Fuzzy accompanied approximation space under fuzzy relation

Document Type : Research Paper


School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, PR China


If there is a fuzzy relation $\widetilde{R}$ between two spaces $U$ and $V$, the fuzzy approximations in both spaces based on $\widetilde{R}$ are widely studied, and they basically reflect only the influence from one space to another. In this paper, on each space of $U$ and $V$, two new fuzzy relations are derived from $\widetilde{R}$, a positive low-value relation and a conservative high-value relation, to reflect the interaction and feedback between the two spaces. So, the fuzzy approximations based on them can reflect the combination of the action and the reaction from one space to another. Therefore, two spaces $U$ and $V$ are closely accompanied, and $(U, V, \widetilde{R})$ is a whole, so it is called a fuzzy accompanied approximation space (FAAS). In an FAAS, the properties of the fuzzy approximation models on each space are studied, the relationships between fuzzy approximation models of two spaces are researched, and examples to show how the approximation operator models in the FAAS to solve practical problems from multiple perspectives are also illustrated. More importantly, when the fuzzy relation $\widetilde{R}$ is a binary relation $R$ or the two spaces are the same, the special cases of FAAS are investigated and some important new models and new results are obtained, which add new ideas and methods to the current research.


[1] E. A. Abo-Tabl, A comparison of two kinds of definitions of rough approximations based on a similarity relation, Information Sciences, 181 (2011), 2587-2596.
[2] F. Angiulli, C. Pizzuti, Outlier mining in large high-dimensional data sets, IEEE Transactions on Knowledge and Data Engineering, 17(2) (2005), 203-215.
[3] Z. Bonikowski, E. Bryniarski, U. Wybraniec-Skardowska, Extensions and intentions in the rough set theory, Information Sciences, 107 (1998), 149-167.
[4] D. N. Chalishajar, R. Rajappa, Fuzzy solutions to second order three point boundary value problem, Applications and Applied Mathematics, 15(2) (2020), 916-927.
[5] D. N. Chalishajar, R. Ramesh, Controllability for impulsive fuzzy neutral functional integro-differential equations, AIP Conference Proceedings, 2159, 030007 (2019). DOI:10.1063/1.5127472.
[6] D. N. Chalishajar, R. Ramesh, Impulsive fuzzy solutions for abstract second order partial neutral functional differential equations, Journal of Applied and Pure Mathematics, 4(1-2) (2022), 71-77.
[7] D. N. Chalishajar, R. Ramesh, S. Vengataasalam, K. Karthikeyan, Existence of fuzzy solutions for nonlocal impulsive
neutral functional differential equations, Journal Nonlinear Analysis and Application, 2017(1) (2017), 1-12.
[8] O. Dalkili, Two novel approaches that reduce the effectiveness of the decision maker in decision making under
uncertainty environments, Iranian Journal of Fuzzy Systems, 19(2) (2022), 105-117.
[9] M. Diker, A. A. U˘gur, Fuzzy rough set models over two universes using textures, Fuzzy Sets and Systems, 442
(2022), 155-195.
[10] D. Dubois, H. Prade, Rough fuzzy sets and fuzzy rough sets, International Journal of General Systems, 17(2-3)
(1990), 191-209.
[11] Q. Gao, L. W. Ma, A novel notion in rough set theory: Invariant subspaces, Fuzzy Sets and Systems, 440 (2022),
[12] R. Jensen, Q. Shen, Semantics-preserving dimensionality reduction: Rough and fuzzy-rough-based approaches, IEEE
Transactions on Knowledge and Data Engineering, 16(12) (2004), 1457-1471.
[13] M. Kondo, On the structure of generalized rough sets, Information Sciences, 176(5) (2005), 589-600.
[14] D. G. Li, D. Q. Miao, Y. Q. Yin, Relation of relative reduct based on nested decision granularity, IEEE International
Conference on Granular Computing, IEEE, 2006.
[15] Z. W. Li, T. S. Xie, Q. G. Li, Topological structure of generalized rough sets, Computers and Mathematics with
Applications, 63 (2012), 1066-1071.
[16] G. Liu, W. Zhu, The algebraic structures of generalized rough set theory, Information Sciences, 178 (2008), 4105-
[17] L. W. Ma, Two fuzzy covering rough set models and their generalizations over fuzzy lattices, Fuzzy Sets and
Systems, 294 (2016), 1-17.
[18] L. W. Ma, The investigation of covering rough sets by Boolean matrices, International Journal of Approximate
Reasoning, 100 (2018), 69-84.
[19] L. W. Ma, Characteristic numbers and approximation operators in generalized rough approximation system, International Journal of Approximate Reasoning, 139 (2021), 166-184.
[20] B. Pang, F. G. Shi, Subcategories of the category of L-convex spaces, Fuzzy Sets and Systems, 313 (2017), 61-74.
[21] Z. Pawlak, Rough sets, International Journal of Computer and Information Sciences, 11 (1982), 341-356.
[22] J. Priyadharsini, P. Balasubramaniam, Solvability of fuzzy fractional stochastic Pantograph differential system,
Iranian Journal of Fuzzy Systems, 19(1) (2022), 47-60.
[23] W. Z. Wu, Y. Leung, J. S. Mi, On characterizations of (I, T)-fuzzy rough approximation operators, Fuzzy Sets and
Systems, 154(1) (2005), 76-102.
[24] J. Xing, C. Gao, J. Zhou, Weighted fuzzy rough sets-based tri-training and its application to medical diagnosis,
Applied Soft Computing, 124 (2022), 109025.
[25] D. Yang, M. Cai, Q. Li, F. Xu, Multigranulation fuzzy probabilistic rough set model on two universes, International
Journal of Approximate Reasoning, 145 (2022), 18-35.
[26] X. Yang, H. Chen, T. Li, C. Luo, A noise-aware fuzzy rough set approach for feature selection, Knowledge-Based
Systems, 250 (2022), 109092.
[27] X. P. Yang, T. J. Li, The minimization of axiom sets characterizing generalized approximation operators, Information Sciences, 176 (2006), 887-899.
[28] Y. Yao, A comparative study of fuzzy sets and rough sets, Information Sciences, 109 (1998), 227-242.
[29] Y. Yao, On generalizing Pawlak approximation operators, in: Lecture Notes in Artificial Intelligence, 1424 (1998),
[30] Z. Zeighami, M. R. Jahed-Motlagh, A. Moarefianpour, G. Heydari, A new stability criterion for high-order dynamic
fuzzy systems, Iranian Journal of Fuzzy Systems, 19(2) (2022), 187-203.
[31] S. Y. Zhang, F. G. Shi, Fuzzy betweenness spaces on continuous lattices, Iranian Journal of Fuzzy Systems, 19(3)
(2022), 39-52.
[32] K. Zhang, J. M. Zhan, W. Z. Wu, et al., Fuzzy β-covering based (φ, T )-fuzzy rough set models and applications to
multi-attribute decision-making, Computers and Industrial Engineering, 128 (2019), 605-621.
[33] H. M. Zhao, L. W. Ma, Several rough set models in quotient spaces, CAAI Transactions on Intelligence Technology,
7(1) (2022), 69-80.