A Genesis of Interval Orders and Semiorders: Transitive NaP-preferences

A NaP-preference (necessary and possible preference) on a set A is a pair ( , P ) of binary relations on A such that its necessary component N is a partial preorder, its possible component P is a completion of , and the two components jointly satisfy natural forms of mixed completeness and mixed transitivity. We study additional mixed transitivity properties of a NaP-preference ( , P ) , which culminate in the full transitivity of its possible component . Interval orders and semiorders are strictly related to these properties, since they are the possible components of suitably transitive NaP-preferences. Further, we introduce strong versions of interval orders and semiorders, which are characterized by enhanced forms of mixed transitivity, and use a geometric approach to compare them to other well known preference relations.