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Introduction to lattices and order. (English) Zbl 0701.06001
Cambridge etc.: Cambridge University Press. viii, 248 p. £20.00/hbk; £9.95/pbk (1990).
This is an excellent introductory textbook on ordered sets and lattices. It covers an up-to-date and clear exposition of the basic theory of ordered sets and lattices, contemporary applications in mathematics, computer science and social science, and an introduction to the representation theory of distributive lattices. A lot of exercises are included. The content of the book is well illustrated by the titles of chapters and sections.
1. Ordered sets: Ordered sets. Examples from mathematics, computer science and social science. Diagrams. Maps between ordered sets. The duality principle: down-sets and up-sets. Maximal and minimal elements; top and bottom. Building new ordered sets.
2. Lattices and complete lattices: Lattices as ordered sets. Complete lattices. Chain conditions and completeness. Completions.
3. CPOs, algebraic lattices and domains: Directed joins and algebraic closure operators. CPOs. Finiteness, algebraic lattices and domains. Information systems.
4. Fixpoint theorems: Fixpoint theorems and their applications. The existence of maximal elements and Zorn’s lemma.
5. Lattices as subalgebraic structures: Lattices as algebraic structures. Sublattices, products and homomorphisms. Congruences.
6. Modular and distributive lattices: Lattices satisfying additional identities. The \(M_ 3-N_ 5\) Theorem.
7. Boolean algebras and their applications: Boolean algebras. Boolean terms and disjunctive normal form. Meet LINDA: the Lindenbaum algebra.
8. Representation theory, the finite case: The representation of finite Boolean algebras. Join-irreducible elements. The representation of finite distributive lattices. Duality between finite distributive lattices and finite ordered sets.
9. Ideals and filters: Ideals and filters. Prime ideals, maximal ideals and ultrafilters. The existence of prime ideals, maximal ideals and ultrafilters.
10. Representation theory, the general case: Representation by lattices of sets. The prime ideal space. Duality. Appendix: a topological toolkit.
11. Formal concept analysis: Contexts and their concepts. The fundamental theorem. From theory to practice.
Reviewer: J.Niederle

MSC:
06-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to ordered structures
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