Violator spaces vs closure spaces

TL;DR

This paper explores the relationship between violator spaces and closure spaces, showing that violator mappings can be defined by a weaker form of closure operators and analyzing properties of violator spaces with unique bases.

Contribution

It establishes a connection between violator spaces and closure spaces, introducing a weak closure operator perspective and characterizing violator spaces with unique bases.

Findings

Violator mapping can be defined by a weak version of closure operators.

Violator spaces with a unique basis satisfy anti-exchange and Krein-Milman properties.

The study bridges concepts between violator spaces and convex geometries.

Abstract

Violator Spaces were introduced by J. Matousek et al. in 2008 as generalization of Linear Programming problems. Convex geometries were invented by Edelman and Jamison in 1985 as proper combinatorial abstractions of convexity. Convex geometries are defined by anti-exchange closure operators. We investigate an interrelations between violator spaces and closure spaces and show that violator mapping may be defined by a week version of closure operators. Moreover, we prove that violator spaces with an unique basis satisfies the anti-exchange and the Krein-Milman properties.