A Representation Theorem for Relatively Complemented Distributive Lattices

Abstract

In this note, we are concerned with the following generalization of a wellknown theorem of M. H. Stone; see (2, 8.2).Theorem 1. Let L be a relatively complemented distributive lattice.(I) If L has no least element, then L is isomorphic to the lattice of non-empty compact-open subsets of an anti-Hausdorff, nearly-Hausdorff, T1-space with a base of open sets consisting of compact-open sets.(II) (3, Theorem 1) If L has a least element, then L is isomorphic to the lattice of all compact-open subsets of a locally compact totally disconnected space. Moreover, the spaces of (I) and (II) are compact if and only if L has a greatest element.The space in question is the space of prime ideals of L with the hull-kernel topology.The author is indebted to M. G. Stanley for several conversations concerning this note.