## 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:

 3e+72 Theory of fuzzy sets, etc.
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### References:

 [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
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