# Book:B.A. Davey/Introduction to Lattices and Order/Second Edition

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## B.A. Davey and H.A. Priestley:

## 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**