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 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.
Keywords
Cite
@article{arxiv.2206.05636,
title = {Convex geometries representable with colors, by ellipses on the plane, and impossible by circles},
author = {Kira Adaricheva and Evan Daisy and Ayush Garg and Zachary King and Grace Ma and Michelle Olson and Cat Raanes and James Thompson},
journal= {arXiv preprint arXiv:2206.05636},
year = {2022}
}
Comments
40 pages, 22 figures, 2 appendices