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Keywords:
closure system; Moore family; convex geometry; (semi)lattice; algorithm
closure system; Moore family; convex geometry; (semi)lattice; algorithm
Summary:
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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