论文标题
凸出的几何形状可用颜色代表，飞机上的椭圆形，而圆圈不可能
Convex geometries representable with colors, by ellipses on the plane, and impossible by circles
论文作者
论文摘要
凸几何形状是满足反交换特性的封闭系统。在K. Adaricheva和M. Bolat（2016）和Polymath REU 2020团队的工作之后，本文继续调查凸几何形式的5元素基础集合。它引入了几个属性：相反的属性，嵌套的三角形属性，Q属性和分离属性，是平面上圆形几何形状的分离属性，从而阻止了5个元素基集中众多凸的几何形状的表示形式。它还表明，正如附录中给出的所有672个凸的几何形状均具有椭圆形的表示，如圆圈所没有已知代表的人，并引入了一种通过定义单个谓词来扩展圆形表示的方法，显示为颜色。
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.
