Majority rule on median semilattices

In (2008), McMorris and Powers [A characterization of majority rule, J. Classif. 25 (2008) 153–158] proved that a hierarchical version of majority rule is the only consensus function satisfying restricted decisive neutrality, symmetry, monotonicity, and is biprofile nontrivial. We show that this result can be extended to any finite median semilattice [Formula: see text] such that [Formula: see text] is not a lattice, the height of [Formula: see text] is greater than one, [Formula: see text] contains at least two atoms, and [Formula: see text] satisfies a condition we call special.