A size-width inequality for distributive lattices

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

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

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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