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Sets and ordered structures. (English) Zbl 1245.06001

Oak Park, IL: Bentham eBooks (ISBN 978-1-60805-338-4/ebook). iv, 253 p. (2012).
In this book the author presents several basis methods and results of the theory of ordered sets that are used in various branches of mathematics.
In the first chapter he starts with a sketch of axiomatic set theory and categories. In the following chapter he deals with partially and totally ordered sets. He discusses statements that are equivalent to the axiom of choice. He establishes the basic properties of well-ordered sets. In the third chapter he introduces ordinals and cardinals and presents the von Neumann construction of ordinals. In Chapter 4 he provides lattice-theoretic tools used in various fields of mathematics: He gives some background necessary in universal algebra. He investigates complete lattices, closure operations and Galois connections. In Chapter 5 he discusses meet and join representations in a lattice and develops the topological duality for distributive lattices. In the last section the author sketches some applications in algebra, topology, universal algebra, analysis and logic.

MSC:

06-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to ordered structures
03E10 Ordinal and cardinal numbers
03E25 Axiom of choice and related propositions
06A05 Total orders
06A06 Partial orders, general
06A15 Galois correspondences, closure operators (in relation to ordered sets)
06B05 Structure theory of lattices
06B15 Representation theory of lattices
06B23 Complete lattices, completions
06D20 Heyting algebras (lattice-theoretic aspects)
06D50 Lattices and duality
06E15 Stone spaces (Boolean spaces) and related structures
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