Representation Theory

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.