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

Abstract

Convex geometry is a closure space (G, φ) 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 φ of convex geometry (G, φ) 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, φ).2010 Mathematics Subject Classification. 05A05, 06A15, 06B99, 52B55.