Uniform Antimatroid Closure Spaces

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. The issue of whether a planar convex geometry, that is a discrete set of points in in the plane, exists in which all convex conngurations are triangles, with no quadrilaterals; or as quadrilaterals and triangles, with no pentagons; etc., has fascinated combinatorial mathematicians for many years. Our results throw light on these kinds of questions, even though they do not resolve them.