The congruences of a lattice form the congruence lattice. Over the last several decades, the study of congruence lattices has established itself as a large and important field with a great number of interesting and deep results, as well as many open problems. Written by one of the leading experts in lattice theory, this text provides a self-contained introduction to congruences of finite lattices and presents the major results of the last 90 years. It features the author’s signature “Proof-by-Picture” method, which is used to convey the ideas behind formal proofs in a visual, more intuitive manner.
Key features include:- an insightful discussion of techniques to construct "nice" finite lattices with given congruence lattices and "nice" congruence-preserving extensions
- complete proofs, an extensive bibliography and index, and over 180 illustrations
- additional chapters covering new results of the last seven years, increasing the size of this edition to 430 pages, 360 statements, and 262 references
This text is appropriate for a one-semester graduate course in lattice theory, and it will also serve as a valuable reference for researchers studying lattices.
Reviews of previous editions:
“[This] monograph…is an exceptional work in lattice theory, like all the contributions by this author. The way this book is written makes it extremely interesting for the specialists in the field but also for the students in lattice theory. ― Cosmin Pelea, Studia Universitatis Babes-Bolyai Mathematica LII (1), 2007 "The book is self-contained, with many detailed proofs presented that can be followed step-by-step. I believe that this book is a much-needed tool for any mathematician wishing a gentle introduction to the field of congruences representations of finite lattices, with emphasis on the more 'geometric' aspects." ― Mathematical Reviews
Author(s): George Grätzer
Edition: 3
Publisher: Birkhäuser
Year: 2023
Language: English
Pages: 439
City: Cham
Short Contents
Contents
Glossary of Notation
Picture Gallery
Preface
Introduction
The topics
Topic A. Congruence lattices of finite lattices
The two types of RTs
Topic B. The ordered set of principal congruences of finite lattices
Topic C. The congruence structure of finite lattices
Topic D. Congruence properties of slim, planar, semimodular (SPS) lattices.
Proof-by-Picture
Outline and notation
Part I. A Brief Introduction to Lattices
Part II. Some Special Techniques
Part III. RTs
Part IV. ETs
Part V. Congruence Lattices of Two Related Lattices
Part VI. The Ordered Set of Principal Congruences
Part VII. Congruence Extensions and Prime Interval
Part VIII. Six Congruence Properties of SPS lattices
Notation
Part I A Brief Introduction to Lattices
Chapter 1 Basic Concepts
1.1. Ordering
1.1.1 Ordered sets
1.1.2 Diagrams
1.1.3 Constructions of ordered sets
1.1.4 Partitions
1.2. Lattices and semilattices
1.2.1 Lattices
1.2.2 Semilattices and closure systems
1.3. Some algebraic concepts
1.3.1 Homomorphisms
1.3.2 Sublattices
1.3.3 Congruences
Chapter 2 Special Concepts
2.1. Elements and lattices
2.2. Direct and subdirect products
2.3. Terms and identities
2.4. Gluing and generalizations
2.4.1 Gluing
2.4.2 Generalizations
2.5. Modular and distributive lattices
2.5.1 The characterization theorems
2.5.2 Finite distributive lattices
2.5.3 Finite modular lattices
Chapter 3 Congruences
3.1. Congruence spreading
3.2. Finite lattices and prime intervals
3.3. Congruence-preserving extensions and variants
Chapter 4 Planar Semimodular Lattices
4.1. Planar lattices
4.2. Two acronyms: SPS and SR
4.3. SPS lattices
4.4. Forks
4.5. Rectangular lattices
Congruences of rectangular lattices
4.6. Rectangular intervals
4.7. Special diagrams for SR lattices
Two approaches
Natural diagrams
C1-diagrams
4.8. Natural diagrams and C1-diagrams
4.9. Discussion
Part II Some Special Techniques
Chapter 5 Chopped Lattices
5.1. Basic definitions
5.2. Compatible vectors of elements
5.3. Compatible vectors of congruences
5.4. From the chopped lattice to the ideal lattice
5.5. Sectional complementation
Chapter 6 Boolean Triples
6.1. The general construction
6.2. The congruence-preserving extension property
6.3. The distributive case
6.4. Two interesting intervals
6.5. Discussion
Chapter 7 Cubic Extensions
7.1. The construction
7.2. The basic property
Part III RTs
Chapter 8 Sectionally Complemented RT
8.1. The Basic RT
8.2. Proof-by-Picture
8.3. Computing
8.4. Sectionally complemented lattices
8.5. The N-relation
Refinements
Join expressions and join covers
The relation N
A closure operator
8.6. Discussion
Sectionally complemented chopped lattices
Congruence class sizes
Spectra
Valuations
Chapter 9 Minimal RT
9.1. The results
9.2. Proof-by-Picture for the minimal construction
9.3. The formal construction
9.4. Proof-by-Picture for minimality
9.5. Computing minimality
9.6. Discussion
History
Improved bounds
Different approaches to minimality
Chapter 10 Semimodular RT
10.1. Semimodular lattices
10.2. Proof-by-Picture
10.3. Construction and proof
10.4. All congruences principal RT for planar semimodular lattices
10.5. Discussion
An addendum
Problems
Chapter 11 Rectangular RT
11.1. Results
11.2. Proof-by-Picture
11.3. All congruences principal RT
11.4. Discussion
Chapter 12 Modular RT
12.1. Modular lattices
12.2. Proof-by-Picture
12.3. Construction and proof
12.4. Discussion
The Independence Theorem for Modular Lattices
Two stronger results
Arguesian lattices
Problems
Chapter 13 Uniform RT
13.1. Uniform lattices
13.2. Proof-by-Picture
13.3. The lattice N(A,B)
The construction
Congruences
Congruence classes
13.4. Formal proof
13.5. Discussion
Isoform lattices
The N(A, B, α) construction
Problems
Part IV ETs
Chapter 14 Sectionally Complemented ET
14.1. Sectionally complemented lattices
14.2. Proof-by-Picture
14.3. Easy extensions
14.4. Formal proof
14.5. Discussion
Chapter 15 Semimodular ET
15.1. Semimodular lattices
15.2. Proof-by-Picture
15.3. The conduit
15.4. The construction
15.5. Formal proof
15.6. Rectangular ET
15.7. Discussion
Chapter 16 Isoform ET
16.1. Isoform lattices
16.2. Proof-by-Picture
16.3. Formal construction
16.4. The congruences
16.5. The isoform property
16.6. Discussion
16.6.1 Variants
Regular lattices
Permutable congruences
Deterministic isoform lattices
Naturally isoform lattices
A generalized construction
16.6.2 Problems
16.6.3 The Congruence Lattice and the Automorphism Group
16.6.4 More problems
Chapter 17 Magic Wands
17.1. Constructing congruence lattices
Bijective maps
Surjective maps
17.2. Proof-by-Picture for bijective maps
17.3. Verification for bijective maps
17.4. 2/3-Boolean triples
17.5. Proof-by-Picture for surjective maps
17.6. Verification for surjective maps
17.7. Discussion
First generalization of Theorem 17.1
Second generalization of Theorem 17.1
A generalization of Theorem 17.2
Magic wands with special properties
Fully invariant congruences
The 1/3-Boolean triple construction
Part V Congruence Lattices of Two Related Lattices
Chapter 18 Sublattices
18.1. The results
18.2. Proof-by-Picture
18.3. Multi-coloring
18.4. Formal proof
18.5. Discussion
History
Applications
Isotone maps
Size and breadth
2-distributive lattices
Chapter 19 Ideals
19.1. The results
19.2. Proof-by-Picture for the main result
19.3. Formal proof
Categoric preliminaries
19.4. Proof-by-Picture for planar lattices
19.5. Discussion
Chapter 20 Two Convex Sublattices
20.1. Introduction
20.2. 19.2. Proof-by-Picture
20.3. Proof
The first triple gluing
The second triple gluing
Completing the proof
20.4. Discussion
Chapter 21 Tensor Extensions
21.1. The problem
21.2. Three unary functions
21.3. Defining tensor extensions
21.4. Computing
Some special elements
An embedding
Distributive lattices
21.5. Congruences
Congruence spreading
Some structural observations
Lifting congruences
The main lemma
21.6. The congruence isomorphism
21.7. Discussion
Part VI The Ordered Set of Principal Congruences
Chapter 22 The RT for Principal Congruences
22.1. Representing the ordered set of principal congruences
22.2. Proving the RT
The lattice Frame
The lattice S(p, q)
The lattice K
22.3. An independence theorem
22.3.1 Frucht lattices
22.3.2 An independence result
22.4. Discussion
Chapter 23 Minimal RTs
23.1. The Minimal RT
23.2. Three or more dual atoms
23.3. Exactly two dual atoms
23.3.1 Constructing the lattice L
23.3.2 Fusion of ordered sets
23.3.3 Splitting an element of an ordered set
23.3.4 Admissible congruences and extensions
23.3.5 The Bridge Theorem
Preliminaries for the bridge construction
The Bridge Theorem, statement and proof
23.3.6 Some technical results and proofs
23.4. Small distributive lattices
Distributive lattices of ≤ 7 size
A distributive lattice of size 8
23.5. Full representability and planarity
23.6. Discussion
Chapter 24 Principal Congruence Representable Sets
24.1. Chain representability
24.2. Proving the Necessity Theorem
24.3. Proof-by-Picture for the Sufficiency Theorem
24.3.1 A colored chain
24.3.2 The frame lattice C and the lattice W(p, q)
24.3.3 Flag lattices
24.4. Construction for the Sufficiency Theorem
24.5. Proving the Sufficiency Theorem
24.5.1 Two preliminary lemmas
24.5.2 The congruences of a W(p, q) lattice
24.5.3 The congruences of flag lattices
24.5.4 The congruences of L
24.5.5 Principal congruences of L
24.6. Discussion
Chapter 25 Isotone Maps
25.1. Two isotone maps
Sublattices
Bounded homomorphisms
25.2. Sublattices, sketching the proof
25.3. Isotone surjective maps
25.4. Proving the Representation Theorem
25.5. Discussion
Part VII Congruence Extensions and Prime Intervals
Chapter 26 The Prime-projectivity Lemma
26.1. Introduction
26.2. Proof
26.3. Discussion
Chapter 27 The Swing Lemma
27.1. The statement
27.2. Proving the Swing Lemma
27.3. Some variants and consequences
27.4. The Two-Cover Condition is not sufficient
27.5. Applying the Swing Lemma to trajectories
Chapter 28 Fork Congruences
28.1. The statements
28.2. Proofs
28.3. Discussion
Part VIII The Six Congruence Properties of SPS lattices
Chapter 29 Six Major Properties
29.1. Introduction
29.2. Czédli’s four properties
29.2.1 Proofs
The Partition Property
The Maximal Cover Property
The No Child Property
The Two-pendant Four-crown Property
29.3. The 3P3C property
29.3.1 Some relations
The V-relation
V-lemma.
The W-relation
The W-relation, Version 1
The W-relation, Version 2
The 3C-relation
The 3C-relation, Version 1
The 3C-relation, Version 2
29.3.2 Proof
Version 1
Version 2
Patch lattices
29.4. Discussion
Lamps
A Meta Theorem
Problems
Bibliography
Index
