Gjonbalaj.md

Title. Description of the closure operator for a convex geometry of convex dimension two

Speaker. Gent Gjonbalaj

Institution. Department of Mathematics, Hofstra University, Hempstead, NY

Abstract. Convex geometry is a closure space $$(G,\phi)$$ with the anti-exchange property. A classical result of Edelman and Jamison (1985) claims that every finite convex geometry is a join of several linear sub-geometries, and the smallest number of such sub-geometries necessary for representation is called a convex dimension. In our work we find necessary and sufficient conditions on a closure operator $$\phi$$ of convex geometry $$(G,\phi)$$ so that its convex dimension is 2, equivalently, they are represented by segments on a line. These conditions can be checked in polynomial time in two parameters: the size of the base set $$|G|$$ and the size of the implicational basis of $$(G,\phi)$$.