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

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Abstract

In this paper we first study what changes occur 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 study the set of all convex geometries which have the same poset of join-irreducible elements. We show that this set—ordered by set inclusion—is a ranked join-semilattice and we characterize its cover relation. 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. Finally, we give an algorithm for computing all convex geometries having the same poset of join-irreducible elements.

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Authors and Affiliations

  1. Departamento de Matematica da Faculdade de Ciencias e CAUL, Universidade de Lisboa, R.Prof.Gama Pinto, 2, 1699, Lisboa, Portugal

    Bordalo Gabriela

  2. LACL, Université Paris 12 Val de Marne, 61 avenue du Général de Gaulle, 940 10, Créteil cedex, France

    Caspard Nathalie

  3. CERMSEM, Université Paris I, Maison des Sciences Économiques, 106-112, Boulevard de l’Hôpital, 756 47, Paris Cedex 13, France

    Monjardet Bernard

Authors
  1. Bordalo Gabriela
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  2. Caspard Nathalie
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  3. Monjardet Bernard
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Correspondence to Bordalo Gabriela.

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Gabriela, B., Nathalie, C. & Bernard, M. Going down in (semi)lattices of finite moore families and convex geometries. Czech Math J 59, 249–271 (2009). https://doi.org/10.1007/s10587-009-0018-2

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  • DOI: https://doi.org/10.1007/s10587-009-0018-2

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