Defining rough sets using tolerances compatible with an equivalence
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We consider tolerances T compatible with an equivalence E on U, meaning that the relational product E ∘ T is included in T. We present the essential properties of E-compatible tolerances and study rough approximations defined by such E and T. We consider rough set pairs (X_E,X^T), where the lower approximation X_E is defined as is customary in rough set theory, but because T is assumed to be E-compatible, X^T allows more elements to be possibly in X, because X^E ⊆ X^T. We also show that each X^T is a union of equivalence classes of E. Motivating examples of E-compatible tolerances are given, and the essential lattice-theoretical properties of the ordered set of rough sets { (X_E,X^T) | X ⊆ U} are established.
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