Optimum basis of finite convex geometry [article]

Kira Adaricheva
2016 arXiv   pre-print
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
more » ... 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
arXiv:1205.3236v5 fatcat:t2aay3p52nbotgpsjspjmtkflq