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There have been two main lines of thought in the hybridization of fuzzy and rough sets, the constructive approach and the axiomatic approach. A general framework for the study of fuzzy-rough sets from both of these viewpoints is presented in [403]. For the constructive approach, generalized lower and upper approximations are de ned based on fuzzy relations. Initially these were fuzzy similarity/equivalence relations [86], but they have since been extended to arbitrary fuzzy relations [403]. The axiomatic approach is primarily for the study of the mathematical properties of fuzzy-rough sets [386]. Here various classes of fuzzy-rough approximation operators are characterized by different sets of axioms that guarantee the existence of types of fuzzy relations producing the same operators. Dubois and Prade de ned the fuzzy P -lower and P -upper approximations as follows [86]: P X (Fi ) = inf max{1 Fi (x), X (x)}
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P X (Fi ) = sup min{ Fi (x), X (x)}
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where Fi is a fuzzy equivalence class and X is the (fuzzy) concept to be approximated. The tuple P X, P X is called a fuzzy-rough set. Also de ned in the
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literature are rough-fuzzy sets [85,349], which can be seen to be a particular case of fuzzy-rough sets. A rough-fuzzy set is a generalization of a rough set, derived from the approximation of a fuzzy set in a crisp approximation space. In [400] it is argued that to be consistent, the approximation of a crisp set in a fuzzy approximation space should be called a fuzzy-rough set, and the approximation of a fuzzy set in a crisp approximation space should be called a rough-fuzzy set, making the two models complementary. In this framework the approximation of a fuzzy set in a fuzzy approximation space is considered to be a more general model, unifying the two theories. However, most researchers consider the traditional de nition of fuzzy-rough sets in [86] as standard. The speci c use of min and max operators in the de nitions above is expanded in [282], where a broad family of fuzzy-rough sets is constructed, each member represented by a particular implicator and t-norm. The properties of three well-known classes of implicators (S-, R-, and QL-implicators) are investigated. Further investigations in this area can be found in [72,359,388,403]. In [39,237] an axiomatic approach is taken, but restricted to fuzzy T-similarity relations (and hence fuzzy T-rough sets). Wu et al. [385] investigated the properties of generalized fuzzy-rough sets, de ning a pair of dual generalized fuzzy approximation operators based on arbitrary fuzzy relations. The approach presented in [228] introduces de nitions for generalized fuzzy lower and upper approximation operators determined by a residual implication. Assumptions are found that allow a given fuzzy set-theoretic operator to represent a lower or upper approximation from a fuzzy relation. Different types of fuzzy relations produce different classes of fuzzy-rough set algebras. The work in [283] generalizes the fuzzy-rough set concept through the use of residuated lattices. An arbitrary residuated lattice is used as a basic algebraic structure, and several classes of L-fuzzy-rough sets and their properties are investigated. In [53] a complete completely distributive (CCD) lattice is selected as the foundation for de ning lower and upper approximations in an attempt to provide a uni ed framework for rough set generalizations. It is demonstrated that the existing fuzzy-rough sets are special cases of the approximations on a CCD lattice for T-similarity relations. The relationships between fuzzy-rough set models and fuzzy ([0, 1]-) topologies on a nite universe have been investigated. The rst such research was reported in [39], where it was proved that the lower and upper approximation operators were fuzzy interior and closure operators, respectively, for fuzzy T-similarity relations. The work carried out in [403] investigated this for arbitrary fuzzy relations. In [278,387] it was shown that a pair of dual fuzzy-rough approximation operators can induce a topological space if and only if the fuzzy relation is re exive and transitive. The suf cient and necessary condition that a fuzzy interior (closure) operator derived from a fuzzy topological space can associate with a fuzzy re exive and transitive relation such that the induced fuzzy lower (upper) approximation operator is the fuzzy interior (closure) operator is also examined.
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