1. Going down in (semi)lattices of finite Moore families and convex geometries
- Creator:
- Bordalo, Gabriela, Caspard, Nathalie, and Monjardet, Bernard
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- closure system, Moore family, convex geometry, (semi)lattice, and algorithm
- Language:
- English
- Description:
- 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.
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public