Poly-antimatroid polyhedra - Kempner, Yulia (Ars mathematica contemporanea, 2014)
The notion of "antimatroid with repetition" was conceived by Bjorner, Lovasz and Shor in 1991 as an extension of the notion of antimatroid in the framework of non-simple languages. Further they were investigated by the name of "poly-antimatroids" (Nakamura, 2005, Kempner and Levit, 2007), where the set system approach was used. If the underlying set of a poly-antimatroid consists of n elements, then the poly-antimatroid may be represented as a subset of the integer lattice ?$\mathbb{Z}^n$?. We concentrate on geometrical properties of two-dimensional ?$(n = 2)$? poly-antimatroids - poly-antimatroid polygons, and prove that these polygons are parallelogram polyominoes. We also show that each two-dimensional poly-antimatroid is a poset poly-antimatroid, i.e., it is closed under intersection. The convex dimension ?$cdim(S)$? of a poly-antimatroid ?$S$? is the minimum number of maximal chains needed to realize ?$S$?. While the convex dimension of an $n$-dimensional poly-antimatroid may be arbitrarily large, we prove that the convex dimension of an ?$n$?-dimensional poset poly-antimatroid is equal to ?$n$?.
Keywords: antimatroid; convex dimension; kombinatorika; lattice animal; matematika; polyhedron; polyomino