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

TL;DR

This paper explores how convex geometries can be represented using ellipses and colors on the plane, identifying limitations with circles and providing methods for ellipse-based representations.

Contribution

It introduces new properties that prevent circle representations of certain convex geometries and demonstrates that all 672 convex geometries on five elements can be represented by ellipses, with a method to expand circle representations using colors.

Findings

Certain convex geometries cannot be represented by circles due to new properties.

All convex geometries on five elements can be represented by ellipses.

A method to expand circle representations with colors is introduced.

Abstract

A convex geometry is a closure system satisfying the anti-exchange property. This paper, following the work of K. Adaricheva and M. Bolat (2016) and the Polymath REU 2020 team, 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.