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Going down in (semi)lattices of finite Moore families and convex geometries

Author

Listed:
  • Gabriela Bordalo

    (Centro de Algebra da Universidade de Lisboa - ULISBOA - Universidade de Lisboa = University of Lisbon)

  • Nathalie Caspard

    (LACL - Laboratoire d'Algorithmique Complexité et Logique - UPEC UP12 - Université Paris-Est Créteil Val-de-Marne - Paris 12 - CNRS - Centre National de la Recherche Scientifique)

  • Bernard Monjardet

    (CES - Centre d'économie de la Sorbonne - UP1 - Université Paris 1 Panthéon-Sorbonne - CNRS - Centre National de la Recherche Scientifique)

Abstract

In this paper we first study the changes occuring in the posets of irreducible elements when one goes from an arbitrary Moore family (respectively, a convex geometry) to one of its lower covers in the lattice of all Moore families (respectively, in the semilattice of all convex geometries) defined on a finite set. Then, we show that the poset of all convex geometries that have the same poset of join-irreducible elements is a ranked join-semilattice, and we give an algorithm for computing it. Finally, we prove that the lattice of all ideals of a given poset P is the only convex geometry having a poset of join-irreducible elements isomorphic to P if and only if the width of P is less than 3.

Suggested Citation

  • Gabriela Bordalo & Nathalie Caspard & Bernard Monjardet, 2009. "Going down in (semi)lattices of finite Moore families and convex geometries," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-00308785, HAL.
    • Handle: RePEc:hal:cesptp:halshs-00308785
    Note: View the original document on HAL open archive server: https://shs.hal.science/halshs-00308785
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    References listed on IDEAS

    as
    1. Monjardet, Bernard & Raderanirina, Vololonirina, 2001. "The duality between the anti-exchange closure operators and the path independent choice operators on a finite set," Mathematical Social Sciences, Elsevier, vol. 41(2), pages 131-150, March.
    2. Caspard, N. & Monjardet, B., 2000. "The Lattice of Closure Systems, Closure Operators and Implicational Systems on a Finite Set : A Survey," Papiers d'Economie Mathématique et Applications 2000.120, Université Panthéon-Sorbonne (Paris 1).
    3. Hausser Bordelo, G. & Monjardet, B., 1999. "The Lattice of Strict Completions of a Poset," Papiers d'Economie Mathématique et Applications 1999.15, Université Panthéon-Sorbonne (Paris 1).
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