TY - JOUR

T1 - Poly-antimatroid polyhedra

AU - Kempner, Yulia

AU - Levit, Vadim E.

PY - 2014

Y1 - 2014

N2 - 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.

AB - 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.

KW - Antimatroid

KW - Convex dimension

KW - Lattice animal

KW - Polyhedron

KW - Polyomino

UR - http://www.scopus.com/inward/record.url?scp=84892160802&partnerID=8YFLogxK

U2 - 10.26493/1855-3974.263.eb4

DO - 10.26493/1855-3974.263.eb4

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AN - SCOPUS:84892160802

SN - 1855-3966

VL - 7

SP - 73

EP - 82

JO - Ars Mathematica Contemporanea

JF - Ars Mathematica Contemporanea

IS - 1

ER -