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

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.
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Submitted on : Friday, August 1, 2008 - 4:21:15 PM
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Gabriela Bordalo, Nathalie Caspard, Bernard Monjardet. Going down in (semi)lattices of finite Moore families and convex geometries. Czechoslovak Mathematical Journal, Akademie věd České republiky, Matematický ústav, 2009, 59 (1), pp.249-271. ⟨halshs-00308785⟩

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