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

Gabriela Bordalo, Nathalie Caspard () and Bernard Monjardet
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Gabriela Bordalo: ULISBOA - Universidade de Lisboa = University of Lisbon = Université de Lisbonne
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

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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.

Keywords: closure system; cover relation; Moore family; poset of irreducible; semilattice; convex geometry; join-irreducible; ensemble ordonné; famille de Moore; fermeture; géométrie convexe; relation de couverture; sup-irréductible; treillis (search for similar items in EconPapers)
Date: 2009
Note: View the original document on HAL open archive server: https://shs.hal.science/halshs-00308785v1
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Published in Czechoslovak Mathematical Journal, 2009, 59 (1), pp.249-271

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Working Paper: Going down in (semi)lattices of finite Moore families and convex geometries (2009)
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