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Axiomatics for fuzzy rough sets. (English) Zbl 0938.03085

Summary: A fuzzy \(T\)-rough set consists of a set \(X\) and a \(T\)-similarity relation \(R\) on \(X\), where \(T\) is a lower semi-continuous triangular norm. We generalize the Farinas-Prade definition for the upper approximation operator \(\overline A: I^X\to I^X\) of a fuzzy \(T\)-rough set \((X,R)\), given originally for the special case \(T= \text{Min}\), to the case of arbitrary \(T\). We propose a new definition for the lower approximation operator \(\underline A: I^X\to I^X\) of \((X,R)\). Our definition satisfies the two important identities \(\overline A\underline A=\underline A\) and \(\underline A\overline A=\overline A\), as well as a number of other interesting properties. We provide an axiomatics to fully characterize these upper and lower approximations.

MSC:

03E72 Theory of fuzzy sets, etc.
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