Description of closure operators in convex geometries of segments on a line

TL;DR

This paper characterizes when a convex geometry of segments on a line has convex dimension 2, providing polynomial-time conditions based on closure operators and implicational bases.

Contribution

It establishes necessary and sufficient conditions for convex geometries to have convex dimension 2, linking them to segments on a line and enabling polynomial-time verification.

Findings

Conditions for convex dimension 2 are characterized.

Polynomial-time checkable criteria are provided.

Connections between closure operators and geometric representations are clarified.

Abstract

Convex geometry is a closure space $(G,\phi)$ with the anti-exchange property. A classical result of Edelman and Jamison (1985) claims that every finite convex geometry is a join of several linear sub-geometries, and the smallest number of such sub-geometries necessary for representation is called the convex dimension. In our work we find necessary and sufficient conditions on a closure operator $\phi$ of convex geometry $(G,\phi)$ so that its convex dimension equals 2, equivalently, they are represented by segments on a line. These conditions can be checked in polynomial time in two parameters: the size of the base set $|G|$ and the size of the implicational basis of $(G,\phi)$.