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

• Published in: Acta scientiarum mathematicarum (Szeged) Vol. 90; no. 1-2; pp. 269 - 322
• Main Authors: Adaricheva, Kira, Daisy, Evan, Garg, Ayush, Ma, Grace, Olson, Michelle, Raanes, Cat, Thompson, James
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.05.2024
• Subjects: Mathematics; Mathematics and Statistics; Original Paper; Convex geometry; Convex dimension; Antimatroid; Convex hull operator; Implications
• ISSN: 0001-6969; 2064-8316
• DOI: 10.1007/s44146-024-00112-2

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.