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2 2011

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Textures and Fuzzy Rough Sets

Diker M.

Fundamenta Informaticae

2011

Vol. 108, nr 3/4

305-336

In this paper, we consider the Alexandroff topology for texture spaces. We prove that there exists a one-to-one correspondence between the Alexandroff ditopologies, and the reflexive and transitive direlations on a given texture. Using textural fuzzy direlations on a fuzzy lattice, we obtain a fuzzy rough set algebra where the inverse fuzzy relation and inverse fuzzy corelation are the upper approximation and lower approximation, respectively. In a special case, this gives us the fuzzy rough sets which are calculated with respect to the min-t norm introduced by S. Gottwald, or the fuzzy rough sets which are considered by D. Pei.

On Some Mathematical Structures of T-Fuzzy Rough Set Algebras in Infinite Universes of Discourse

Wu W. Z.

Fundamenta Informaticae

2011

Vol. 108, nr 3/4

337-369

In this paper, a general framework for the study of fuzzy rough approximation operators determined by a triangular norm in infinite universes of discourse is investigated. Lower and upper approximations of fuzzy sets with respect to a fuzzy approximation space in infinite universes of discourse are first introduced. Essential properties of various types of T -fuzzy rough approximation operators are then examined. An operator-oriented characterization of fuzzy rough sets is also proposed, that is, T -fuzzy rough approximation operators are defined by axioms. Different axiom sets of upper and lower fuzzy set-theoretic operators guarantee the existence of different types of fuzzy relations which produce the same operators. A comparative study of T -fuzzy rough set algebras with some other mathematical structures are presented. It is proved that there exists a one-to-one correspondence between the set of all reflexive and T -transitive fuzzy approximation spaces and the set of all fuzzy Alexandrov spaces such that the lower and upper T -fuzzy rough approximation operators are, respectively, the fuzzy interior and closure operators. It is also shown that a reflexive fuzzy approximation space induces a measurable space such that the family of definable fuzzy sets in the fuzzy approximation space forms the fuzzy -algebra of the measurable space. Finally, it is explored that the fuzzy belief functions in the Dempster-Shafer of evidence can be interpreted by the T -fuzzy rough approximation operators in the rough set theory, that is, for any fuzzy belief structure there must exist a probability fuzzy approximation space such that the derived probabilities of the lower and upper approximations of a fuzzy set are, respectively, the T -fuzzy belief and plausibility degrees of the fuzzy set in the given fuzzy belief structure.