On a dependence relation in finite lattices

AbstractWe study a dependence relation between the join-irreducible elements of a finite lattice that generalizes the classical perspectivity relation between the atoms of a geometric lattice. This relation is useful in the axiomatic approach of latticial consensus problems, since if it is strongly connected one can get arrowian theorems. First we show that the dependence relation is linked with the arrows relations between the irreducible elements of the lattice. Then we characterize the join-prime and the strong join-irreducible elements of the lattice by means of properties of the dependence relation, which induces characterizations of distributive and strong lattices by means of this relation. Then we characterize the sinks of the dependence relation which allows to show that this relation cannot be strongly connected in some classes of lattices. In the final note we point out the fact that this dependence relation generalizes the dependence relation introduced in the study of finite sublattices of free lattices