Pp. 163-210 (48)
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.
Ideal, filter, prime filter, maximal filter, irreducibility, decomposition,
set-theoretical embedding, clopen set, Stone space, homomorphism,
Priestley space, Priestley duality.