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

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