Poly-antimatroid polyhedra

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 & 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 representedas a subset of the integer lattice Zⁿ. 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-dimensionalpoly-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-dimensionalpoly-antimatroid may be arbitrarily large, we prove that the convex dimension of an n-dimensional poset poly-antimatroid is equal to n.