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arXiv · 2206.05636

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

Kira Adaricheva·Evan Daisy·Ayush Garg·Zachary King·Grace Ma·Michelle Olson·Cat Raanes·James Thompson
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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.

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Kira Adaricheva, Evan Daisy, Ayush Garg, Zachary King, Grace Ma, Michelle Olson, Cat Raanes, James Thompson. 2022-06-12. Convex geometries representable with colors, by ellipses on the plane, and impossible by circles. https://arxiv.org/abs/2206.05636

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