2022/06/12 by Kira Adaricheva, Adaricheva, Kira, Evan Daisy +13
Computer Science · Engineering · #Combinatorics (math.CO) #Computational Geometry and Mesh Generation #Digital Image Processing Techniques #FOS: Mathematics #Modular Robots and Swarm Intelligence #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.2206.05636
openalex publication_date 2022/06/12 · openalex created_date 2022/06/16 · openalex updated_date 2026/07/28
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.