## An axiomatic approach of fuzzy rough sets based on residuated lattices.(English)Zbl 1189.03059

Summary: Rough set theory was developed by Pawlak as a formal tool for approximate reasoning about data. Various fuzzy generalizations of rough approximations have been proposed in the literature. As a further generalization of the notion of rough sets, $$L$$-fuzzy rough sets were proposed by A. M. Radzikowska and E. E. Kerre [Fuzzy Sets Syst. 126, 137–155 (2002; Zbl 1004.03043)]. In this paper, we present an operator-oriented characterization of $$L$$-fuzzy rough sets, that is, $$L$$-fuzzy approximation operators are defined by axioms. The methods of axiomatization of $$L$$-fuzzy upper and $$L$$-fuzzy lower set-theoretic operators guarantee the existence of corresponding $$L$$-fuzzy relations which produce the operators. Moreover, the relationship between $$L$$-fuzzy rough sets and $$L$$-topological spaces is obtained. The sufficient and necessary condition for the conjecture that an $$L$$-fuzzy interior (closure) operator derived from an $$L$$-fuzzy topological space can associate with an $$L$$-fuzzy reflexive and transitive relation such that the corresponding $$L$$-fuzzy lower (upper) approximation operator is the $$L$$-fuzzy interior (closure) operator is examined.

### MSC:

 3e+72 Theory of fuzzy sets, etc.

Zbl 1004.03043
Full Text:

### References:

 [1] Pawlak, Z., Rough sets, International journal of computer and information science, 11, 341-356, (1982) · Zbl 0501.68053 [2] Pawlak, Z., Rough sets—theoretical aspects to reasoning about data, (1991), Kluwer Academic Publisher Boston · Zbl 0758.68054 [3] Quafatou, M., $$\alpha -$$RST: A generalization of rough set theory, Information sciences, 124, 201-316, (2000) [4] Skowron, A.; Stepaniuk, J., Generalized approximation spaces, (), 18-21 [5] Slowinski, R.; Vanderpooten, D., Similarity relation as a basis for rough approximations, (), 17-33 [6] Slowinski, R.; Vanderpooten, D., A generalized definition of rough approximations based on similarity, IEEE transactions on knowledge and data engineering, 12, 2, 331-336, (2000) [7] Yao, Y.Y., Two views of the theory of rough sets in finite universe, International journal of approximate reasoning, 15, 291-317, (1996) · Zbl 0935.03063 [8] Ziarko, W., Variable precision rough set model, Journal of computer and system science, 46, 1, 39-59, (1993) · Zbl 0764.68162 [9] Lin, T.Y., Topological and fuzzy rough sets, (), 287-304 [10] Lin, T.Y., Granular computing: fuzzy logic and rough sets, (), 183-200 · Zbl 0949.68067 [11] Lin, T.Y., A rough logic formalism for fuzzy controllers: A hard and soft computing view, International journal of approximative reasoning, 15, 395-414, (1996) · Zbl 0947.93512 [12] Nguyen, H.T., Intervals in Boolean rings: approximation and logic, Foundations of computer and decision sciences, 17, 131-138, (1992) · Zbl 0781.06011 [13] Zakowski, W., On a concept of rough sets, Demonstration mathematic, XV, 1129-1133, (1982) · Zbl 0526.04005 [14] Comer, S., An algebraic approach to approximation of information, Fundamenta informaticae, 14, 492-502, (1991) · Zbl 0727.68114 [15] Comer, S., On connections between information system rough sets and algebraic logic, (), 117-124 · Zbl 0793.03074 [16] Lin, T.Y.; Liu, Q., Rough approximate operators: axiomatic rough set theory, (), 256-260 · Zbl 0818.03028 [17] Wiweger, R., On topological rough sets, Bulletin of the Polish Academy of sciences and mathematics, 37, 89-93, (1989) · Zbl 0755.04010 [18] Thiele, H., On axiomatic characterizations of crisp approximation operators, Information sciences, 129, 221-226, (2000) · Zbl 0985.03044 [19] Yao, Y.Y., Constructive and algebraic methods of the theory of rough sets, Journal of information sciences, 109, 21-47, (1998) · Zbl 0934.03071 [20] Yao, Y.Y., Generalized rough set model, (), 286-318 · Zbl 0946.68137 [21] Morsi, N.N.; Yakout, M.M., Axiomatics for fuzzy rough sets, Fuzzy sets and systems, 100, 327-342, (1998) · Zbl 0938.03085 [22] H. Thiele, On axiomatic characterization of fuzzy approximation operators. I, the fuzzy rough set based case, RSCTC2000, Banff Park Lodge, Bariff, Canada, October 19, in: Conf. Proc., 2000, pp. 239-247 [23] H. Thiele, On axiomatic characterization of fuzzy approximation operators II, The rough fuzzy set based case, in: Proc. 31st IEEE Internat. Symp. on Multiple-Valued Logic, 2001, pp. 330-335 [24] H. Thiele, On axiomatic characterization of fuzzy approximation operators III, The fuzzy diamond and fuzzy box cases, in: The 10th IEEE Internat. Conf. on Fuzzy Systems, vol. 2, 2001, pp. 1148-1151 [25] Radzikowska, A.M.; Kerre, E.E., A comparative study of fuzzy rough sets, Fuzzy sets and systems, 126, 137-155, (2002) · Zbl 1004.03043 [26] Wu, W.Z.; Mi, J.S.; Zhang, W.X., Generalized fuzzy rough sets, Information sciences, 151, 263-282, (2003) · Zbl 1019.03037 [27] Wu, W.Z.; Zhang, W.X., Constructive and axiomatic approaches of fuzzy approximation operators, Information sciences, 159, 233-254, (2004) · Zbl 1071.68095 [28] Wu, W.Z.; Leung, Y.; Mi, J.S., On characterizations of $$(\mathcal{I}, \mathcal{T}) -$$fuzzy rough approximation operators, Fuzzy sets and systems, 154, 76-102, (2005) [29] Mi, J.S.; Zhang, W.X., An axiomatic characterization of a fuzzy generalization of rough sets, Information sciences, 160, 235-249, (2004) · Zbl 1041.03038 [30] Liu, G.L., The axiomatization of the rough set upper approximation operations, Fundamenta informaticae, 69, 23, 331-342, (2006) · Zbl 1096.68150 [31] Liu, G.L., Generalized rough sets over fuzzy lattices, Information sciences, 178, 6, 1651-C1662, (2008) [32] Zhu, W.; Wang, F.Y., Reduction and axiomization of covering generalized rough sets, Information sciences, 152, 217-C230, (2003) · Zbl 1069.68613 [33] Zhu, W.; Wang, F.Y., Axiomatic systems of generalized rough sets, (), 216-C221 [34] Radzikowska, A.M.; Kerre, E.E., Fuzzy rough sets based on residuated lattices, (), 278-296 · Zbl 1109.68118 [35] Dubois, D.; Prade, H., Rough fuzzy sets and fuzzy rough sets, International journal of general systems, 17, 191-209, (1990) · Zbl 0715.04006 [36] Dubois, D.; Prade, H., Putting fuzzy sets and rough sets together, (), 203-232 [37] Dilworth, R.P.; Ward, N., Residuated lattices, Transactions of the American mathematical society, 45, 335-354, (1939) · Zbl 0021.10801 [38] Pei, D.W., The characterization of residuated lattices and regular residuated lattices, Acta Mathematica sinica, 45, 271-278, (2002), (in Chinese) · Zbl 1010.03052 [39] Eesteva, F.; Godo, L., Monoidal t-norm based logic: towards a logic for left-continuous t-norms, Fuzzy sets and systems, 124, 271-288, (2001) · Zbl 0994.03017 [40] Hájek, P., Metamathematics of fuzzy logic, (1998), Kluwer Academic Publishers Dordrecht · Zbl 0937.03030 [41] Pei, D.W., On equivalent forms of fuzzy logic systems NM and IMTL, Fuzzy sets and systems, 138, 187-195, (2003) · Zbl 1031.03047 [42] Wang, G.J., Non-classical mathematical logic and approximate reasoning, (2000), Science in China Press, (in Chinese) [43] Goguen, J.A., $$L$$-fuzzy sets, Journal of mathematical analysis and applications, 18, 145-174, (1967) · Zbl 0145.24404 [44] Chuchro, M., On rough sets in topological Boolean algebras, (), 157-160 · Zbl 0822.68106 [45] Chuchro, M., A certain conception of rough sets in topological Boolean algebras, Bulletin of the section of logic, 22, 1, 9-12, (1993) · Zbl 0776.04004 [46] Kortelainen, J., On relationship between modified sets, topological space and rough sets, Fuzzy sets and systems, 61, 91-95, (1994) · Zbl 0828.04002 [47] Boixader, D.; Jacas, J.; Recasens, J., Upper and lower approximations of fuzzy sets, International journal of general systems, 29, 555-568, (2000) · Zbl 0955.03056 [48] Wu, W.Z., (), 167-174 [49] Qin, K.Y.; Pei, Z., On the topological properties of fuzzy rough sets, Fuzzy sets and systems, 151, 601-613, (2005) · Zbl 1070.54006 [50] Wang, G.J., Theory of $$L$$-fuzzy topological spaces, (1988), Shaanxi Normal University Press Xi’an, (in Chinese)
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.