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1 2025

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Convex geometries yielded by transit functions

Changat Manoj, Sheela Lekshmi Kamal K., Peterin Iztok, Shanavas Ameera Vaheeda

Opuscula Mathematica

2025

Vol. 45, no. 4

423--450

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.