A genesis of interval orders and semiorders: transitive NaP-preferences

A NaP-preference (necessary and possible preference) on a set A is a pair (N,P) of binary relations on A such that its necessary component N is a partial preorder, its possible component P is a completion of N, and the two components jointly satisfy natural forms of mixed completeness and mixed transitivity. We study additional mixed transitivity properties of a NaP-preference (N,P), which culminate in the full transitivity of its possible component P. 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.