Violator spaces vs closure spaces

Yulia Kempner, Vadim E. Levit

Research output: Contribution to journalArticlepeer-review

1 Scopus citations


The main goal of the paper is to make connections between two well-known, but, up to now, independently developed theories: the theory of violator spaces and the theory of closure spaces. Violator spaces were introduced by Matoušek et al. in 2008 as generalization of linear programming problems. The notion of closure arises in many disciplines, including topology, algebra, convexity analysis, logic etc. In this work, we investigate interrelations between violator spaces and closure spaces. We show that a violator mapping may be defined by a weak version of a closure operator. Interrelations between violator spaces and closure spaces give new insights on a number of well known findings. For example, we prove that violator spaces with a unique basis satisfy both the anti-exchange and the Krein–Milman properties. Finally, based on subsequent relaxations of the closure operator notion we introduce convex spaces as a generalization of violator spaces.

Original languageEnglish
Pages (from-to)203-213
Number of pages11
JournalEuropean Journal of Combinatorics
StatePublished - Aug 2019


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