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Matchings and Radon transforms in lattices. II: Concordant sets. (English) Zbl 0626.06008

[Part I cf. Order 2, 105-112 (1985; Zbl 0582.06008).]
A sufficient condition for the existence of a matching of a subset of a finite lattice into another subset of the same lattice (i.e. an injection of the first one into the second with the property that the image of any element is not smaller than the element itself) is that the first subset is ”concordant” with the second one. Using the finite Radon transform, the author proves that the incidence matrix of concordant subsets has maximum rank which implies the existence of a matching. The notion of concordance is further used to derive several rank and covering inequalities in finite lattices among which 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.
Reviewer: M.Pirlot

MSC:

06B05 Structure theory of lattices
06C10 Semimodular lattices, geometric lattices
05B20 Combinatorial aspects of matrices (incidence, Hadamard, etc.)
44A15 Special integral transforms (Legendre, Hilbert, etc.)
06C05 Modular lattices, Desarguesian lattices

Citations:

Zbl 0582.06008
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References:

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