Definability of Approximations for a Generalization of the Indiscernibility Relation

We discuss a generalization of the indiscernibility relation, i.e., a relation R that is not necessarily reflexive, symmetric, or transitive. On the basis of granules, defined by R, we introduce the idea of definability. Twelve different basic definitions of approximations are discussed. Since four of these approximations do not satisfy, in general, the inclusion property, four additional modified approximations are introduced. Furthermore, eight other approximations are constructed by duality. The main objective is to study definability of approximations. We study definability of all approximations for reflexive, symmetric, or transitive relations. In particular, for reflexive relations the set of these twenty four approximations is reduced, in general, to the set of fourteen approximations