Normalized and strict NaP-preferences

A NaP-preference (necessary and possible preference) is a pair of nested reflexive binary relations having a preorder as its smaller component, and satisfying natural forms of mixed completeness and mixed transitivity. A NaP-preference is normalized if its smaller component is a partial order. Dually, a strict NaP-preference is a pair of nested asymmetric binary relations having a strict partial order as its smaller component and satisfying suitable mixed transitivity properties. We show that normalized and strict NaP-preferences on the same ground set are in a one-to-one correspondence. It is known that a NaP-preference can be characterized by the existence of a set of total preorders whose intersection and union are respectively equal to its two components. In the same spirit, we characterize normalized and strict NaP-preferences by means of suitable families of order relations, respectively called injective and projective. The properties of injectivity and projectivity are a collectionwise extension of the antisymmetry and the completeness of a single binary relation.