Choice Functions and Extensive Operators

Abstract

The paper puts forth a theory of choice functions in a neat way connecting it to a theory of extensive operators and neighborhood systems. We consider classes of heritage choice functions satisfying conditions M, N, W, and C, or combinations of these conditions. In terms of extensive operators these classes can be considered as generalizations of symmetric, anti-symmetric and transitive binary relations. Among these classes we meet the well-known classes of matroids and convex geometries. Using a ‘topological’ language we discuss these classes of monotone extensive operators (or heritage choice functions) in terms of neighborhood systems. A remarkable inversion on the set of choice functions is introduced. Restricted to the class of heritage choice functions the inversion turns out to be an involution, and under this involution the axiom N is auto-inverse, whereas the axioms W and M change places.