A Theory of Finite Closure Spaces Based on Implications

Abstract

Although many notions familiar from topology and matroid theory make sense for arbitrary closure spaces, we claim that "implicational bases" are most worthwhile studying. They are applied in the theory of relational data bases and in formal concept analysis. Further applications in combinatorics and algebra are foreseeable. The main theorem describes the structure of minimum, respectively optimal, implicational bases of any finite closure space. It has strong corollaries for specific closure spaces, e.g., having a geometric, respectively modular, respectively lower distributive, lattice of closed sets.