$L$-fuzzifying rough sets based on $KM$-fuzzy pseudo metrics

Abstract

In this paper, we construct an $L$-fuzzifying lower and an $L$-fuzzifying upper approximation operator ($L$ denotes a completely distributive De Morgan algebra) based on a $KM$-fuzzy pseudo metric and study their properties.Then we discuss some equivalent characterizations of the corresponding axiom of $KM$-fuzzy pseudo metrics via its induced $L$-fuzzifying rough set, respectively.Moreover, we explore the relationships between the crisp rough set induced by a $KM$-fuzzy pseudo metric $M$ and the crisp rough set induced by the corresponding crisp pseudo metrics $d_{a}^{M}$.Besides, we investigate the relationships between the $L$-fuzzifying rough set induced by a family of crisp pseudo metrics $\mathcal{D}$ and the $L$-fuzzifying rough set induced by the corresponding $KM$-fuzzy pseudo metric ${M}^{\mathcal{D}}$. Finally, we get the conclusions that$\underline{R}^{\mathcal{D}}\leq {\underline{R}^{M}}^{\mathcal{D}}$ and ${\overline{R}^{M}}^{\mathcal{D}}\leq \overline{R}^{\mathcal{D}}$.

Keywords

• L-fuzzifying lower approximation operator
• L-fuzzifying upper approximation operator
• L-fuzzifying rough set
• KM-fuzzy pseudo metric.

Supporting Institution

North China University of Technology; Jinyinhu high school; Nanjing Institute of Technology

Project Number

National Natural Science Foundation of China (No.11901007), North China University of Technology organized scientific research project (No. 2023YZZKY19,~No.2024NCUTYXCX104), Natural Science Foundation of Jiangsu Province (BK20230411)

Ethical Statement

The authors declare that there are no Ethical Data in our paper.

Thanks

The authors would like to express their sincere thanks to the anonymous reviewers for their careful reading and constructive comments.

References

1. [1] Y.L. Bao, H.L. Yang and Y.H. She, Using one axiom to characterize L-fuzzy rough approximation operators based on residuated lattices, Fuzzy Sets Syst. 336, 87–115, 2018.
2. [2] C.L. Chang, Fuzzy topological spaces, J. Math. Anal. Appl. 24, 182–190, 1968.
3. [3] D. Dubois and H. Prade, Rough fuzzy sets and fuzzy rough sets, Internat. J. Gen. Systems 17, 191–209, 1990.
4. [4] D. Dubois and H. Prade, Putting rough sets and fuzzy sets together, Intelligent Decision Support 11, 203–232, 1992.
5. [5] J.A. Goguen, L-fuzzy sets, J. Math. Anal. Appl. 18, 145–174, 1967.
6. [6] I. Kramosil and J. Michalek, Fuzzy metrics and statistical metric spaces, Kybernetika 11, 336–344, 1975.
7. [7] T.J. Li, Y. Leung and W.X. Zhang, Generalized fuzzy rough approximation operators based on fuzzy coverings, Int. J. Approx. Reason. 48, 836–856, 2008.
8. [8] L.Q. Li, B.X. Yao, J.M. Zhan and Q. Jin, L-fuzzifying approximation operators derived from general L-fuzzifying neighborhood systems, Int. J. Mach. Learn. Cybern. 12, 1343–1367, 2021.

9. [9] G.L. Liu and Y. Sai, A comparison of two types of rough sets induced by coverings, Int. J. Approx. Reason. 50, 521–528, 2009.
10. [10] Z.M. Ma and B.Q. Hu, Topological and lattice structures of L-fuzzy rough sets determined by lower and upper sets, Inform. Sci. 218, 194–204, 2013.
11. [11] L.W. Ma, Two fuzzy covering rough set models and their generalizations over fuzzy lattices, Fuzzy Sets Syst. 294, 1–17, 2016.
12. [12] P. Mandal and A.S. Ranadive, Multi-granulation fuzzy probabilistic rough sets and their corresponding three-way decisions over two universes, Iran. J. Fuzzy Syst. 16, 61–76, 2019.
13. [13] I. Mardones-Pérez and M.A. de Prada Vicente, A representation theorem for fuzzy pseudometrics, Fuzzy Sets Syst. 195, 90–99, 2012.
14. [14] J.S. Mi and W.X. Zhang, An axiomatic characterization of a fuzzy generalization of rough sets, Inform. Sci. 160, 235–249, 2004.
15. [15] J.S. Mi, Y. Leung, H.Y. Zhao and T. Feng, Generalized fuzzy rough sets determined by a triangular norm, Inform. Sci. 178, 3203–3213, 2008.
16. [16] N.N. Morsi and M.M. Yakout, Axiomatics for fuzzy rough sets, Fuzzy Sets Syst. 100, 327–342, 1998.
17. [17] B. Pang, J.S. Mi and Z.Y. Xiu, L-fuzzifying approximation operators in fuzzy rough sets, Inform. Sci. 480, 14–33, 2019.
18. [18] B. Pang, J.S. Mi and W. Yao, L-fuzzy rough approximation operators via three new types of L-fuzzy relations, Soft Comput. 23, 11433–11446, 2019.
19. [19] B. Pang and J.S. Mi, Using single axioms to characterize L-rough approximate operators with respect to various types of L-relations, Int. J. Mach. Learn. Cybern. 11, 1061–1082, 2020.
20. [20] Z. Pawlak, Rough sets, Internat. J. Comput. Inform. Sci. 11, 341–356, 1982.
21. [21] Z. Pawlak, Rough Sets: Theoretical Aspects of Reasoning about Data, Kluwer Academic Publishers, Boston, 1991.
22. [22] W.J. Pervin, Foundations of General Topology, Academic Press, New York, 1964.
23. [23] A.M. Radzikowska and E.E. Kerre, A comparative study of fuzzy rough sets, Fuzzy Sets Syst. 126, 137–155, 2002.
24. [24] A.M. Radzikowska and E.E. Kerre, Fuzzy Rough Sets Based on Residuated Lattices, in Trans. Rough Sets II, 3135, 278–296, 2004.
25. [25] M.V. Rosa, On fuzzy topology fuzzy convexity spaces and fuzzy local convexity, Fuzzy Sets Syst. 62, 97–100, 1994.
26. [26] F.G. Shi and Z.Y. Xiu, A new approach to the fuzzification of convex structures, J. Appl. Math. 2014, 1–12, 2014.
27. [27] Y. Sun, B. Pang and J.S. Mi, Axiomatic characterizations of (G,O)-fuzzy rough approximation operators via overlap and grouping functions on a complete lattice, Internat. J. Gen. Systems 52 (6), 664–693, 2023.
28. [28] X.W. Wei, B. Pang and J.S. Mi, Axiomatic characterizations of L-valued rough sets using a single axiom, Inform. Sci. 580, 283–310, 2021.
29. [29] X.W. Wei, B. Pang and J.S. Mi, Axiomatic characterizations of L-fuzzy rough sets by L-fuzzy unions and L-fuzzy intersections, Internat. J. Gen. Systems 51 (3), 277–312, 2022.
30. [30] Y. Chen and X.W. Wei, Axiomatic approaches to three types of L-valued rough sets, Int. J. Mach. Learn. Cybern. 15, 5469–5493, 2024.
31. [31] W.Z. Wu and W.X. Zhang, Constructive and axiomatic approaches of fuzzy approximation operators, Inform. Sci. 159, 233–254, 2004.
32. [32] W.Z. Wu, Y. Leung and J.S. Mi, On characterizations of (I, T)-fuzzy rough approximation operators, Fuzzy Sets Syst. 154, 76–102, 2005.
33. [33] Y.Y. Yao, Relational interpretations of neighborhood operators and rough set approximation operators, Inform. Sci. 111, 239–259, 1998.
34. [34] 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.
35. [35] Y. Zhong and F.G. Shi, Characterizations of (L,M)-fuzzy topology degrees, Iran. J. Fuzzy Syst. 15, 129–149, 2018.
36. [36] Y. Zhong, S. Lin and F.H. Chen, An L-fuzzy convex structure induced by Lsubuniverse degrees, J. Nonlinear Convex Anal. 21, 2795–2804, 2020.
37. [37] Y. Zhong and A.P. Šostak, A new definition of fuzzy k-pseudo metric and its induced fuzzifying structures, Iran. J. Fuzzy Syst. 18, 55–66, 2021.
38. [38] Y. Zhong and S. Lin, M-fuzzifying k-pseudo metric space and its induced M-fuzzifying structures, Comput. Appl. Math. 41, 41–70, 2022.
39. [39] Y. Zhong, X. Wu, A.P. Šostak and F.G. Shi, (L,M)-fuzzy k-pseudo metric space, Mathematics 10, 1151, 2022.
40. [40] Y. Zhong, S. Lin and J. Chang, Characterizations of M-fuzzifying topological operators and its applications, Filomat 39 (3), 905–919, 2025.
41. [41] Y. Zhong, Y.X. Han, F.F. Zhao and Z.Y. Yang, Fuzzification of the category of (L,M)- fuzzy convex spaces, Filomat 39 (14), 4857–4871, 2025.
42. [42] Y. Zhong, Y.H. Guo and Z.H. Yang, Research on k-norms and fuzzy k-norms, Comput. Appl. Math. 44, 112, 2025.