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

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 σ-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. (This work was supported by grants from the National Natural Science Foundation of China (Nos. 61075120, 60673096 and 60773174), and the Natural Science Foundation of Zhejiang Province (No. Y107262).)

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