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On characterizations of \((\mathcal J,\mathcal T)\)-fuzzy rough approximation operators. (English) Zbl 1074.03027

Summary: In rough set theory, the lower and upper approximation operators defined by a fixed binary relation satisfy many interesting properties. Various fuzzy generalizations of rough approximations have been made in the literature. This paper proposes a general framework for the study of \((\mathcal J,\mathcal T)\)-fuzzy rough approximation operators within which both constructive and axiomatic approaches are used. In the constructive approach, a pair of lower and upper generalized fuzzy rough approximation operators, determined by an implicator \(\mathcal J\) and a triangular norm \(\mathcal T\), is first defined. Basic properties of \((\mathcal J,\mathcal T)\)-fuzzy rough approximation operators are investigated. The connections between fuzzy relations and fuzzy rough approximation operators are further established. In the axiomatic approach, an operator-oriented characterization of rough sets is proposed, that is, \((\mathcal J,\mathcal T)\)-fuzzy approximation operators are defined by axioms. Different axiom sets of \(\mathcal T\)-upper and \(\mathcal J\)-lower fuzzy set-theoretic operators guarantee the existence of different types of fuzzy relations which produce the same operators. Finally, an open problem posed by A. M. Radzikowska and E. Kerre [Fuzzy Sets Syst. 126, 137–155 (2002; Zbl 1004.03043)] is solved.

MSC:

03E72 Theory of fuzzy sets, etc.
68T37 Reasoning under uncertainty in the context of artificial intelligence

Citations:

Zbl 1004.03043
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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] Boixader, D.; Jacas, J.; Recasens, J., Upper and lower approximations of fuzzy sets, Internat. J. Gen. Systems, 29, 555-568 (2000) · Zbl 0955.03056
[3] Comer, S., An algebraic approach to the approximation of information, Fund. Inform., 14, 492-502 (1991) · Zbl 0727.68114
[4] Comer, S., On connections between information systems, rough sets, and algebraic logic, (Rauszer, C., Algebraic Methods in Logic and Computer Science, Banach Center Publisher, vol. 28 (1993), Polish Academy of Sciences), 117-127 · Zbl 0793.03074
[5] Cornelis, C.; Deschrijver, G.; Kerre, E. E., Implication in intuitionistic fuzzy and interval-valued fuzzy set theory: construction, classification, application, Internat. J. Approx. Reason., 35, 55-95 (2004) · Zbl 1075.68089
[6] Dubois, D.; Prade, H., Rough fuzzy sets and fuzzy rough sets, Internat. J. Gen. Systems, 17, 191-208 (1990) · Zbl 0715.04006
[7] Klir, G. J.; Yuan, B., Fuzzy Logic: Theory and Applications (1995), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ · Zbl 0915.03001
[8] Kuncheva, L. I., Fuzzy rough sets: application to feature selection, Fuzzy Sets and Systems, 51, 147-153 (1992)
[9] Lin, T. Y.; Liu, Q., Rough approximate operators: axiomatic rough set theory, (Ziarko, W., Rough Sets Fuzzy Sets and Knowledge Discovery (1994), Springer: Springer Berlin), 256-260 · Zbl 0818.03028
[10] Mi, J.-S.; Zhang, W.-X., An axiomatic characterization of a fuzzy generalization of rough sets, Inform. Sci., 160, 235-249 (2004) · Zbl 1041.03038
[11] Morsi, N. N.; Yakout, M. M., Axiomatics for fuzzy rough sets, Fuzzy Sets and Systems, 100, 327-342 (1998) · Zbl 0938.03085
[12] Nanda, S.; Majumda, S., Fuzzy rough sets, Fuzzy Sets and Systems, 45, 157-160 (1992) · Zbl 0749.04004
[13] Pawlak, Z., Rough sets, Internat. J. Comput. Inform. Sci., 11, 341-356 (1982) · Zbl 0501.68053
[14] Pawlak, Z., Rough Sets: Theoretical Aspects of Reasoning about Data (1991), Kluwer Academic Publishers: Kluwer Academic Publishers Boston · Zbl 0758.68054
[15] Radzikowska, A. M.; Kerre, E. E., A comparative study of fuzzy rough sets, Fuzzy Sets and Systems, 126, 137-155 (2002) · Zbl 1004.03043
[16] Ruan, D.; Kerre, E. E., Fuzzy implication operators and generalized fuzzy method of cases, Fuzzy Sets and Systems, 54, 23-37 (1993) · Zbl 0784.68078
[17] Slowinski, R.; Vanderpooten, D., A Generalized definition of rough approximations based on similarity, IEEE Trans. Knowledge and Data Eng., 12, 331-336 (2000)
[18] Thiele, H., On axiomatic characterisations of crisp approximation operators, Inform. Sci., 129, 221-226 (2000) · Zbl 0985.03044
[20] Thiele, H., On axiomatic characterisation of fuzzy approximation operators II, the rough fuzzy set based case, (Proc. 31st IEEE Internat. Symp. on Multiple-Valued Logic (2001)), 330-335
[21] Thiele, H., On axiomatic characterization of fuzzy approximation operators III the fuzzy diamond and fuzzy box cases, (The 10th IEEE Internat. Conf. on Fuzzy Systems, vol. 2 (2001)), 1148-1151
[22] Wiweger, R., On topological rough sets, Bull. Polish Acad. Sci. Math., 37, 89-93 (1989) · Zbl 0755.04010
[23] Wu, W.-Z.; Leung, Y.; Zhang, W.-X., Connections between rough set theory and Dempster-Shafer theory of evidence, Internat. J. Gen. Systems, 31, 405-430 (2002) · Zbl 1007.03049
[24] Wu, W.-Z.; Mi, J.-S.; Zhang, W.-X., Generalized fuzzy rough sets, Inform. Sci., 151, 263-282 (2003) · Zbl 1019.03037
[25] Wu, W.-Z.; Zhang, W.-X., Neighborhood operator systems and approximations, Inform. Sci., 144, 201-217 (2002) · Zbl 1019.68109
[26] Wu, W.-Z.; Zhang, W.-X., Constructive and axiomatic approaches of fuzzy approximation operators, Inform. Sci., 159, 233-254 (2004) · Zbl 1071.68095
[27] Wybraniec-Skardowska, U., On a generalization of approximation space, Bull. Polish Acad. Sci. Math., 37, 51-61 (1989) · Zbl 0755.04011
[28] Yao, Y. Y., Two views of the theory of rough sets in finite universes, Internat. J. Approx. Reason., 15, 291-317 (1996) · Zbl 0935.03063
[29] Yao, Y. Y., Combination of rough and fuzzy sets based on alpha-level sets, (Lin, T. Y.; Cercone, N., Rough Sets and Data Mining: Analysis for Imprecise Data (1997), Kluwer Academic Publishers: Kluwer Academic Publishers Boston), 301-321 · Zbl 0859.04005
[30] Yao, Y. Y., Constructive and algebraic methods of the theory of rough sets, J. Inform. Sci., 109, 21-47 (1998) · Zbl 0934.03071
[31] Yao, Y. Y., Generalized rough set model, (Polkowski, L.; Skowron, A., Rough Sets in Knowledge Discovery 1. Methodology and Applications (1998), Physica-Verlag: Physica-Verlag Heidelberg), 286-318 · Zbl 0946.68137
[32] Yao, Y. Y., Relational interpretations of neighborhood operators and rough set approximation operators, Inform. Sci., 111, 239-259 (1998) · Zbl 0949.68144
[33] Yao, Y. Y.; Lin, T. Y., Generalization of rough sets using modal logic, Intelligent Automat. Soft Comput. an Internat. J., 2, 103-120 (1996)
[34] Zakowski, W., On a concept of rough sets, Demonstratio Math., XV, 1129-1133 (1982) · Zbl 0526.04005
[35] Zhang, W.-X.; Leung, Y.; Wu, W.-Z., Information Systems and Knowledge Discovery (2003), Science Press: Science Press Beijing
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