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

A NaP-preference (necessary and possible preference) on a set A is a pair ({left(succsim^{^{_N}}!,,succsim^{^{_P}}!right)}) of binary relations on A such that its necessary component ({succsim^{^{_N}} !!}) is a partial preorder, its possible component ({succsim^{^{_P}} !!}) is a completion of ({succsim^{^{_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 ({left(succsim^{^{_N}}!,,succsim^{^{_P}}!right)}) , which culminate in the full transitivity of its possible component ({succsim^{^{_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.