Convex geometries representable with colors, by ellipses on the plane, and impossible by circles

Abstract

A convex geometry is a closure system satisfying the anti-exchange property. This paper, following the work of Adaricheva and Bolat (Discrete Math 342(N3):726–746, 2019) and the Polymath REU 2020 team (Convex geometries representable by at most 5 circles on the plane. arXiv:2008.13077), continues to investigate representations of convex geometries on a 5-element base set. It introduces several properties: the opposite property, nested triangle property, area Q property, and separation property, of convex geometries of circles on a plane, preventing this representation for numerous convex geometries on a 5-element base set. It also demonstrates that all 672 convex geometries on a 5-element base set have a representation by ellipses, as given in the appendix for those without a known representation by circles, and introduces a method of expanding representation with circles by defining unary predicates, shown as colors.