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 examples, because they correspond exactly to the intervals of a binary convex structure which has a certain separation properly. In one direction, this result relies on a study of so-called base-point orders induced by a convex structure. Thesis ordering 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.