A size-width inequality for distributive lattices

Combinatorica volume 6, pages29–33(1986)Cite this article

Abstract

We show that every collection ofw sets such that none contains any other generates at least 3w-2 sets under the operations of taking intersections and unions. In particular, we prove that if the finite distributive lattice ℒ contains an antichain of sizew, then |ℒ| ≧3w, forw≠1, 2, 3, 6, where the minimal exceptional cases arise from the Boolean algebras ℬn withn=0, 1, 2, 3, 4 atoms.

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Reference

  1. [1]

    G. Birkhoff,Lattice Theory, 3rd ed., Amer. Math. Soc. Colloq. Publ. 25, Providence, R. I., 1967.

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Author information

Affiliations

  1. Institut für Ökonometrie und Op. Research, Rheinische Friedrich-Wilhelms-Universität, 5300, Bonn 1, Germany

    U. Faigle

  2. Department of Mathematics, University of Calgary, T2N 1N4, Calgary, Alberta, Canada

    B. Sands

Authors
  1. U. Faigle
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  2. B. Sands
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Additional information

Supported by Sonderforschungbereich 21 (DFG), Institut für Operations Research, Universität Bonn. The research was completed while the first author visited the University of Calgary, whose hospitality is gratefully acknowledged.

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Faigle, U., Sands, B. A size-width inequality for distributive lattices. Combinatorica 6, 29–33 (1986). https://doi.org/10.1007/BF02579406

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AMS subject classification (1980)

  • 05 C 65
  • 06 D 99