L-valued Fuzzy Rough Sets

Document Type: Original Manuscript

Authors

1 Department of Mathematics, Beijing Forestry University

2 Department of Mathematics, Ocean University of China, 238 Songling Road, 266100, Qingdao, P.R.China

Abstract

In this paper, we take a GL-quantale as the truth value table to study a new rough set model—L-valued fuzzy rough sets. The three key components of this model are: an L-fuzzy set A as the universal set, an L-valued relation of A and an L-fuzzy set of A (a fuzzy subset of fuzzy sets). Then L-valued fuzzy rough sets are completely characterized via both constructive and axiomatic approaches. 

Keywords


[1] M. Banerjee, S. K. Pal, Roughness of fuzzy set, Information Sciences, 93 (1996), 235–246.

[2] J. G. Bazan, A comparison of Dynamic and Non-dynamic Rough Set Methods for Extracting Laws From Decision Table, Heidelberg: PhysicA-Verlag (1998), 321–365.

[3] X. Chen, Q. Li, Construction of rough approximations in fuzzy setting, Fuzzy Sets and Systems, 158 (2007), 2641– 2653.

[4] D. Dubois, H. Prade, Rough fuzzy sets and fuzzy rough sets, International Journal of General Systems, 17 (1990), 191–208.

[5] D. Dubois, H. Prade, Putting Fuzzy Sets and Rough Sets Together, International Decision Support: Handbook of Applications and Advances of Rough Sets Theory, Kluwer, 1992.

[6] A. Ekins, A. Šostak and I. Ujane, On a Category of Extensional Fuzzy Rough Approximation Operators, In: International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems, (2016), 48–60.

[7] J. A. Goguen, L-fuzzy sets, Journal of Mathematical Analysis and Applications, 18 (1967), 145–174.

[8] S. E. Han, I. S. Kim, A. Šostak, On approximate-type systems generated by L-relations, Information Sciences, 281 (2014), 8–20.

[9] U. Höhle, M-valued sets and sheaves over integral commutative GL-monoids, in: S. E. Rodabaugh, E. P. Klement, U. Höhle (Eds.), Applications of Category Theory to Fuzzy Subsets, Kluwer Academic Publishers, Dordrecht, Boston, London, 1992.

[10] U. Höhle, Many-valued Topology and Its Applications, Kluwer Academic Publishers, Dordrecht, Boston, London, 2001.

[11] U. Höhle, Many-valued equalities and their representations, in: E. P. Klement, R. Mesiar(Eds.), Logic, Algebraic, Analytic, and Probabilistic Aspects of Triangular Norms, Amsterdam, Boston, Heidelberg, 2005.

[12] U. Höhle, T. Kubiak, A non-commutative and non-idempotent theory of quantale sets, Fuzzy Sets and Systems, 166 (2011), 1–43.

[13] J. Kortelainen, On relationship between modified sets, topological spaces and rough sets, Fuzzy Sets and Systems 61 (1994), 91–95.

[14] M. Liu, D. G. Chen, C. Wu, H. X. Li, Fuzzy reasoning based on a new fuzzy rough set and its application to scheduling problems, Computers and Mathematics with Applications, 51 (2006), 1507–1518.

[15] F. Li, Y. Q. Yin, Approaches to knowledge reduction of covering decision systems based on information theory, Information sciences, 179 (2009), 1694–1704.

[16] Z. W. Li, R. C. Cui, Similarity of fuzzy relations based on fuzzy topologies induced by fuzzy rough approximation operators, Information sciences, 305 (2015), 219–233.

[17] Z. W. Li, T. S. Xie, Q. G. Li, Topological structures of generalized rough sets, Computers and Mathematics with Applications, 63 (2012), 1066–1071.

[18] Z. M. Ma, B. Q. Hu, Topological and lattice structures of L-fuzzy rough sets determined by lower and upper sets, Information Sciences, 218 (2013), 194–204.

[19] J. S. Mi, W. X. Zhang, An axiomatic characterization of a fuzzy generalization of rough sets, Information Sciences, 160 (2004), 235–249.

[20] N. N. Morsi, M. M. Yakout, Axiomatics for fuzzy rough sets, Fuzzy sets and systems, 100 (1998), 327–342.

[21] S. G. Matthews, Partial metric topology, Annals of the New York Academy of Sciences, 728 (1994), 183–197.

[22] A. Nakamura, Fuzzy rough sets, Note on Multiple-valued Logic in Japan, 9 (1988), 1–8.

[23] Z. Pawlak, Rough sets, International Journal of Computer and Information Sciences, 11 (1982), 341–356.

[24] Z. Pawlak, Rough Sets: Theoretical Aspects of Reasoning about Data, Kluwer Academic Publishers, Dordrecht, Boston, London, 1991.

[25] Q. Pu, D. X. Zhang, Preordered sets valued in a GL-monoid, Fuzzy Sets and Systems, 187 (2012), 1–32.

[26] A. M. Radzikowska, E. E. Kerre, A comparative study of fuzzy rough sets, Fuzzy sets and systems, 126 (2002), 137–155.

[27] A. M. Radzikowska, E. E. Kerre, Fuzzy rough sets based on residuated lattice, Transactions on Rough Sets II, LNCS, 3135 (2004), 278–296.

[28] A. Šostak, Towards the theory of M-approximate systems: fundamentals and examples, Fuzzy Sets and Systems, 161 (2010), 2440–2461.

[29] Y. H. She, G. J. Wang, An axiomatic approach of fuzzy rough sets based on residuated lattices, Computers and Mathematics with Applications, 58 (2009), 189–201.

[30] H. Thiele, On Axiomatic Characterization of Fuzzy Approximation Operators I, the Rough Fuzzy Set Based Case, in: Proceedings of the 31st IEEE International Symposium on Multiple-Valued Logic, (2001), 330–335.

[31] C. Y. Wang, B. Q. Hu, Fuzzy rough sets based on generalized residuated lattices, Information Sciences, 248 (2013), 31–49.

[32] W. Z. Wu, Y. Leung, J. S. Mi, On characterizations of (I,T)-fuzzy rough approximation operators, Fuzzy Sets and Systems, 154 (2005), 76–102.

[33] W. Z. Wu, Y. H. Xu, M. W. Shao, G. Wang, Axiomatic characterizations of (S,T)-fuzzy rough approximation operators, Information Sciences, 334 (2016), 17–43.

[34] Z. Wu, W. Du, K. Qin, The properties of L-fuzzy rough set based on complete residuated lattice,in: International Symposium on Information Science and Engineering, (2008), 617–621.

[35] Y. Y. Yao, Generalization of rough sets using modal logic, Intelligent Automation and Soft Computing, 2 (1996), 103–120.

[36] Y. Y. Yao, Combination of Rough and Fuzzy Sets Based on Alpha-level Sets, in: T.Y. Lin, N. Cercone (Eds.), Rough Sets and Data Mining: Analysis for Imprecise Data, Kluwer Academic Publishers, Boston, 1997.

[37] L. A. Zadeh, Fuzzy sets, Information and Control, 8(1965), 338-358.

[38] L. Zhou, W. Z. Wu, On generalized intuitionistic fuzzy rough approximation operators, Information Sciences, 178 (2008), 2448–2465.