Convex geometries representable with colors, by ellipses on the plane, and impossible by circles
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 t...
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Published in | Acta scientiarum mathematicarum (Szeged) Vol. 90; no. 1-2; pp. 269 - 322 |
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Main Authors | , , , , , , |
Format | Journal Article |
Language | English |
Published |
Cham
Springer International Publishing
01.05.2024
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Subjects | |
Online Access | Get full text |
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Summary: | 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. |
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ISSN: | 0001-6969 2064-8316 |
DOI: | 10.1007/s44146-024-00112-2 |