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Dilworth’s Completion, Submodular Functions, and Combinatorial Optimization

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The Dilworth Theorems

Part of the book series: Contemporary Mathematicians ((CM))

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Abstract

The motivation behind Dilworth’s investigation in [5] is the question whether a lattice can be imbedded into a geometric lattice. This paper concentrates on quasimodular point lattices, i. e., point lattices L of finite length such that all maximal chains of L share the same length and the rank function f of L satisfies the weakened submodularity condition for all a, b ∈ L,

$$a \wedge b \ne 0\,implies f\left( {a \vee b} \right) + f\left( {a \wedge b} \right) \leqslant f\left( a \right) + f\left( b \right)$$
(1)

.

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Author information

Authors and Affiliations

  1. University of Twente, 7500 AE, Enschede, The Netherlands

    Ulrich Faigle

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  1. Ulrich Faigle
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Editor information

Editors and Affiliations

  1. Department of Mathematics and Computer Science, Dartmouth College, Hanover, NH, 03755, USA

    Kenneth P. Bogart

  2. Department of Mathematics, University of Hawaii, Honolulu, HI, 96822, USA

    Ralph Freese

  3. Department of Mathematics, University of North Texas, Denton, TX, 76203-5116, USA

    Joseph P. S. Kung

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Faigle, U. (1990). Dilworth’s Completion, Submodular Functions, and Combinatorial Optimization. In: Bogart, K.P., Freese, R., Kung, J.P.S. (eds) The Dilworth Theorems. Contemporary Mathematicians. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4899-3558-8_28

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  • DOI: https://doi.org/10.1007/978-1-4899-3558-8_28

  • Publisher Name: Birkhäuser, Boston, MA

  • Print ISBN: 978-1-4899-3560-1

  • Online ISBN: 978-1-4899-3558-8

  • eBook Packages: Springer Book Archive

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