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.


03E72 Theory of fuzzy sets, etc.
Full Text: DOI


[1] Abdel-Hamid, A.A.; Morsi, N.N., On the relationship of extended necessity measures to implication operators on the unit interval, Inform. sci., 82, 129-145, (1995) · Zbl 0870.68139
[2] Farinas, L.; Prada, H., Rough sets, twofold fuzzy sets and modal logic, fuzziness in indiscrenibility and partial information, (), 103-120
[3] Kerre, E.E., Fuzzy sets and approximate reasoning, () · Zbl 0435.54003
[4] Morsi, N.N., Fuzzy T-locality spaces, Fuzzy sets and systems, 69, 193-219, (1995) · Zbl 0857.54008
[5] Morsi, N.N., Dual fuzzy neighbourhood spaces II, J. fuzzy math., 3, 29-67, (1995) · Zbl 0858.54004
[6] N.N. Morsi, Properties and characterizations of the residuation implications of triangular norms, Internat. J. Approx. Reason., to appear.
[7] Ovchinnikov, S., Similarity relations, fuzzy partitions, and fuzzy orderings, Fuzzy sets and systems, 40, 107-126, (1991) · Zbl 0725.04003
[8] Pawlak, Z., Rough sets, Internat. J. comput. inform. sci., 11, 341-356, (1982) · Zbl 0501.68053
[9] Pawlak, Z., Rough sets and fuzzy sets, Fuzzy sets and systems, 17, 99-102, (1985) · Zbl 0588.04004
[10] Schweizer, B.; Sklar, A., Probabilistic metric spaces, (1983), North-Holland Amsterdam · Zbl 0546.60010
[11] Zadeh, L.A., Similarity relations and fuzzy orderings, Inform. sci., 3, 177-200, (1971) · Zbl 0218.02058
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.