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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

MSC:

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

[1] G.Birkhoff (1967) Lattice Theory, 3rd edn., American Mathematical Society, Providence, Rhode Island. · Zbl 0153.02501
[2] T.Brylawski (1975) Modular constructions for combinatorial geometries, Trans. Amer. Math. Soc. 203, 1-44. · Zbl 0299.05023
[3] R. P.Dilworth (1954) Proof of a conjecture on finite modular lattices, Ann. Math. (2) 60, 359-364. · Zbl 0056.26203
[4] T. A.Dowling and R. M.Wilson (1975) Whitney number inequalities for geometric lattices, Proc. Amer. Math. Soc. 47, 504-512. · Zbl 0297.05010
[5] D.Duffus (1982) Matchings in modular lattices, J. Combin. Theory, Ser. A 32, 303-314. · Zbl 0479.06011
[6] J. P. S.Kung (1979) The Radon transform of a combinatorial geometry, I., J. Combin. Theory, Ser. A 26, 97-102. · Zbl 0406.05023
[7] J. P. S. Kung (to appear) Matchings and Radon transforms in lattices. II. Concordant sets. · Zbl 0626.06008
[8] K. Reuter (to appear) A matching result for modular lattices.
[9] I. Rival (1974) Contributions to combinatorial lattice theory, Doctoral thesis, University of Manitoba.
[10] I.Rival (1975) Maximal sublattices of finite distributive lattices. II., Proc. Amer. Math. Soc. 44, 263-268. · Zbl 0255.06003
[11] I.Rival (1976) Combinatorial inequalities for semimodular lattices of breadth two, Algebra Universalis 6, 303-311. · Zbl 0423.06008
[12] G.-C.Rota (1964) On the foundations of combinatorial theory. I. Theory of Möbius functions, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 2, 340-368. · Zbl 0121.02406
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