Generalization of rough fuzzy sets based on a fuzzy ideal

Document Type : Research Paper

Authors

1 Department of Mathematics, Faculty of Science, Benha University, Benha, Egypt

2 Department of Mathematics, Faculty of Science, Sohag University, Sohag, Egypt

Abstract

Since Pawlak defined the notion of rough sets in 1982, many authors made wide research studying rough sets in the ordinary case and the fuzzy case. This paper introduced a new style of rough fuzzy sets based on %arbitrary fuzzy relation $R$ and a fuzzy ideal $\ell$ on a universal finite set $X$. New lower and new upper fuzzy sets are introduced, and consequently, fuzzy interior and  fuzzy closure operators of a rough fuzzy set are discussed. These definitions, if $\ell$ is restricted to $\ell^{\circ} = \{\overline{0}\}$, imply the fuzzification of previous definitions given in the ordinary case, and moreover in the crisp case, we get exactly these previous definitions. The new style gives us a better accuracy value of roughness than the previous styles. Rough fuzzy connectedness is introduced as a sample of applications on the recent style of roughness.

Keywords

References

[1] A. A. Allam, M. Y. Bakeir, E. A. Abo-Tabl, New approach for basic rough set concepts, In: International Workshop on ”rough sets, fuzzy sets”, Data Mining and Granular Computing. Lecture Notes in Artificial Intelligence 3641, Springer, Regina, (2005), 64-73.
[2] A. A. Allam, M. Y. Bakeir, E. A. Abo-Tabl, New approach for closure spaces by relations, Acta Mathematica Academiae Paedagogicae Nyregyhziensis, 22 (2006), 285-304.
[3] S. H. Alsulami, I. Ibedou, S. E. Abbas, Fuzzy roughness via ideals, Journal of Intelligent and Fuzzy Systems, 39 (2020), 6869-6880.
[4] D. Boixader, J. Jacas, J. Recasens, Upper and lower approximations of fuzzy sets, International Journal of General Systems, 29 (2000), 555-568.
[5] D. Dubois, H. Prade, Rough fuzzy sets and fuzzy rough sets, International Journal of General Systems, 17(2-3) (1990), 191-209.
[6] E. Ekici, T. Noiri, Connectedness in ideal topological spaces, Novi Sad Journal of Mathematics, 38(2) (2008), 65-70.
[7] I. Ibedou, S. E. Abbas, Fuzzy topological concepts via ideals and grills, Annals of Fuzzy Mathematics and Informatics, 15(2) (2018), 137-148.
[8] A. Kandil, S. A. El-Sheikh, M. Hosny, M. Raafat, Bi-ideal approximation spaces and their applications, Soft Computing, 24 (2020), 12989-13001.
[9] A. Kandil, M. M. Yakout, A. Zakaria, Generalized rough sets via ideals, Annals of Fuzzy Mathematics and Informatics, 5 (2013), 525-532.
[10] A. Koam, I. Ibedou, S. E. Abbas, Fuzzy ideal topological spaces, Journal of Intelligent and Fuzzy Systems, 36 (2019), 5919-5928.
[11] A. M. Kozae, S. A. El-Sheikh, M. Hosny, On generalized rough sets and closure spaces, International Journal of Applied Mathematics, 23 (2010), 997-1023.
[12] G. Liu, Generalized rough sets over fuzzy lattices, Information Sciences, 178 (2008), 1651-1662.
[13] N. N. Morsi, M. M. Yakout, Axiomatics for fuzzy rough sets, Fuzzy Sets and Systems, 100 (1998), 327-342.
[14] Z. Pawlak, Rough sets, International Journal of Computer Science and Information Technologies, 11 (1982), 341-356.
[15] K. Qin, Z. Pei, On the topological properties of fuzzy rough sets, Fuzzy Sets and Systems, 151 (2005), 601-613.
[16] D. Sarkar, Fuzzy ideal theory: Fuzzy local function and generated fuzzy topology, Fuzzy Sets and Systems, 87 (1997), 117-123.
[17] C. Y. Wang, L. Wan, B. Zhang, Topological structures induced by L-fuzzifying approximation operators, Iranian Journal of Fuzzy Systems, 18(5) (2021), 141-154.
[18] W. Z. Wu, J. S. Mi, W. X. Zhang, Generalized fuzzy rough sets, Information Sciences, 151 (2003), 263-282.
[19] Y. Y. Yao, Two views of the theory of rough sets in finite universes, International Journal of Approximate Reasoning, 15 (1996), 291-317.
[20] L. A. Zadeh, Similarity relation and fuzzy orderings, Information Sciences, 3 (1971), 177-200.
[21] P. Zhi, P. Pei, L. Zheng, Topology vs generalized rough sets, Fuzzy Sets and Systems, 52(2) (2011), 231-239.
Iranian Journal of Fuzzy Systems
Volume 20, Issue 1
January and February 2023
Pages 27-38

Files

Share

How to cite

Statistics