A topological characterization of complete distributive lattices

An ordered compact space is a compact topological space X, endowed with a partially ordered relation, whose graph is a closed set of X × X (cf. [4]). An important subclass of these spaces is that of Priestley spaces, characterized by the following property: for every x, y ? X with x ? y there is an increasing clopen set A (i.e. A is closed and open and such that a ? A, a ? z implies that z?A) which separates x from y, i.e., x ? A and y ? A. It is known (cf. [5, 6]) that there is a dual equivalence between the category Ld01 of distributive lattices with least and greatest element and the category P of Priestley spaces.In this paper we shall prove that a lattice L ? Ld01 is complete if and only if the associated Priestley space X verifies the condition: (E0) D ? X, D is increasing and open implies $\text{D}{}^1$ is increasing clopen (where A¹ denotes the least increasing set which includes A).This result generalizes a well-known characterization of complete Boolean algebras in terms of associated Stone spaces (see [2, Ch. III, Section 4, Lemma 1], for instance).We shall also prove that an ordered compact space that fulfils (E0) is necessarily a Priestley space.