Sets and Ordered Structures

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This e-book presents several basic methods and results of order theory that are currently used in various branches of mathematics. It presents topics that require a broad explanation in a concise and ...
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Representation Theory

Pp. 163-210 (48)

DOI: 10.2174/978160805338411201010163

Author(s): Sergiu Rudeanu

Abstract

The title of this chapter summarizes three types of representation theorems dealt with: representations of the elements of certain lattices as meets/joins of elements from prescribed subsets, isomorphic representation of several types of posets (semilattices, lattices) as posets (semilattices, lattices) of sets with inclusion as partial order, and finally a more sophisticated development of the latter representations in the case of distributive and Boolean lattices: the duality between these categories and certain categories of topological spaces. These types of problems are treated in §§ 2, 3 and 5, respectively. The first section is devoted to ideals and filters both as a preparation to the subsequent sections and in view of the numerous other applications. The topological prerequisites necessary to §5 are collected in §4.

Keywords:

Ideal, filter, prime filter, maximal filter, irreducibility, decomposition, set-theoretical embedding, clopen set, Stone space, homomorphism, Priestley space, Priestley duality.