Closure spaces that are not uniquely generated

Because antimatroid closure spaces satisfy the anti-exchange axiom, it is easy to show that they are uniquely generated. That is, the minimal set of elements determining a closed set is unique. A prime example is a discrete convex geometry in Euclidean space where closed sets are uniquely generated by their extreme points. But, many of the geometries arising in computer science, e.g. the world wide web or rectilinear VLSI layouts are not uniquely generated. Nevertheless, these closure spaces still illustrate a number of fundamental antimatroid properties which we demonstrate in this paper. In particular, we examine both a pseudo-convexity operator and the Galois closure of formal concept analysis. In the latter case, we show how these principles can be used to automatically convert a formal concept lattice into a system of implications.