^{1}Department of Mathematics, Faculty of Science and Arts at Belqarn, P. O. Box 60, Sabt Al-Alaya 61985, Bisha University, Saudi Arabia

^{2}Department of Mathematics, Beni-suef University, Beni-suef, Egypt

Abstract

ur aim of this paper is to introduce the concept of $L$-double fuzzy rough sets in which both constructive and axiomatic approaches are used. In constructive approach, a pair of $L$-double fuzzy lower (resp. upper) approximation operators is defined and the basic properties of them are studied. From the viewpoint of the axiomatic approach, a set of axioms is constructed to characterize the $L$-double fuzzy upper (resp. lower) approximation of $L$-double fuzzy rough sets. Finally, from $L$-double fuzzy approximation operators, we generated Alexandrov $L$-double fuzzy topology.

[1] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20(1) (1986), 87-96. [2] K. Atanassov, Intuitionistic fuzzy sets: theory and applications, Physica-Verlag, Heidelberg, 1999. [3] C. L. Chang, Fuzzy topological spaces, J. Math. Anal. Appl., 24 (1968), 182-190. [4] J. K. Chen and J. J. Li, An application of rough sets to graph theory, Information Sciences, 201 (2012), 114-127. [5] D. C oker, An introduction to intuitionistic fuzzy topological spaces, Fuzzy Sets and Systems, 88 (1997), 81-89. [6] D. C oker and M. Demirci, An introduction to intuitionistic fuzzy topological spaces in Sostak's sense, Busefal, 67 (1996), 67-76.

[7] D. C oker, Fuzzy rough sets are intuitionistic L-fuzzy sets, Fuzzy Sets and Systems, 96(3) (1998), 381-383. [8] D. Dubois and H. Prade, Rough fuzzy sets and fuzzy rough sets, International Journal of General System, 17 (1990), 191-208. [9] A. A. Estaji, M. R. Hooshmandasl and B. Davvaz, Rough set theory applied to lattice theory, Information Sciences, 200 (2012), 108-122. [10] J. G. Garcia and S. E. Rodabaugh, Order-theoretic, topological, categorical redundancides of intervalvalued sets, grey sets, vague sets, interval-valued intuitionistic sets, intuitionistic fuzzy sets and topologies, Fuzzy Sets and Systems, 156 (2005), 445-484. [11] J. A. Goguen, L-fuzzy sets, J. Math. Anal. Appl., 18 (1967), 145-174. [12] J. A. Goguen, The fuzzy tychono theorem, J. Math. Anal. Appl., 34 (1973), 734-742. [13] J. Hao and Q. Li, The relationship between L-fuzzy rough set and L-topology, Fuzzy Sets and Systmes, 178 (2011), 74-83. [14] U. Hohle and A. P. Sostak, A general theory of fuzzy topological spaces, Fuzzy Sets and Systmes, 73(1) (1995), 131-149. [15] S. P. Jena and S. K. Ghosh, Intuitionistic fuzzy rough sets, Notes on Intuitionistic Fuzzy Sets, 8 (2002), 1-18. [16] Y. B. Jun, Roughness of ideals in BCK-algebras, Scientiae Mathematicae Japonicae, 75(1) (2003), 165-169. [17] M. Kryszkiewicz, Rough set approach to incomplete information systems, Information Sciences, 112 (1998), 39-49. [18] T. Kubiak, On fuzzy topologies, Ph.D. Thesis, Adam Mickiewicz, Poznan, Poland, (1985). [19] L. Lin, X. H. Yuan and Z. Q. Xia, Multicriteria fuzzy decision-making methods based on intuitionistic fuzzy sets, Journal of Computer and System Sciences, 73(1) (2007), 84-88. [20] G. Liu, Generalized rough sets over fuzzy lattices, Information Sciences, 178 (2008), 1651- 1662. [21] Z. Pawlak, Information system theoretical foundations, Information Sciences, 6(1981), 205- 218 . [22] Z. Pawlak, Rough sets, International Journal of Computer and Information Sciences, 11 (1982), 341-356. [23] Z. Pawlak, Rough sets: theoretical aspects of reasoning about data, Kluwer Academic Publishers, Boston, 1991. [24] Z. Pawlak and A. Skowron, Rudiments of rough sets, Information Sciences, 177 (2007), 3-27. [25] Z. Pawlak and A. Skowron, Rough sets: some extensions, information sciences, Information Sciences, 177 (2007), 28-40. [26] W. Pedrycz, Granular computing: aanalysis and design of intelligent systems, CRC Press, Boca Raton, 2013. [27] K. Qin and Z. Pei, On the topological properties of fuzzy rough sets, Fuzzy Sets and Systmes, 151 (2005), 601-613. [28] M. Quafafou, -RST: a generalization of rough set theory, Information Sciences, 124 (2000), 301-316. [29] A. M. Radzikowska, Rough approximation operations based on IF sets, Lecture Notes in Computer Science , 4029 (2006), 528-537. [30] A. M. Radzikowska and E. E. Kerre, Fuzzy rough sets based on residuated lattices, Transac- tions on Rough Sets, Lecture Notes in Computer Sciences, 3135 (2004), 278-296. [31] A. A. Ramadan, Smooth topological spaces, Fuzzy Sets and Systmes, 48 (1992), 371-375. [32] A. A. Ramadanm and A. A. Abd El-latif, Supra fuzzy convergence of fuzzy lter, Bull. Korean Math. Soc., 45(2) (2008), 207-220. [33] A. A. Ramadanm and A. A. Abd El-latif, Images and preimages of L-fuzzy ideal bases, J. Fuzzy Math., 17(3) (2009), 611-632. [34] S. K. Samanta and T. K. Mondal, On intuitionistic gradation of openness, Fuzzy Sets and Systmes, 131 (2002), 323-336.

[35] S. K. Samanta and T. K. Mondal, Intuitionistic fuzzy rough sets and rough intuitionistic fuzzy sets, Journal of Fuzzy Mathematics, 9 (2001), 561-582. [36] M. H. Shahzamanian, M. Shirmohammadi and B. Davvaz, Roughness in Cayley graphs, Information Sciences, 180 (2010), 3362-3372. [37] A. P. Sostak, On a fuzzy topological structure, Rend. Circ. Mat. Palermo 2 Suppl., 11 (1985), 89-103. [38] S. P. Tiwari and A. K. Srivastava, Fuzzy rough sets, fuzzy preorders and fuzzy topologies, Fuzzy Sets and Systmes, 210 (2013), 63-68. [39] L. K. Vlachos and G. D. Sergiadis, Intuitionistic fuzzy information applications to pattern recognition, Pattern Recognition Letters, 177(11) (2007), 197-206. [40] U. Wybraniec-Skardowska, On a generalization of approximation space, Bulletin of the Polish Academy of Sciences: Mathematics, 37 (1989), 51-61. [41] Z. S. Xu, Intuitionistic preference relations and their application in group decision making, Information Sciences, 177 (11) (2007), 2363-2379. [42] Y. Y. Yao, Relational interpretations of neighborhood operators and rough set approximation operators, Information Sciences, 111 (1998), 239-259. [43] D. S. Yeung, D. Chen, E. C. C. Tsang, J. W. T. Lee and X. Z. Wang, On the generalization of fuzzy rough sets, IEEE Trans. Fuzzy Syst., 13 (2005), 343-361. [44] L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353. [45] Z. Zhang, Generalized intuitionistic fuzzy rough sets based on intuitionistic fuzzy coverings, Information Sciences, 198 (2012), 186-206. [46] X. Zhang, B. Zhou and P. Li, A general frame for intuitionistic fuzzy rough sets, Information Sciences, 216 (2012), 34-49. [47] L. Zhou, W. Z. Wu and W. X. Zhang, On intuitionistic fuzzy rough sets and their topological structures, International Journal of General Systems, 38 (2009), 589-616. [48] W. Ziarko, Variable precision rough set model, Journal of Computer and System Sciences, 46 (1993), 39-59.