Indexed rough approximations and generalized possibility theory

Indexed rough approximations that generalize the fuzzy rough sets are proposed. A family of indexed relations between objects with the set of indices being a lattice is considered. Relations in the family are ordered by the inclusion, and moreover the ordering is assumed to be consistent with the ordering of the lattice. Thus, a collection of rough approximations, each of which is induced from a relation in the family, is obtained. The corresponding modal operators with the indices are defined; the completeness between the axiomatic system and the Kripke model which is the above collection of rough approximations is proved. A generalized possibility theory is derived from the collection of rough approximations