Abstract
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 represented as a subset of the integer lattice 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 c d i m ( 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 .
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