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

Gabriela, Bordalo, Nathalie, Caspard, and Bernard, Monjardet. "Going down in (semi)lattices of finite Moore families and convex geometries." Czechoslovak Mathematical Journal 59.1 (2009): 249-271. <http://eudml.org/doc/37921>.

@article{Gabriela2009,
abstract = {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.},
author = {Gabriela, Bordalo, Nathalie, Caspard, Bernard, Monjardet},
journal = {Czechoslovak Mathematical Journal},
keywords = {closure system; Moore family; convex geometry; (semi)lattice; algorithm; closure system; Moore family; convex geometry; semilattice; algorithm},
language = {eng},
number = {1},
pages = {249-271},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Going down in (semi)lattices of finite Moore families and convex geometries},
url = {http://eudml.org/doc/37921},
volume = {59},
year = {2009},
}

TY - JOUR
AU - Gabriela, Bordalo
AU - Nathalie, Caspard
AU - Bernard, Monjardet
TI - Going down in (semi)lattices of finite Moore families and convex geometries
JO - Czechoslovak Mathematical Journal
PY - 2009
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 59
IS - 1
SP - 249
EP - 271
AB - 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.
LA - eng
KW - closure system; Moore family; convex geometry; (semi)lattice; algorithm; closure system; Moore family; convex geometry; semilattice; algorithm
UR - http://eudml.org/doc/37921
ER -