Well-graded families of NaP-preferences

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 components jointly satisfy natural forms of mixed completeness and transitive coherence. A NaP-preference is normalized if its smaller component is a partial order. We show that normalized NaP-preferences on a finite set are well-graded in the sense of Doignon and Falmagne (1997).