PRESERVING CONSENSUS HIERARCHIES

In numerical taxonomy we often have the task of finding a consensus hi erarchy for a given set of hierarchies. This consensus hierarchy shoul d reflect the substructures which are common to all hierarchies of the set. Because there are several kinds of substructures in a hierarchy, the general axiom to preserve common substructures leads to different axioms for each kind of substructure. In this paper we consider the t hree substructures cluster, separation, and nesting, and we give sever al characterizations of hierarchies preserving these substructures. Th ese characterizations facilitate interpretation of axioms for preservi ng substructures and the examination of properties of consensus method s. Finally some extensions concerning the preserving of qualified subs tructures are discussed.