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

Year: 2022
Type: preprint
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.
DOI: 10.48550/arxiv.2206.05636
Authors: Adaricheva, Kira, Daisy, Evan, Garg, Ayush, King, Zachary, Ma, Grace, Olson, Michelle, Raanes, Cat, Thompson, James
Institutions: Hofstra University, Amherst College, Indian Institute of Technology Delhi, University of Notre Dame, California State University, Fullerton, University of North Carolina at Chapel Hill, Carnegie Mellon University
Language: English
Open Access status: green
OpenAlex ID: W4282921815
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