Matchings and Radon transforms in lattices. I: Consistent lattices. (English) Zbl 0582.06008

An element in a lattice is join-irreducible if \(x=a\vee b\) implies \(x=a\) or \(x=b\). A meet-irreducible is a join-irreducible in the order dual. A lattice is consistent if for every element x and every join-irreducible j, the element \(x\vee j\) is a join-irreducible in the upper interval [x,1]. This paper shows that in a finite consistent lattice, the incidence matrix of meet-irreducibles versus join-irreducibles has rank the number of join-irreducibles. Since modular lattices and their order duals are consistent, this settles a conjecture of Rival on matchings in modular lattices.
Reviewer: Ph.Vincke


06B05 Structure theory of lattices
06C05 Modular lattices, Desarguesian lattices
05B20 Combinatorial aspects of matrices (incidence, Hadamard, etc.)
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