Fuzzy rough sets based on Morsi fuzzy hemimetrics

Abstract

In this paper, we introduce a notion of Morsi fuzzy hemimetrics, a common generalization of hemimetrics and Morsi fuzzy metrics, as the basic structure to define and study fuzzy rough sets. We define a pair of fuzzy upper and lower approximation operators and investigate their properties. It is shown that upper definable sets, lower definable sets and definable sets are equivalent. Definable sets form an Alexandrov fuzzy topology such that the upper and lower approximation operators are the closure and the interior operators respectively.

Keywords

• Morsi fuzzy hemimetric
• fuzzy rough set
• approximation operator
• definable set
• Alexandrov fuzzy topology

References

1. [1] Z. Bonikowski, E. Bryniarski and U. Wybraniec-Skardowska, Extensions and intentions in the rough set theory, Inf. Sci. 107 (1–4), 149–167, 1998.
2. [2] G. Cattaneo and D. Ciucci, Algebraic structures for rough sets, Lect. Notes. Comput. Sci. 3135, 208–252, 2004.
3. [3] B.A. Davey and H.A. Priestley, Introduction to Lattices and Order, 2nd. ed., Cambridge University Press, Cambridge, 2002.
4. [4] L. D’eer, C. Cornelis and Y.Y. Yao, A semantically sound approach to Pawlak rough sets and covering-based rough sets, Int. J. Approx. Reason. 78 (11), 62–72, 2016.
5. [5] T.Q. Deng, Y.M. Chen, W.L. Xu and Q.H. Dai, A novel approach to fuzzy rough sets based on a fuzzy covering, Inf. Sci. 177 (11), 2308–2326, 2007.
6. [6] D. Dubois and H. Prade, Rough fuzzy sets and fuzzy rough sets, Int. J. Gen. Syst. 17 (2–3), 191–209, 1990.
7. [7] D. Dubois and H. Prade, Putting rough sets and fuzzy sets together, Intell. Decis. Support. 11, 203–232, 1992.
8. [8] A. George and P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets Syst. 64, 395–399, 1994.

9. [9] J. Goubault-Larrecq, Non-Hausdorff Topology and Domain Theory, Cambridge University Press, Cambridge, 2013.
10. [10] O. Grigorenko, J.J. Miñana, A. Šostak and O. Valero, On t-conorm based fuzzy (pseudo) metrics, Axioms 9 (78), 2020, doi: 10.3390/axioms9030078.
11. [11] P. Hajek, Metamathematics of Fuzzy Logic, Kluwer Academic Publishers, London, 1998.
12. [12] E.P. Klement, R. Mesiar and E. Pap, Triangular Norms, Kluwer Academic Publishers, Dordrecht, 2000.
13. [13] I. Kramosil and J. Michálek, Fuzzy metrics and statistical metric spaces, Kybernetika 11, 336–344, 1975.
14. [14] L.Q. Li, Q. Jin, B.X. Yao and J.C. Wu, A rough set model based on fuzzifying neighborhood systems, Soft. Comput. 24 (8), 6085–6099, 2020.
15. [15] T.J. Li, Y. Leung and W.X. Zhang, Generalized fuzzy rough approximation operators based on fuzzy coverings, Int. J. Approx. Reason. 48 (3), 836–856, 2008.
16. [16] G.L. Liu, The axiomatization of the rough set upper approximation operations, Fundam. Inform. 69 (3), 331–342, 2006.
17. [17] L.W. Ma, Two fuzzy covering rough set models and their generalizations over fuzzy lattices, Fuzzy Sets Syst. 294, 1–17, 2016.
18. [18] J.S. Mi and W.X. Zhang, An axiomatic characterization of a fuzzy generalization of rough sets, Inf. Sci. 160 (1–4), 235–249, 2004.
19. [19] N.N. Morsi, On fuzzy pseudo-normed vector spaces, Fuzzy Sets Syst. 27, 351-372, 1988.
20. [20] Z. Pawlak, Rough sets, Int. J. Comput. Inform. Sci. 11 (5), 341–356, 1982.
21. [21] A.M. Radzikowska, On lattice-based fuzzy rough sets, In: C. Cornelis, G. Deschrijver, M. Nachtegael, S. Schockaert, Y. Shi (Eds.), 35 Years of Fuzzy Set Theory, Springer, 107–126, 2010.
22. [22] A.M. Radzikowska and E.E. Kerre, A comparative study of fuzzy rough sets, Fuzzy Sets Syst. 126 (2), 137–155, 2002.
23. [23] A.M. Radzikowska and E.E. Kerre, Fuzzy rough sets based on residuated lattices, Lect. Notes. Comput. Sc. 3135, 278–296, 2005.
24. [24] Y. She and G. Wang, An axiomatic approach of fuzzy rough sets based on residuated lattices, Comput. Math. Appl. 58 (1), 189–201, 2009.
25. [25] A. Skowron and J. Stepaniuk, Tolerance approximation spaces, Fundam. Inform. 27 (2–3), 245–253, 1996.
26. [26] R. Slowinski and D. Vanderpooten, A generalized definition of rough approximations based on similarity, IEEE Trans. Knowl. Data Eng. 12 (2), 331–336, 2000.
27. [27] H. Thiele, On axiomatic characterization of fuzzy approximation operators I, Proceedings of the 2nd International Conference of Rough Sets and Current Trends in Computing, RSCTC 2000 Banff, Canada, 277–285, 2000.
28. [28] X.W. Wei, B. Pang and J.S. Mi, Axiomatic characterizations of L-valued rough sets using a single axiom, Inf. Sci. 580, 283–310, 2021.
29. [29] W.Z. Wu, Y. Leung and J.S. Mi, On characterizations of $(I,T)$-fuzzy rough approximation operators, Fuzzy Sets Syst. 154 (1), 76–102, 2005.
30. [30] B. Yang and B.Q. Hu, A fuzzy covering-based rough set model and its generalization over fuzzy lattice, Inf. Sci. 367–368, 463–486, 2016.
31. [31] Y.Y. Yao, The two sides of the theory of rough sets, Knowl.-Based. Syst. 80, 67–77, 2015.
32. [32] W. Yao, Y.H. She and L.X Lu, Metric-based $L$-fuzzy rough sets: Approximation operators and definable sets, Knowl.-Based. Syst. 163, 91–102, 2019.
33. [33] W. Yao, G. Zhang and C.-J. Zhou, Real-valued hemimetric-based fuzzy rough sets and an application to contour extraction of digital surfaces, Fuzzy Sets Syst. 459, 201–219,2023 .
34. [34] W. Zakowski, Approximations in the space $(u, p)$, Demonstr. Math. 16 (3), 761–769, 1983.