattice
L2
is a map of
L1
into
L2
satisfying both
(1)
1.3. Some algebraic concepts 13
and
(2)
. A homomorphism of a lattice into itself is called an endomorphism.
A one-to-one homomorphism will also be called an embedding.
Note that meet-homomorphisms, join-homomorphisms, and (lattice) homo-
morphisms are all isotone.
Figure 1.7: Morphisms.
Figure 1.7 shows three maps of the four-element lattice B2into the three-
element chain C
3
. The first map is isotone but it is neither a meet- nor a
join-homomorphism. The second map is a join-homomorphism but is not a
meet-homomorphism, thus not a homomorphism. The third map is a (lattice)
homomorphism.
Various versions of homomorphisms and embeddings will be used. For in-
stance, for lattices and join-semilattices, there are also
{∨,
0
}
-homomorphism,
and so on, with obvious meanings. An onto homomorphism
ϕ
is also called
surjective, while a one-to-one homomorphism is called injective; it is the
same as an embedding. For bounded lattices, we often use bounded homo-
morphisms and bounded embeddings, that is,
{
0
,
1
}
-homomorphisms and
{
0
,
1
}
-embeddings. (In the literature, bounded homomorphisms sometimes
have a different definition; this is unlikely to cause any confusion.)
It should always be clear from the context what kind of homomorphism we
are considering. If we say, “let
ϕ
be a homomorphism of
K
into
L
”, where
K
and
L
are lattices, then
ϕ
is a lattice homomorphism, unless otherwise stated.
1.3.2 Sublattices
Asublattice (
K, ,
) of the lattice (
L, ,
) is defined on a nonempty subset
K
of
L
with the property that
a
,
bK
implies that
ab, abK
(the operations
,
are formed in (
L, ,
)), and the
and the
of (
K, ,
) are restrictions
to
K
of the
and the
of (
L, ,
), respectively. Instead of “(
K, ,
) is a
sublattice of (
L, ,
)”, we will simply say that “
K
is a sublattice of
L
”—in
symbols,
KL
. Of course, a sublattice of a lattice is again a lattice. If
K
is
a sublattice of L, then we call Lan extension of K—in symbols, LK.
For every
HL
,
H6
=
, there is a smallest sublattice
sub
(
H
)
L
containing
H
called the sublattice of
L
generated by
H
. We say that
H
is a
generating set of sub(H).
For a bounded lattice
L
, the sublattice
K
is bounded (also called a
{
0
,
1
}
-
sublattice) if the 0 and 1 of
L
are in
K
. Similarly, we can define a
{
0
}
-sublattice,
14 1. Basic Concepts
bounded extension, and so on.
The subset
K
of the lattice
L
is called convex iff
a, b K
,
cL
, and
acb
imply that
cK
. We can add the adjective “convex” to sublattices,
extensions, and embeddings. A sublattice
K
of the lattice
L
is convex if it a
convex subset of
L
. Let
L
be an extension of
K
; then
L
is a convex extension
if
K
is a convex sublattice. An embedding is convex if the image is a convex
sublattice.
For a, b L,ab, the interval
I= [a, b] = {x|axb}
is an important example of a convex sublattice. We will use the notation 1
I
for the largest element of
I
, that is,
b
and 0
I
for the smallest element of
I
,
that is, a.
An interval [
a, b
] is trivial if
a
=
b
. The smallest nontrivial intervals are
called prime; that is, [
a, b
] is prime iff
ab
. Another important example of a
convex sublattice is an ideal. A nonempty subset
I
of
L
is an ideal iff it is a
down set with the property:
(Id) a, b Iimplies that abI.
An ideal
I
of
L
is proper if
I6
=
L
. Since the intersection of any number of
ideals is an ideal, unless empty, we can define
id
(
H
), the ideal generated by a
subset
H
of the lattice
L
, provided that
H6
=
. If
H
=
{a}
, we write
id
(
a
)
for
id
(
{a}
), and call it a principal ideal. Obviously,
id
(
a
) =
{x|xa}
=
a
.
So instead of
id
(
a
), we could use
a
; many do, who work in categorical aspects
of lattice theory—and use id for the identity map.
The set
Id L
of all ideals of
L
is an ordered set under set inclusion, and as
an ordered set it is a lattice. In fact, for
I, J Id L
, the lattice operations in
Id L
are
IJ
=
id
(
IJ
) and
IJ
=
IJ
. So we obtain the formula for the
ideal join:
xIJiff xijfor some iI,jJ.
We call Id Lthe ideal lattice of L. Now observe the formulas: id(a)id(b) =
id
(
ab
),
id
(
a
)
id
(
b
) =
id
(
ab
). Since
a6
=
b
implies that
id
(
a
)
6
=
id
(
b
), these
yield:
The map a7→ id(a)embeds Linto Id L.
Since the definition of an ideal uses only
and
, it applies to any join-
semilattice
S
. The ordered set
Id S
is a join-semilattice and the same join
formula holds as the one for lattices. Since the intersection of two ideals could
be empty,
Id S
is not a lattice, in general. However, for a
{∨,
0
}
-semilattice (a
join-semilattice with zero), Id Sis a lattice.
1.3. Some algebraic concepts 15
For lattices (join-semilattices)
S
and
T
, let
ε:ST
be an embedding.
We call
ε
an ideal-embedding if
εS
is an ideal of
T
. Then, of course for any
ideal
I
of
S
, we have that
εI
is an ideal of
T
. Ideal-embeddings play a major
role in Chapter 20.
By dualizing, we get the concepts of filter,
fil
(
H
), the filter generated by a
subset Hof the lattice L, provided that H6=,principal filter fil(a), and so
on.
1.3.3 Congruences
An equivalence relation
α
on a lattice
L
is called a congruence relation, or
congruence, of Liff ab(mod α) and cd(mod α) imply that
acbd(mod α),(SP)
acbd(mod α)(SP)
(Substitution Properties). Trivial examples are the relations
0
and
1
(introduced
in Section 1.1.4). As in Section 1.1.4, for
aL
, we write
a/α
for the congruence
class (congruence block) containing
a
; observe that
a/α
is a convex sublattice.
If
L
is a lattice,
KL
, and
α
a congruence on
L
, then
αeK
, the restriction
of αto K,is a congruence of K. Formally, for x, y K,
xy(mod αeK) iff xy(mod α) in L.
We call αdiscrete on Kif αeK=0.
Sometimes it is tedious to compute that a binary relation is a congruence
relation. Such computations are often facilitated by the following result
(G. Gr¨atzer and E. T. Schmidt [
116
] and F. Maeda [
164
]), referred to as the
Technical Lemma in the literature.
Lemma 1.1.
A reflexive binary relation
α
on a lattice
L
is a congruence
relation iff the following three properties are satisfied for x, y, z, t L:
(i) xy(mod α)iff xyxy(mod α).
(ii) xyz
,
xy
(
mod α
), and
yz
(
mod α
)imply that
xz
(mod α).
(iii) xy
and
xy
(
mod α
)imply that
xtyt
(
mod α
)and
xtyt(mod α).
Let
Con L
denote the set of all congruence relations on
L
ordered by set
inclusion (remember that we can view αCon Las a subset of L2).
We use the Technical Lemma to prove the following result.
16 1. Basic Concepts
Theorem 1.2. Con Lis a lattice. For α,βCon L,
αβ=αβ.
The join, αβ, can be described as follows:
xy(mod αβ)iff there is a sequence
xy=z0z1≤ ··· ≤ zn=xy
of elements of
L
such that
zizi+1
(
mod α
)or
zizi+1
(
mod β
)for every
iwith 0i<n.
Remark. For the binary relations
γ
and
δ
on a set
A
, we define the binary
relation
γδ
, the product of
γ
and
δ
, as follows: for
a, b A
, the relation
a γ δ b
holds iff
a γ x
and
x δ b
for some
xA
. The relation
αβ
is formed
by repeated products. Theorem 1.2 strengthens this statement.
The integer
n
in Theorem 1.2 can be restricted for some congruence joins.
We call the congruences
α
and
β
permutable if
αβ
=
αβ
. A lattice
L
is
congruence permutable if any pair of congruences of
L
are permutable. The
chain Cnis congruence permutable iff n2.
Con L
is called the congruence lattice of
L
. Observe that
Con L
is a
sublattice of
Part L
; that is, the join and meet of congruence relations as
congruence relations and as equivalence relations (partitions) coincide.
If
L
is nontrivial, then
Con L
contains the two-element sublattice
{0,1}
.
If
Con L
=
{0,1}
, we call the lattice
L
simple. All the nontrivial lattices of
Figure 1.5 are simple. Of the many lattices of Figure 1.6, only M3is simple.
Given
a, b L
, there is a smallest congruence
con
(
a, b
)—called a principal
congruence—under which ab. The formula
(3) α=_( con(a, b)|ab(mod α) )
is trivial but important. For HL, the smallest congruence under which H
is in one class is formed as con(H) = W( con(a, b)|a, b H).
Homomorphisms and congruence relations express two sides of the same
phenomenon. Let
L
be a lattice and let
α
be a congruence relation on
L
. Let
L/α
=
{a/α|aL}
. Define
and
on
L/α
by
a/αb/α
= (
ab
)
/α
and
a/αb/α
= (
ab
)
/α
. The lattice axioms are easily verified. The lattice
L/αis the quotient lattice of Lmodulo α.
Lemma 1.3. The map
ϕα:x7→ x/α,for xL,
is a homomorphism of Lonto L/α.
1.3. Some algebraic concepts 17
The lattice
K
is a homomorphic image of the lattice
L
iff there is a
homomorphism of
L
onto
K
. Theorem 1.4 (illustrated in Figure 1.8) states
that any quotient lattice is a homomorphic image. To state it, we need one
more concept: Let
ϕ:LL1
be a homomorphism of the lattice
L
into the
lattice
L1
, and define the binary relation
α
on
L
by
xαy
iff
ϕx
=
ϕy
; the
relation
α
is a congruence relation of
L
, called the kernel of
ϕ
, in notation,
ker(ϕ) = α.
Theorem 1.4
(Homomorphism Theorem)
.
Let
L
be a lattice. Any homomor-
phic image of
L
is isomorphic to a suitable quotient lattice of
L
. In fact, if
ϕ:LL1
is a homomorphism of
L
onto
L1
and
α
is the kernel of
ϕ
, then
L/α
=L1
; an isomorphism (see Figure 1.8)is given by
ψ:x/α7→ ϕx
for
xL.
onto
L
ϕ
L1
L/
x
7→
x/α
α
ψ:x/α7→ ϕx
Figure 1.8: The Homomorphism Theorem.
We also know the congruence lattice of a homomorphic image:
Theorem 1.5
(Second Isomorphism Theorem)
.
Let
L
be a lattice and let
α
be a congruence relation of
L
. For any congruence
β
of
L
such that
βα
,
define the relation β/αon L/αby
x/αy/α(mod β/α)iff xy(mod β).
Then
β/α
is a congruence of
L/α
. Conversely, every congruence
γ
of
L/α
can
be (uniquely)represented in the form
γ
=
β/α
for some congruence
βα
of
L
. In particular, the congruence lattice of
L/α
is isomorphic with the interval
[α,1]of the congruence lattice of L.
Let
L
be a bounded lattice. A congruence
α
of
L
separates 0 if 0
/α
=
{
0
}
,
that is,
x
0 (
mod α
) implies that
x
= 0. Similarly, a congruence
α
of
L
separates 1 if 1
/α
=
{
1
}
, that is,
x
1 (
mod α
) implies that
x
= 1. We call
the lattice
L
non-separating if 0 and 1 are not separated by any congruence
α6=0.
Similarly, a homomorphism
ϕ
of the lattices
L1
and
L2
with zero is 0-
separating if ϕ0 = 0, but ϕx 6= 0 for x6= 0.
Chapter
2
Special Concepts
In this chapter we introduce special elements, constructions, and classes of
lattices that play an important role in the representation of finite distributive
lattices as congruence lattices of finite lattices.
2.1. Elements and lattices
In a nontrivial finite lattice
L
, an element
a
is join-reducible if
a
= 0 or if
a
=
bc
for some
b<a
and
c<a
; otherwise, it is join-irreducible. Let
J
(
L
)
denote the set of all join-irreducible elements of
L
, regarded as an ordered set
under the ordering of L. By definition, 0 /J(L). For aL, set
J(a) = {x|xa, x J(L)}= id(a)J(L),
that is,
J
(
a
) is
a
formed in
J
(
L
). Note that, by definition, 0 is not a
join-irreducible element; and similarly, 1 is not a meet-irreducible element.
In a finite lattice, every element is a join of join-irreducible elements (indeed,
a=WJ(a)), and similarly for meets.
Dually, we define meet-reducible