A topological characterization of complete distributive lattices

✍ Scribed by Lucian Beznea

• Publisher: Elsevier Science
• Year: 1984
• Tongue: English
• Weight: 181 KB
• Volume: 49
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/0012-365x(84)90063-3

✦ Synopsis

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 X (of. [4]). An important subclass of these spaces is that of Priest/ey spaces, characterized by the following property: for every x, y ~X with x~y there is an increasing ciopen set A (i.e. A is closed and open and such that a e A, a ~< z implies that z ~ A) which separates x from y, i.e., x ~ A and y¢~ A. It is known (eft. [5,6]) that there is a dual equivalence between the category Ldl01 of distributive lattices with least and greatest element and the category IP 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: (EO) De_X, D is increasing and open implies/5" is increasing clopen (where A* denotes the least increasing set which includes A). This result generalizes a welt-known characterization of complete Boolean algebras in terms of associated Stone spaces (see [2, Ch. 11-I, Section 4, 1.emma 1], for instance).

We shall also prove that an ordered compact space that fulfils rE0) is necessarily a Priestley space.