Representation of algebraic convex geometries

Abstract

Convex geometry is a set system generated by the closure operator with the antiexchange axiom. These systems model the concept of convexity in various settings. They are also closely connected to anti-matroids, which are set systems with the property of accessibility. In particular, the latter were used in modelling the states of human learners and found practical applications in designing the automatic tutoring systems. In current work we develop the theoretical foundations of infinite convex geometries in case their closure operator satisfies the finitary property: closure of any subset is a union of closures of its finite subsets. In such case, the convex geometry is called algebraic.