Convex geometries yielded by transit functions

Let V be a finite nonempty set. A transit function is a map R : V x V → 2V such that R(u, u) = {u}, R(u, v) = R(v, u) and u ∈ R(u, v) holds 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 the 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.

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Opracowanie rekordu ze środków MNiSW, umowa nr POPUL/SP/0154/2024/02 w ramach programu "Społeczna odpowiedzialność nauki II" - moduł: Popularyzacja nauki (2025)