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Generalized fuzzy rough approximation operators determined by fuzzy implicators. (English) Zbl 1316.68198

Summary: In this paper, a general framework for the study of dual fuzzy rough approximation operators determined by a fuzzy implication operator \(\mathcal I\) in infinite universes of discourse is investigated. Lower and upper approximations of fuzzy sets with respect to a fuzzy approximation space in infinite universes of discourse are first introduced. Properties of \(\mathcal I\)-fuzzy rough approximation operators are then examined. An operator-oriented characterization of fuzzy rough sets is further proposed, that is, \(\mathcal I\)-fuzzy rough approximation operators are defined by axioms. Different axiom sets of lower and upper \(\mathcal I\)-fuzzy set-theoretic operators guarantee the existence of different types of fuzzy relations which produce the same operators. Finally, a comparative study of \(\mathcal I\)-fuzzy rough sets with fuzzy topological spaces is presented. It is proved that there exists a one-to-one correspondence between the set of all reflexive and \(\mathcal T\)-transitive fuzzy approximation spaces and the set of all fuzzy Alexandrov spaces such that the lower and upper \(\mathcal I\)-fuzzy rough approximation operators in a fuzzy approximation space are, respectively, the fuzzy interior and closure operators in a fuzzy topological space.

MSC:

68T37 Reasoning under uncertainty in the context of artificial intelligence
03B52 Fuzzy logic; logic of vagueness
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[1] Abdel-Hamid, A. A.; Morsi, N. N., On the relationship of extended necessity measures to implication operators on the unit interval, Information Sciences, 82, 129-145 (1995) · Zbl 0870.68139
[2] Abu-Donia, H. M.; Salama, A. S., Generalization of Pawlakʼs rough approximation spaces by using δ β-open sets, International Journal of Approximate Reasoning, 53, 1094-1105 (2012) · Zbl 1264.54005
[3] Arenas, F. G., Alexandroff spaces, Acta Mathematica Universitatis Comenianae, 68, 17-25 (1999) · Zbl 0944.54018
[4] 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
[5] Chang, C. L., Fuzzy topological spaces, Journal of Mathematical Analysis and Applications, 24, 182-190 (1968) · Zbl 0167.51001
[6] Chen, D. G.; Zhao, S. Y., Local reduction of decision system with fuzzy rough sets, Fuzzy Sets and Systems, 161, 1871-1883 (2010) · Zbl 1192.68683
[7] Chuchro, M., On Rough Sets in Topological Boolean Algebras, (Ziarko, W., Rough Sets, Fuzzy Sets and Knowledge Discovery (1994), Springer-Verlag: Springer-Verlag Berlin), 157-160 · Zbl 0822.68106
[8] Chuchro, M., A certain conception of rough sets in topological Boolean algebras, Bulletin of the Section of Logic, 22, 9-12 (1993) · Zbl 0776.04004
[9] Cock, M. D.; Cornelis, C.; Kerre, E. E., Fuzzy rough sets: The forgotten step, IEEE Transactions on Fuzzy Systems, 15, 121-130 (2007)
[10] Cornelis, C.; Deschrijver, G.; Kerre, E. E., Implication in intuitionistic fuzzy and interval-valued fuzzy set theory: Construction, classification, application, International Journal of Approximate Reasoning, 35, 55-95 (2004) · Zbl 1075.68089
[11] Dubois, D.; Prade, H., Rough fuzzy sets and fuzzy rough sets, International Journal of General Systems, 17, 191-208 (1990) · Zbl 0715.04006
[12] Fang, J. M.; Chen, P. W., One-to-one correspondence between fuzzifying topologies and fuzzy preorders, Fuzzy Sets and Systems, 158, 1814-1822 (2007) · Zbl 1137.54005
[13] Fang, J. M., I-fuzzy Alexandrov topologies and specialization orders, Fuzzy Sets and Systems, 158, 2359-2374 (2007) · Zbl 1136.54002
[14] Hao, J.; Li, Q. G., The relationship between \(L\)-fuzzy rough set and \(L\)-topology, Fuzzy Sets and Systems, 178, 74-83 (2011) · Zbl 1238.54005
[15] Hooshmandasl, M. R.; Karimi, A.; Almbardar, M.; Davvaz, B., Axiomatic systems for rough set-valued homomorphisms of associative rings, International Journal of Approximate Reasoning, 54, 297-306 (2013) · Zbl 1266.03059
[16] Hu, Q. H.; Yu, D. R.; Pedrycz, W.; Chen, D. G., Kernelized fuzzy rough sets and their applications, IEEE Transactions on Knowledge and Data Engineering, 23, 1649-1667 (2011)
[17] Hu, Q. H.; Zhang, L.; An, S.; Zhang, D.; Yu, D. R., On robust fuzzy rough set models, IEEE Transactions on Fuzzy Systems, 20, 636-651 (2012)
[18] Hu, Q. H.; Zhang, L.; Chen, D. G.; Pedrycz, W.; Yu, D. R., Gaussian kernel based fuzzy rough sets: Model, uncertainty measures and applications, International Journal of Approximate Reasoning, 51, 453-471 (2010) · Zbl 1205.68424
[19] Jensen, R.; Shen, Q., Fuzzy-rough attributes reduction with application to web categorization, Fuzzy Sets and Systems, 141, 469-485 (2004) · Zbl 1069.68609
[20] Jensen, R.; Shen, Q., Semantics-preserving dimensionality reduction: Rough and fuzzy-rough based approaches, IEEE Transactions on Knowledge and Data Engineering, 16, 1457-1471 (2004)
[21] Jensen, R.; Shen, Q., Fuzzy-rough sets assisted attribute selection, IEEE Transactions on Fuzzy Systems, 15, 73-89 (2007)
[22] Jensen, R.; Shen, Q., New approaches to fuzzy-rough feature selection, IEEE Transactions on Fuzzy Systems, 17, 824-838 (2009)
[23] Klement, E. P.; Mesiar, R.; Pap, E., Triangular Norms, Trends in Logic, vol. 8 (2000), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht
[24] Klir, G. J.; Yuan, B., Fuzzy Logic: Theory and Applications (1995), Prentice Hall: Prentice Hall Englewood Cliffs, NJ · Zbl 0915.03001
[25] Kuncheva, L. I., Fuzzy rough sets: Application to feature selection, Fuzzy Sets and Systems, 51, 147-153 (1992)
[26] Lai, H. L.; Zhang, D. X., Fuzzy preorder and fuzzy topology, Fuzzy Sets and Systems, 157, 1865-1885 (2006) · Zbl 1118.54008
[27] Li, T. J.; Zhang, W. X., Rough fuzzy approximations on two universes of discourse, Information Sciences, 178, 892-906 (2008) · Zbl 1128.68099
[28] Liu, G. L., Axiomatic systems for rough sets and fuzzy rough sets, International Journal of Approximate Reasoning, 48, 857-867 (2008) · Zbl 1189.03056
[29] Liu, X. D.; Pedrycz, W.; Chai, T. Y.; Song, M. L., The development of fuzzy rough sets with the use of structures and algebras of axiomatic fuzzy sets, IEEE Transactions on Knowledge and Data Engineering, 21, 443-462 (2009)
[30] Lowen, R., Fuzzy topological spaces and fuzzy compactness, Journal of Mathematical Analysis and Applications, 56, 621-633 (1976) · Zbl 0342.54003
[31] Mi, J.-S.; Leung, Y.; Zhao, H.-Y.; Feng, T., Generalized fuzzy rough sets determined by a triangular norm, Information Sciences, 178, 3203-3213 (2008) · Zbl 1151.03344
[32] 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
[33] Morsi, N. N.; Yakout, M. M., Axiomatics for fuzzy rough sets, Fuzzy Sets and Systems, 100, 327-342 (1998) · Zbl 0938.03085
[34] Naturman, C. A., Interior Algebras and Topology (1991), Department of Mathematics, University of Cape Town, Ph.D. Thesis
[35] Ouyang, Y.; Wang, Z. D.; Zhang, H.-P., On fuzzy rough sets based on tolerance relations, Information Sciences, 180, 532-542 (2010) · Zbl 1189.68131
[36] Pawlak, Z., Rough sets, International Journal of Computer and Information Science, 11, 341-356 (1982) · Zbl 0501.68053
[37] Pawlak, Z., Rough Sets: Theoretical Aspects of Reasoning about Data (1991), Kluwer Academic Publishers: Kluwer Academic Publishers Boston · Zbl 0758.68054
[38] Qin, K. Y.; Pei, Z., On the topological properties of fuzzy rough sets, Fuzzy Sets and Systems, 151, 601-613 (2005) · Zbl 1070.54006
[39] Radzikowska, A. M.; Kerre, E. E., A comparative study of fuzzy rough sets, Fuzzy Sets and Systems, 126, 137-155 (2002) · Zbl 1004.03043
[40] Radzikowska, A. M.; Kerre, E. E., Fuzzy rough sets based on residuated lattices, (Transactions on Rough Sets II. Transactions on Rough Sets II, Lecture Notes in Computer Science, vol. 3135 (2004)), 278-296 · Zbl 1109.68118
[41] 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
[42] Tiwari, S. P.; Srivastava, A. K., Fuzzy rough sets, fuzzy preorders and fuzzy topologies, Fuzzy Sets and Systems, 210, 63-68 (2013) · Zbl 1260.54024
[43] Wang, L. D.; Liu, X. D.; Qiu, W. R., Nearness approximation space based on axiomatic fuzzy sets, International Journal of Approximate Reasoning, 53, 200-211 (2012) · Zbl 1250.68260
[44] Wang, X. Z.; Tsang, E. C.C.; Zhao, S. Y.; Chen, D. G.; Yeung, D. S., Learning fuzzy rules from fuzzy samples based on rough set technique, Fuzzy Sets and Systems, 177, 4493-4514 (2007) · Zbl 1129.68069
[45] Wiweger, R., On topological rough sets, Bulletin of Polish Academy of Sciences: Mathematics, 37, 89-93 (1989) · Zbl 0755.04010
[46] Wu, W.-Z., On some mathematical structures of \(T\)-fuzzy rough set algebras in infinite universes of discourse, Fundamenta Informaticae, 108, 337-369 (2011) · Zbl 1241.03066
[47] Wu, W.-Z.; Leung, Y.; Mi, J.-S., On generalized fuzzy belief functions in infinite spaces, IEEE Transactions on Fuzzy Systems, 17, 385-397 (2009)
[48] Wu, W.-Z.; Leung, Y.; Mi, J.-S., On characterizations of \((I, T)\)-fuzzy rough approximation operators, Fuzzy Sets and Systems, 15, 76-102 (2005) · Zbl 1074.03027
[49] Wu, W.-Z.; Mi, J.-S., Some mathematical structures of generalized rough sets in infinite universes of discourse, (Transactions on Rough Sets XIII. Transactions on Rough Sets XIII, Lecture Notes in Computer Science, vol. 6499 (2011)), 175-206 · Zbl 1242.68341
[50] Wu, W.-Z.; Xu, Y.-H., On fuzzy topological structures of rough fuzzy sets, (Transactions on Rough Sets XVI. Transactions on Rough Sets XVI, Lecture Notes in Computer Science, vol. 7736 (2013)), 125-143 · Zbl 1377.68253
[51] Wu, W.-Z.; Mi, J.-S.; Zhang, W.-X., Generalized fuzzy rough sets, Information Sciences, 151, 263-282 (2003) · Zbl 1019.03037
[52] Wu, W.-Z.; Zhang, W.-X., Constructive and axiomatic approaches of fuzzy approximation operators, Information Sciences, 159, 233-254 (2004) · Zbl 1071.68095
[53] 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
[54] Yeung, D. S.; Chen, D. G.; Tsang, E. C.C.; Lee, J. W.T.; Wang, X. Z., On the generalization of fuzzy rough sets, IEEE Transactions on Fuzzy Systems, 13, 343-361 (2005)
[55] Zhang, X. H.; Zhou, B.; Li, P., A general frame for intuitionistic fuzzy rough sets, Information Sciences, 216, 34-49 (2012) · Zbl 1251.03064
[56] Zhao, S. Y.; Tsang, E. C.C., On fuzzy approximation operators in attribute reduction with fuzzy rough sets, Information Sciences, 178, 3163-3176 (2008) · Zbl 1154.68532
[57] Zhao, S. Y.; Tsang, E. C.C.; Chen, D. G., The model of fuzzy variable precision rough sets, IEEE Transactions on Fuzzy Systems, 17, 451-467 (2009)
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