Undominated Maximals: General Definition and Characterizations

This paper proposes a general definition of an undominated maximal of a relation on a constraint set. No specific requirement is imposed on either the asymmetry of the objective relation or the constraint set (which might, or might not, coincide with the ground set of the objective relation). Several characterizations are formulated that express undominated maximals of an objective relation as maximals of some trace associated with that objective relation. By means of some of these characterizations, the structure of the entire set of undominated maximals is examined in the particular case of relations induced by open and closed convex cones among them, the weak and strong Pareto dominance—and, in the case of semiorders, that admit certain types of representability. The results of the last part of the examination allow the construction of many examples of relations whose entire sets of maximals and undominated maximals are completely identifiable in an elementary way.