Geodesic
Poly-antimatroid polyhedra
Ars Mathematica Contemporanea, Tome 7 (2014) no. 1, pp. 73-82.

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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 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-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.
DOI : 10.26493/1855-3974.263.eb4
Keywords: antimatroid, polyhedron, convex dimension, lattice animal, polyomino 
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Yulia Kempner; Vadim E. Levit. Poly-antimatroid polyhedra. Ars Mathematica Contemporanea, Tome 7 (2014) no. 1, pp. 73-82. doi : 10.26493/1855-3974.263.eb4. http://geodesic.mathdoc.fr/articles/10.26493/1855-3974.263.eb4/

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