TY - JOUR

T1 - A new matching property for posets and existence of disjoint chains

AU - Logan, Mark J.

AU - Shahriari, Shahriar

N1 - Copyright:
Copyright 2004 Elsevier B.V., All rights reserved.

PY - 2004/10

Y1 - 2004/10

N2 - Let P be a graded poset. Assume that x1,...,xm are elements of rank k and y1,...,ym are elements of rank l for some ki≤yi, for 1≤i≤m. Lehman and Ron (J. Combin. Theory Ser. A 94 (2001) 399) proved that, if P is the subset lattice, then there exist m disjoint skipless chains in P that begin with the x's and end at the y's. One complication is that it may not be possible to have the chains respect the original matching and hence, in the constructed set of chains, xi and yi may not be in the same chain. In this paper, by introducing a new matching property for posets, called shadow-matching, we show that the same property holds for a much larger class of posets including the divisor lattice, the subspace lattice, the lattice of partitions of a finite set, the intersection poset of a central hyperplane arrangement, the face lattice of a convex polytope, the lattice of noncrossing partitions, and any geometric lattice.

AB - Let P be a graded poset. Assume that x1,...,xm are elements of rank k and y1,...,ym are elements of rank l for some ki≤yi, for 1≤i≤m. Lehman and Ron (J. Combin. Theory Ser. A 94 (2001) 399) proved that, if P is the subset lattice, then there exist m disjoint skipless chains in P that begin with the x's and end at the y's. One complication is that it may not be possible to have the chains respect the original matching and hence, in the constructed set of chains, xi and yi may not be in the same chain. In this paper, by introducing a new matching property for posets, called shadow-matching, we show that the same property holds for a much larger class of posets including the divisor lattice, the subspace lattice, the lattice of partitions of a finite set, the intersection poset of a central hyperplane arrangement, the face lattice of a convex polytope, the lattice of noncrossing partitions, and any geometric lattice.

KW - Boolean lattice

KW - Disjoint chains

KW - Geometric lattice

KW - Matching property

KW - Noncrossing partitions

KW - Shadow-matching

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U2 - 10.1016/j.jcta.2004.06.002

DO - 10.1016/j.jcta.2004.06.002

M3 - Article

AN - SCOPUS:9244259635

VL - 108

SP - 77

EP - 87

JO - Journal of Combinatorial Theory - Series A

JF - Journal of Combinatorial Theory - Series A

SN - 0097-3165

IS - 1

ER -