Path-independence and closure operators with the anti-exchange property

Abstract

Recently in Koshevoy (Math. Social Sci. 38 (1999) 35), it was established a connection between the theory of choice functions satisfying path-independence condition and closure operators with the anti-exchange property. Closure operators with the anti-exchange property are a combinatorial abstraction of usual convex hull closure in Euclidean spaces. Interest in these structures has its sources in different fields of mathematics. We demonstrate that path-independent choice functions provide another source for this structure. Specifically, we associate to a choice function f a collection of expanding maps. We prove that a function f is path-independent if and only if all the maps of this collection are coinciding anti-exchange closure operators. Consequences of such a characterization are demonstrated.