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 [1] 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. [1] K. Adaricheva and J.B.Nation, On implicational bases of closure systems with unique critical sets, arxiv:1205.2881

Presented as a part of invited talk at Conference on Universal Algebra and Lattice Theory, Szeged, Hungary, 21-25 June, 2012

References in corpus (2)

• Representing finite convex geometries by relatively convex sets
• On implicational bases of closure systems with unique critical sets