Decisive neutrality, restricted decisive neutrality, and split decisive neutrality on median semilattices and median graphs.

Consensus functions on finite median semilattices and finite median graphs are studied from an axiomatic point of view. We start with a new axiomatic characterization of majority rule on a large class of median semilattices we call sufficient. A key axiom in this result is the restricted decisive neutrality condition. This condition is a restricted version of the more well-known axiom of decisive neutrality given in [4]. Our theorem is an extension of the main result given in [7]. Another main result is a complete characterization of the class of consensus on a finite median semilattice that satisfies the axioms of decisive neutrality, bi-idempotence, and symmetry. This result extends the work of Monjardet [9]. Moreover, by adding monotonicity as a fourth axiom, we are able to correct a mistake from the Monjardet paper. An attempt at extending the results on median semilattices to median graphs is given, based on a new axiom called split decisive neutrality. We are able to show that majority rule is the only consensus function defined on a path with three vertices that satisfies split decisive neutrality and symmetry.