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Matchings and Radon transforms in lattices II. Concordant sets

Published online by Cambridge University Press:  24 October 2008

Joseph P. S. Kung
Joseph P. S. Kung
Affiliation:
Department of Mathematics, North Texas State University, Denton, TX 76203, U.S.A.
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Abstract

Let and ℳ be subsets of a finite lattice L. is said to be concordant with ℳ if, for every element x in L, either x is in ℳ or there exists an element x+ such that (CS1) the Möbius function μ(x, x+) ≠ 0 and (CS2) for every element j in , x ∨ j ≠ x+. We prove that if is concordant with ℳ, then the incidence matrix I(ℳ | ) has maximum possible rank ||, and hence there exists an injection σ: → ℳ such that σ(j) ≥ j for all j in . Using this, we derive several rank and covering inequalities in finite lattices. Among the results are generalizations of the Dowling-Wilson inequalities and Dilworth's covering theorem to semimodular lattices, and a refinement of Dilworth's covering theorem for modular lattices.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 101 , Issue 2 , March 1987 , pp. 221 - 231
Copyright
Copyright © Cambridge Philosophical Society 1987

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