Optimum basis of finite convex geometry

Abstract

Convex geometries form a subclass of closure systems with unique criticals, or UC-systems. We show that the F-basis introduced in Adaricheva and Nation (2014) for UC-systems, becomes optimum in convex geometries, in two essential parts of the basis: right sides (conclusions) of binary implications and left sides (premises) of non-binary ones. The right sides of non-binary implications can also be optimized, when the convex geometry either satisfies the Carousel property, or does not have D-cycles. The latter generalizes a result of P.L. Hammer and A. Kogan for acyclic Horn Boolean functions. Convex geometries of order convex subsets in a poset also have tractable optimum basis. The problem of tractability of optimum basis in convex geometries in general remains to be open.