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An axiomatic characterization of a fuzzy generalization of rough sets. (English) Zbl 1041.03038

Summary: In rough set theory, the lower and upper approximation operators defined by a fixed binary relation satisfy many interesting properties. Several authors have proposed various fuzzy generalizations of rough approximations. In this paper, we introduce definitions for generalized fuzzy lower and upper approximation operators determined by a residual implication. Then we find assumptions which permit a given fuzzy set-theoretic operator to represent an upper (or lower) approximation derived from a special fuzzy relation. Different classes of fuzzy rough set algebras are obtained from different types of fuzzy relations. And different sets of axioms of a fuzzy set-theoretic operator guarantee the existence of different types of fuzzy relations which produce the same operator. Finally, we study the composition of two approximation spaces. It is proved that the approximation operators in the composition space are just the composition of the approximation operators in the two fuzzy approximation spaces.

MSC:

03E72 Theory of fuzzy sets, etc.
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