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

361 Pages·2016·7.25 MB·English

by  George Grätzer (auth.)

George Grätzer
The Congruences
of a Finite Lattice
A “Proof-by-Picture” Approach
Second Edition

George Gra¨tzer
The Congruences of a Finite
Lattice
A “Proof-by-Picture” Approach
Second Edition
GeorgeGra¨tzer
Toronto,ON,Canada
ISBN978-3-319-38796-3 ISBN978-3-319-38798-7 (eBook)
DOI10.1007/978-3-319-38798-7
LibraryofCongressControlNumber:2016941750
MathematicsSubjectClassification(2010):06B10,06D05,06C05,06C10
©SpringerInternationalPublishingSwitzerland2016
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To La´szlo´ 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 xxiii
Preface to the Second Edition xxv
Introduction xxvii
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
vii
viii SHORT CONTENTS
III Congruence Lattices of Finite Lattices 87
8 The Dilworth Theorem 89
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 283
SHORT CONTENTS ix
VII Congruence Structure 289
24 Prime Intervals and Congruences 291
25 Some Applications of the Swing Lemma 315
Bibliography 323
Index 337