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

Subject Matter

 * Lattice Theory
 * Order Theory

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