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

## 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

### 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
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