Geodesic
Going down in (semi)lattices of finite Moore families and convex geometries
Czechoslovak Mathematical Journal, Tome 59 (2009) no. 1, pp. 249-271.

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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.
Classification : 06A12
Keywords: closure system; Moore family; convex geometry; (semi)lattice; algorithm 
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     title = {Going down in (semi)lattices of finite {Moore} families and convex geometries},
     journal = {Czechoslovak Mathematical Journal},
     pages = {249--271},
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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, Tome 59 (2009) no. 1, pp. 249-271. http://geodesic.mathdoc.fr/item/CMJ_2009__59_1_a17/