Medians and Majorities in Semimodular Lattices

Given a $\nu $-tuple $(x_1 , \cdots ,x_\nu )$ in a metric space $(X,d)$, a median is an element m of X minimizing the sum $\sum_i d(m, x_i )$. One of the basic facts about medians in a finite space is that their obtainment by the majority rule is characteristic of a distributive-like structure on X, called a median semilattice. It is shown here that, in the case where X is a finite lattice, the semimodularity property is characterized by a weaker relation between the medians and the majority rule. Some consequences of this result are investigated.