Convex Geometries yielded by Transit Functions

Let V be a finite nonempty set. A transit function is a map R:V× V→ 2V such that R(u,u)=\, $R(u,v)=R(v,u)$ and u∈ R(u,v) hold for every u,v∈ V. A set K⊆ V is R-convex if R(u,v)⊂ K for every u,v∈ K and all R-convex subsets of V form a convexity CR. We consider Minkowski-Krein-Milman property that every R-convex set K in a convexity CR is the convex hull of the set of extreme points of K from axiomatic point of view and present a characterization of it. Later we consider several well-known transit functions on graphs and present the use of the mentioned characterizations on them.