The congruences of a finite lattice : a proof-by-picture approach /
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| Author / Creator: | Gratzer, George A. |
|---|---|
| Imprint: | Boston ; Basel : Birkhäuser, c2006. |
| Description: | xxii, 281 p. : ill. ; 24 cm. |
| Language: | English |
| Subject: | |
| Format: | Print Book |
| URL for this record: | http://pi.lib.uchicago.edu/1001/cat/bib/5789333 |
Table of Contents:
- Preface
- Glossary of Notation
- Picture Gallery
- Acknowledgments
- I. A Brief Introduction to Lattices
- 1. Basic Concepts
- 1.1. Ordering
- 1.1.1. Orders
- 1.1.2. Diagrams
- 1.1.3. Order constructions
- 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
- 2. Special Concepts
- 2.1. Elements and lattices
- 2.2. Direct and subdirect products
- 2.3. Polynomials and identities
- 2.4. Gluting
- 2.5. Modular and distributive lattices
- 2.5.1. The characterization theorems
- 2.5.2. Finite distributive lattices
- 2.5.3. Finite modular lattices
- 3. Congruences
- 3.1. Congruence spreading
- 3.2. Prime intervals
- 3.3. Congruence-preserving extensions and variants
- II. Basic Techniques
- 4. Chopped Lattices
- 4.1. Basic definitions
- 4.2. Compatible vectors of elements
- 4.3. Compatible vectors of congruences
- 4.4. From the chopped lattice to the ideal lattice
- 4.5. Sectional complementation
- 5. Boolean Triples
- 5.1. The general construction
- 5.2. The congruence-preserving extension property
- 5.3. The distributive case
- 5.4. Two interesting intervals
- 6. Cubic Extensions
- 6.1. The construction
- 6.2. The basic property
- III. Representation Theorems
- 7. The Dilworth Theorem
- 7.1. The representation theorem
- 7.2. Proof-by-Picture
- 7.3. Computing
- 7.4. Sectionally complemented lattices
- 7.5. Discussion
- 8. Minimal Representations
- 8.1. The results
- 8.2. Proof-by-Picture for minimal construction
- 8.3. The formal construction
- 8.4. Proof-by-Picture for minimality
- 8.5. Computing minimality
- 8.6. Discussion
- 9. Semimodular Lattices
- 9.1. The representation theorem
- 9.2. Proof-by-Picture
- 9.3. Construction and proof
- 9.4. Discussion
- 10. Modular Lattices
- 10.1. The representation theorem
- 10.2. Proof-by-Picture
- 10.3. Construction and proof
- 10.4. Discussion
- 11. Uniform Lattices
- 11.1. The representation theorem
- 11.2. Proof-by-Picture
- 11.3. The lattice N (A, B)
- 11.4. Formal proof
- 11.5. Discussion
- IV. Extensions
- 12. Sectionally Complemented Lattices
- 12.1. The extension theorem
- 12.2. Proof-by-Picture
- 12.3. Simple extensions
- 12.4. Formal proof
- 12.5. Discussion
- 13. Semimodular Lattices
- 13.1. The extension theorem
- 13.2. Proof-by-Picture
- 13.3. The conduit
- 13.4. The construction
- 13.5. Formal proof
- 13.6. Discussion
- 14. Isoform Lattices
- 14.1. The result
- 14.2. Proof-by-Picture
- 14.3. Formal construction
- 14.4. The congruences
- 14.5. The isoform property
- 14.6. Discussion
- 15. Independence Theorems
- 15.1. Results
- 15.2. Proof-by-Picture
- 15.2.1. Frucht lattices
- 15.2.2. An automorphism-preserving simple extension
- 15.2.3. A congruence-preserving rigid extension
- 15.2.4. Merging the two extensions
- 15.2.5. The representation theorems
- 15.3. Formal proofs
- 15.3.1. An automorphism-preserving simple extension
- 15.3.2. A congruence-preserving rigid extension
- 15.3.3. Proof of the independence theorems
- 15.4. Discussion
- 16. Magic Wands
- 16.1. Constructing congruence lattices
- 16.1.1. Bijective maps
- 16.1.2. Surjective maps
- 16.2. Proof-by-Picture for bijective maps
- 16.3. Verification for bijective maps
- 16.4. 2/3-boolean triples
- 16.5. Proof-by-Picture for surjective maps
- 16.6. Verification for surjective maps
- 16.7. Discussion
- V. Two Lattices
- 17. Sublattices
- 17.1. The results
- 17.2. Proof-by-Picture
- 17.3. Multi-coloring
- 17.4. Formal proof
- 17.5. Discussion
- 18. Ideals
- 18.1. The results
- 18.2. Proof-by-Picture for the main result
- 18.3. A very formal proof: Main result
- 18.3.1. Categoric preliminaries
- 18.3.2. From Di to Or
- 18.3.3. From Or to He
- 18.3.4. From Ch to Di
- 18.3.5. From He to Ch
- 18.3.6. From Ch to La
- 18.3.7. The final step
- 18.4. Proof for sectionally complemented lattices
- 18.5. Proof-by-Picture for planar lattices
- 18.6. Discussion
- 19. Tensor Extensions
- 19.1. The problem
- 19.2. Three unary functions
- 19.3. Defining tensor extensions
- 19.4. Computing
- 19.4.1. Some special elements
- 19.4.2. An embedding
- 19.4.3. Distributive lattices
- 19.5. Congruences
- 19.5.1. Congruence spreading
- 19.5.2. Some structural observations
- 19.5.3. Lifting congruences
- 19.5.4. The main lemma
- 19.6. The congruence isomorphism
- 19.7. Discussion
- Bibliography
- Index
