## Optimum basis of finite convex geometry.(English)Zbl 1423.06012

Summary: Convex geometries form a subclass of closure systems with unique criticals, or $$U C$$-systems. We show that the $$F$$-basis introduced in [the author and J. B. Nation, Discrete Appl. Math. 162, 51–69 (2014; Zbl 1341.06003)] for $$U C$$-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 [“Quasi-acyclic propositional Horn knowledge bases: optimal compression”, IEEE Trans. Knowl. Data Eng. 7, No. 5, 751–762, (1995; doi:10.1109/69.469822)] 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.

### MSC:

 06A15 Galois correspondences, closure operators (in relation to ordered sets) 52A01 Axiomatic and generalized convexity 06E30 Boolean functions

Zbl 1341.06003
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### References:

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