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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. [ABSTRACT FROM AUTHOR]