The interplay between transitivity and completeness: Generalized NaP-preferences

We analyze the interaction between two classical tenets of economic rationality: transitivity and completeness. For a single binary relation, the combination of these properties yields a total preorder. For two binary relations, their interplay entails a NaP-preference: this is a nested pair (N,P) of relations satisfying mixed forms of transitivity and completeness. It is known that a NaP-preference (N,P) is characterized by a family of total preorders having N and P as their intersection and union, respectively. This paper examines extensions of this approach to more than two relations. First we define generalized NaP-preferences (GNaP-preferences): these are diamond-shaped quadruples of binary relations on a set such that transitivity and completeness are suitably spread throughout all four components. Then we characterize a GNaP-preference by a family of total preorders witnessing all relations by different combinations of union and intersection. We also show that a suitable amalgamation of NaP-preferences generates a GNaP-preference, and vice versa each GNaP-preference arises by this procedure. Finally, we introduce modal preference structures: these are lattices of binary relations such that the satisfaction of transitivity and completeness is guided by the action of the universal and the existential quantifiers over a family of total preorders. Applications of these novel preference structures to economics and psychology emerge naturally.