Binary Convexities and Distributive Lattices

A convex structure is binary if every finite family of pairwise intersecting convex sets has a non-empty intersection. Distributive lattices with the convexity of all order-convex sublattices are a prominent type of example, because they correspond exactly to the intervals of a binary convex structure which has a certain separation property. In one direction, this result relies on a study of so-called base-point orders induced by a convex structure. These orderings are used to construct an 'intrinsic' topology. For binary convexities, certain basic questions are answered with the aid of some results on completely distributive lattices. Several applications are given. Dimension problems are studied in a subsequent paper. 0. Introduction A convex structure consists of a set X, together with a collection ^ of subsets of X, henceforth called convex sets, such that (1.1) the empty set and the set X are convex; (1.2) the intersection of convex sets is convex; (1.3) the union of an updirected collection of convex sets is convex. The collection € itself is called a convexity on X. Axiom (1.2) allows the construction of an associated (convex) hull operator (usually denoted by h) in the obvious way. The hull of a finite set is called a polytope, and the hull of a two-point set is also called an interval. A half-space is a convex set with a convex complement. The following separation axioms—comparable with the axioms Tl 5 . . . ,T4, in topology—are used frequently: Sx: singletons are convex (which we will assume throughout); S2: two distinct points are in complementary half-spaces; S3: a convex set is an intersection of half-spaces; S4: two disjoint convex sets extend to complementary half-spaces. In this paper we will concentrate largely on a particular, though fundamental, class of convexities with the following binarity property: each finite collection of pairwise intersecting convex sets has a non-empty intersection. The basic types of examples are described in §1. Many of these examples arise in a topological context; for other examples, a natural topology can be constructed. Binary convex structures on a topological space have been studied extensively in the past years [15, 17, 19, 22, 29, 34, 38]. The topology and the convexity are always assumed to be compatible in the sense that polytopes are closed. A triple, consisting of a set X, a convexity €, and a compatible topology ST, is called a topological convex Proc. London Math. Soc. (3), 48 (1984), 1-33. 5388.3.48 A