Necessary and possible indifferences

A NaP-preference (necessary and possible preference) is a pair of nested reflexive relations on a set such that the smaller is transitive, the larger is complete, and the two relations jointly satisfy properties of transitive coherence and mixed completeness. It is known that a NaP-preference is characterized by the existence of a set of total preorders whose intersection and union give its two components. We introduce the symmetric counterpart of a NaP-preference, called a NaP-indifference: this is a pair of nested symmetric relations on a set such the smaller is an equivalence relation, and the larger is a transitively coherent extension of the first. A NaP-indifference can be characterized by the existence of a set of equivalence relations whose intersection and union give its two components. NaP-indifferences naturally arise in applications: for instance, in the field of individual choice theory, suitable pairs of similarity relations revealed by a choice correspondence yield a NaP-indifference. We classify NaP-indifferences in two categories, according to their genesis: (i) derived, which are canonically obtained by taking the symmetric part of a NaP-preference; (ii) primitive, which arise independently of the existence of an underlying NaP-preference. This partition into two classes turns out to be related to the notion of incomparability graph.