Uniform Antimatroid Closure Spaces

Abstract

Often the structure of discrete sets can be described in terms of a closure operator. When each closed set has a unique minimal generating set (as in convex geometries in which the extreme points of a convex set generate the closed set), we have an antimatroid closure space. In this paper, we show there exist antimatroid closure spaces of any size, of which convex geometries are only a sub-family, all of whose closed sets are generated by precisely the same number of points. We call them uniform closure spaces.