Medians for weight metrics in the covering graphs of semilattices

Abstract

We consider the undirected covering graph G of a finite (meet) semilattice X endowed with a lower valuation. More precisely, our main concerns are the lower valuations associated to a weighting of the join-irreducible elements of X and the corresponding minimum path length metrics in G, which are frequently considered in the literature. Some results on the medians for such metrics are obtained, in relation with a lattice majority rule. Especially, these medians are characterized in the case where X is distributive. The unanimity, or Pareto, property is also investigated for such medians.