Book:B.A. Davey/Introduction to Lattices and Order/Second Edition
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B.A. Davey and H.A. Priestley: Introduction to Lattices and Order (2nd Edition)
Published $\text {2002}$, Cambridge University Press
- ISBN 978-0-52136-766-0
Subject Matter
Contents
- Preface to the second edition
- Preface to the first edition
- 1. Ordered sets
- Ordered sets
- Examples from social science and computer science
- Diagrams: the art of drawing ordered sets
- Down-sets and up-sets
- Maps between ordered sets
- Exercises
- 2. Lattices and complete lattices
- Lattices as ordered sets
- Lattices as algebraic structures
- Sublattices, products and homomorphisms
- Ideals and filters
- Complete lattices and $\bigcap$-structures
- Chain conditions and completeness
- Join-irreducible elements
- Exercises
- 3. Formal concept analysis
- Contexts and their concepts
- The fundamental theorem of concept lattices
- From theory to practice
- Exercises
- 4. Modular, distributive and Boolean lattices
- Lattices satisfying additional identities
- The $M_3$–$N_5$ Theorem
- Boolean lattices and Boolean algebras
- Boolean terms and disjunctive normal form
- Exercises
- 5. Representation: the finite case
- Building blocks for lattices
- Finite Boolean algebras are powerset algebras
- Finite distributive lattices are down-set lattices
- Finite distributive lattices are finite ordered sets in partnership
- Exercises
- 6. Congruences
- Introducing congruences
- Congruences and diagrams
- The lattice of congruences of a lattice
- Exercise
- 7. Complete lattices and Galois connections
- Closure operations
- Complete lattices coming from algebra: algebraic lattices
- Galois connections
- Completions
- Exercises
- 8. CPOs and fixpoint theorems
- CPOs
- CPOs of partial maps
- Fixpoint theorems
- Calculating with fixpoints
- Exercises
- 9. Domains and information systems
- Domains for computing
- Domains re-modelled. information systems
- Using fixpoint theorems to solve domain equations
- Exercises
- 10. Maximality principles
- Do maximal elements exist? – Zorn's Lemma and the Axiom of Choice
- Prime and maximal ideals
- Powerset algebras and down-set lattices revisited
- Exercises
- 11. Representation: the general case
- Stone's representation theorem for Boolean algebras
- Meet LINDA: the Lindenbaum algebra
- Priestley's representation theorem for distributive lattices
- Distributive lattices and Priestley spaces in partnership
- Exercises
- Appendix A: a topological toolkit
- Appendix B: further reading
- Notation index
- Index