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A survey on topological structures on fuzzy rough sets

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Abstract

Fuzzy rough set theory gives a mathematical tool for studying unsettled knowledge that is beclouded, inexact, and mutually exclusive. The perception and conclusions of fuzzy rough sets theory are inextricably linked to topological perception. The topological appearance and its applications in fuzzy rough sets theory have been extensively discussed by researchers. The underlying subordinate of topology and classic fuzzy rough sets theory, as well as the expressive work done in this area over the previous years, are highlighted in this research.

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References

  1. Acharjya, D.P., Tripathy, B.K.: Intuitionistic fuzzy rough set on two universal sets and knowledge representation. Math. Sci. Int. Res. J. 1(2), 584–598 (2012)

    Google Scholar 

  2. Atanassov, K.T.: Intuitionistic fuzzy sets. Fuzzy Sets Syst. 20, 87–96 (1986)

    Article  Google Scholar 

  3. Bodenhofer, U.: A unified framework of opening and closure operators with respect to arbitrary fuzzy relations. Soft Comput. 7, 220–227 (2003)

    Article  Google Scholar 

  4. Bodenhofer, U., De Cock, M., Kerre, E.E.: Openings and closures of fuzzy preorderings, Theoretical basics and applications to fuzzy rule-based system. Int. J. Gen. Syst. 32(4), 343–360 (2003)

    Article  MathSciNet  Google Scholar 

  5. Chang, C.L.: Fuzzy topological spaces. J. Math. Anal. Appl. 24, 182–190 (1968)

    Article  MathSciNet  Google Scholar 

  6. Chen, N., Xu, Z.S., Xia, M.M.: Correlation coefficients of hesitant fuzzy sets and their applications to clustering analysis. Appl. Math. Model. 37, 2197–2211 (2013)

    Article  MathSciNet  Google Scholar 

  7. Cock, M.D., Cornelis, C., Kerre, E.E.: Fuzzy rough sets: the forgotten step. IEEE Trans. Fuzzy Syst. 15(1), 121–130 (2007)

    Article  Google Scholar 

  8. Coker, D., Demirci, M.: An introduction to intuitionistic fuzzy topological spaces in Sostak’s sense. Busefal 67, 67–76 (1996)

    Google Scholar 

  9. Coker, D.: An introduction of intuitionistic fuzzy topological spaces. Fuzzy Sets Syst. 88, 81–89 (1997)

    Article  MathSciNet  Google Scholar 

  10. Coker, D.: Fuzzy rough sets are intuitionistic L-fuzzy sets. Fuzzy Sets Syst. 96(3), 381–383 (1998)

    Article  MathSciNet  Google Scholar 

  11. Deepak, D., John, S.J.: Hesitant fuzzy rough sets through hesitant fuzzy relations. Ann. Fuzzy Math. Inform. 8, 33–46 (2014)

    MathSciNet  Google Scholar 

  12. Dubois, D., Prade, H.: Rough fuzzy set and fuzzy rough sets. Int. J. Gen. Syst. 17, 191–209 (1990)

    Article  Google Scholar 

  13. Farhadinia, B.: Information measures for hesitant fuzzy sets and interval-valued hesitant fuzzy sets. Inform. Sci. 240, 129–144 (2013)

    Article  MathSciNet  Google Scholar 

  14. Figueira, J., Greco, S., Ehrogott, M.: Multiple Criteria Decision Analysis: State of the Art Surveys. Springer, Berlin (2005)

    Book  Google Scholar 

  15. Garcia, J.G., Rodabaugh, S.E.: Order-theoretic, topological, categorical redundancides of intervalvalued sets, grey sets, vague sets, interval-valued intuitionistic sets, intuitionistic fuzzy sets and topologies. Fuzzy Sets Syst. 156, 445–484 (2005)

    Article  Google Scholar 

  16. Ghanim, M.H.: Pseudo-closure operators in fuzzy topological spaces. Fuzzy Sets Syst. 39, 339–346 (1991)

    Article  MathSciNet  Google Scholar 

  17. Goguen, J.A.: L-Fuzzy sets. J. Math. Anal. Appl. 18, 145–174 (1967)

    Article  MathSciNet  Google Scholar 

  18. Hao, J., Li, Q.: The relationship between L-fuzzy rough set and L-topology. Fuzzy Sets Syst. 178, 74–83 (2011)

    Article  MathSciNet  Google Scholar 

  19. Isbell, J.R.: Uniform spaces. American Mathematical Society, Providence (1964)

  20. Kelley, J.L.: General Topology. Van Nostrand Company, New York (1995)

    Google Scholar 

  21. Khare, M., Tiwari, S.: L-approach merotopies and their categorical perspective. Demonstr. Math. 45(3), 699–716 (2012)

    MathSciNet  Google Scholar 

  22. Khare, M., Tiwari, S.: Completion in a common supercategory of Met, UAP, wsAP and near. Demonstr. Math. 46(1), 209–27 (2013)

    MathSciNet  Google Scholar 

  23. Kortelainen, J.: On relationships between modified sets, topological spaces and rough sets. Fuzzy Sets Syst. 61, 91–95 (1994)

    Article  MathSciNet  Google Scholar 

  24. Kumar, V., Tiwari, S.: Čech L-fuzzy rough proximity spaces. New Math. Natl. Comput. 1–15 (2023). https://doi.org/10.1142/S1793005724500315

  25. Lai, H., Zhang, D.: Fuzzy preorder and fuzzy topology. Fuzzy Sets Syst. 157, 1865–1885 (2006)

    Article  MathSciNet  Google Scholar 

  26. ABD El-Latif, A.A., Ramadan, A.A.: On L-double fuzzy rough sets. Iran. J. Fuzzy Syst. 13(3), 125–142 (2016)

    MathSciNet  Google Scholar 

  27. Liang, D.C., Liu, D.: A novel risk decision-making based on decision-theoretic rough sets under hesitant fuzzy inform. IEEE Trans. Fuzzy Syst. 23, 237–247 (2015)

    Article  Google Scholar 

  28. Liu, G.: Generalized rough sets over fuzzy lattices. Inform. Sci. 178, 1651–1662 (2008)

    Article  MathSciNet  Google Scholar 

  29. Lowen, R.: Fuzzy topological spaces and fuzzy compactness. J. Math. Anal. Appl. 56, 621–633 (1976)

    Article  MathSciNet  Google Scholar 

  30. Ma, Z.M., Bao Qing, Hu.: Topological and lattice structures of L-fuzzy rough sets determined by lower and upper sets. Inform. Sci. 218, 194–204 (2013)

    Article  MathSciNet  Google Scholar 

  31. Morsi, N.N., Yakout, M.M.: Axiomatics for fuzzy rough sets. Fuzzy Sets Syst. 100(1–3), 327–342 (1998)

    Article  MathSciNet  Google Scholar 

  32. Naessens, H., De Meyer, H., De Baets, B.: Algorithms for the computation of T-transitive closures. IEEE Trans. Fuzzy Syst. 10(4), 541–551 (2002)

    Article  Google Scholar 

  33. Nanda, S.: Majumdar: Fuzzy rough sets. Fuzzy Set. Syst. 45(2), 157–160 (1992)

    Article  MathSciNet  Google Scholar 

  34. Panga, B., Mi, J.-S., Xiuc, Z.-Y.: L-Fuzzifying approximation operators in fuzzy rough sets. Inform. Sci. 480, 14–33 (2019)

    Article  MathSciNet  Google Scholar 

  35. Pawlak, Z.: Rough sets. Int. J. Inform. Comput. Sci. 11(5), 341–356 (1982)

    Article  Google Scholar 

  36. Pawlak, Z.: Rough Sets, Theoretical Aspects of Reasoning About Data. Kluwer Academic Publishers, Boston (1991)

    Google Scholar 

  37. Pawlak, Z., Skowron, A.: Rudiments of rough sets. Inform. Sci. 177(1), 3–27 (2007)

    Article  MathSciNet  Google Scholar 

  38. Pawlak, Z., Skowron, A.: Rough sets: some extensions. Inform. Sci. 177(1), 28–40 (2007)

    Article  MathSciNet  Google Scholar 

  39. Pawlak, Z., Skowron, A.: Rough sets and Boolean reasoning. Inform. Sci. 177(1), 41–73 (2007)

    Article  MathSciNet  Google Scholar 

  40. Pei, D.: A gereralized model of fuzzy rough sets. Int. J. Gen. Syst. 34(5), 603–613 (2005)

    Article  MathSciNet  Google Scholar 

  41. Peters, J.F.: Near sets. Special theory about nearness of objects. Fund. Inform. 75(3–4), 407–433 (2007)

    MathSciNet  Google Scholar 

  42. Peters, J.F., Skowron, A., Stepaniuk, J.: Nearness of objects: extension of approximation space model. Fund. Inform. 79(3–4), 497–512 (2007)

    MathSciNet  Google Scholar 

  43. Peters, J.F., Skowron, A., Stepaniuk, J.: Nearness of visual objects. Application of rough sets in proximity spaces. Fund. Inform. 128, 159–176 (2013)

    MathSciNet  Google Scholar 

  44. Qiao, J., Hu, B.Q.: A short note on L-fuzzy approximation spaces and L-fuzzy pretopological spaces. Fuzzy Sets Syst. 321, 126–134 (2017)

    Article  MathSciNet  Google Scholar 

  45. Qin, K., Pei, Z.: On the topological properties of fuzzy rough sets. Fuzzy Sets Syst. 151(3), 601–613 (2005)

    Article  MathSciNet  Google Scholar 

  46. Radzikowska, A.M., Kerre, E.E.: A comparative study of fuzzy rough sets. Fuzzy Sets Syst. 126, 137–156 (2002)

    Article  MathSciNet  Google Scholar 

  47. Radzikowska, A.M., Kerre, E.E.: Fuzzy rough sets based on residuated lattices. Trans. Rough Sets Lect. Notes Comput. Sci. 3135, 278–296 (2004)

    Article  Google Scholar 

  48. Samanta, S.K., Mondal, T.K.: Intuitionistic fuzzy rough sets and rough intuitionistic fuzzy sets. J. Fuzzy Math. 9(3), 561–582 (2001)

    MathSciNet  Google Scholar 

  49. Samanta, S.K., Mondal, T.K.: On intuitionistic gradation of openness. Fuzzy Sets Syst. 131, 323–336 (2002)

    Article  MathSciNet  Google Scholar 

  50. Sebastian, S., Ramakrishnan, T.V.: Multi-fuzzy sets: an extension of fuzzy sets. Fuzzy Inform. Eng. 3(1), 35–43 (2011)

    Article  MathSciNet  Google Scholar 

  51. Sebastian, S., Ramakrishnan, T.V.: Multi-fuzzy topology. Int. J. App. Math. 24(1), 117–129 (2011)

    MathSciNet  Google Scholar 

  52. Singh, P.K., Tiwari, S.: A fixed point theorem in rough semi-linear uniform spaces. Theor. Comput. Sci. 851, 111–120 (2021)

    Article  MathSciNet  Google Scholar 

  53. Singh, P.K., Tiwari, S.: Topological structures in rough set theory: a survey. Hacet. J. Math. Stat. 49(4), 1270–1294 (2020)

    Article  MathSciNet  Google Scholar 

  54. Tang, W., Wu, J., Zheng, D.: On fuzzy rough sets and their topological structures. Math. Prob. Eng. 2014, 546372 (2014). https://doi.org/10.1155/2014/546372

  55. Tiwari, S.: Ultrafilter completeness in \(\varepsilon \)-approach nearness spaces. Math. Comput. Sci. 7, 107–111 (2013)

    Article  MathSciNet  Google Scholar 

  56. Tiwari, S.P., Srivastwa, A.K.: Fuzzy rough sets, fuzzy preorders and fuzzy topologies. Fuzzy Sets Syst. 210, 63–68 (2013)

    Article  MathSciNet  Google Scholar 

  57. Tiwari, S., Singh, P.K.: An approach of proximity in rough set theory. Fund. Inform. 166(3), 251–271 (2019)

    MathSciNet  Google Scholar 

  58. Tiwari, S., Singh, P.K.: Rough semi-uniform spaces and its image proximities. Electron. Res. Arch. 28(2), 1095–1106 (2020)

    Article  MathSciNet  Google Scholar 

  59. Tiwari, S., Singh, P.K.: Čech rough proximity spaces. Mat. Vesn. 72(1), 6–16 (2020)

    Google Scholar 

  60. Tiwari, S., Singh, P.K.: Smirnov type compactification of rough pseudo metric spaces using proximity approach. Afr. Mat. 32, 1–11 (2021)

  61. Torra, V., Narukawa, Y.: On hesitant fuzzy sets and decision. In: The 18th IEEE International Conference on Fuzzy Systems, Jeju Island, Korea, pp. 1378–1382 (2009)

  62. Torra, V.: Hesitant fuzzy sets. Int. J. Intell. Syst. 25, 529–539 (2010)

    Google Scholar 

  63. Varma, G., John, S.J.: Rough approximations of multi-fuzzy sets, International granular computing. Rough Sets Intell. Syst. 3(4), 327–344 (2014)

    Google Scholar 

  64. Wang, C.Y.: Topological structures of L-fuzzy rough sets and similarit sets of L-fuzzy relations. Int. J. Approx. Reason. 83, 160–175 (2017)

    Article  Google Scholar 

  65. Wang, C.Y.: Topological characterizations of generalized fuzzy rough sets. Fuzzy Sets Syst. 312, 109–125 (2017)

    Article  MathSciNet  Google Scholar 

  66. Wu, W.-Z., Mi, J.S., Zhang, W.-X.: Generalized fuzzy rough sets. Inform. Sci. 151, 263–282 (2003)

    Article  MathSciNet  Google Scholar 

  67. Wu, W.-Z., Zhang, W.-X.: Constructive and axiomatic approaches of fuzzy approximation operators. Inform. Sci. 159, 233–254 (2004)

    Article  MathSciNet  Google Scholar 

  68. Yang, X., Yang, Y.: Independence of axiom sets on intuitionistic fuzzy rough approximation operators. Int. J. Mach. Learn. Cybern. 4(5), 505–513 (2013)

    Article  MathSciNet  Google Scholar 

  69. Yang, X.B., Song, X.N., Qi, Y.S., et al.: Constructive and axiomatic approaches to hesitant fuzzy rough set. Soft Comput. 18, 1067–1077 (2014)

    Article  Google Scholar 

  70. Yao, Y.: Two views of theory of rough sets in finite universes. Int. J. Approx. Reason. 15, 291–317 (1996)

    Article  MathSciNet  Google Scholar 

  71. Yeung, D.S., Chen, D., Tsang, E.C.C., Lee, J.W.T., Wang, X.-Z.: On the generalization of fuzzy rough sets. IEEE Trans. Fuzzy Syst. 13, 343–361 (2005)

    Article  Google Scholar 

  72. Ying-Ming, L., Mao-Kong, L.: Fuzzy Topology. World Scientific Publishing, Singapore (1998)

    Book  Google Scholar 

  73. Yun, S.M., Lee, S.J.: Intuitionistic fuzzy rough approximation operators. Int. J. Fuzzy Log. Intell. Syst. 15(3), 208–215 (2015)

    Article  Google Scholar 

  74. Yun, S.M., Lee, S.J.L: Ituitionistic fuzzy approximation spaces induced by intuitionistic fuzzy topologies. In: Proceedings of 2016 Joint 8th International Conference on Soft Computing and Intelligent Systems (SCIS) and 17th International Symposium on Advanced Intelligent Systems (ISIS), Sapporo, Japan, pp. 774–777 (2016)

  75. Yun, S.M., Lee, S.J.: Intuitionistic fuzzy topologies induced by intuitionistic fuzzy approximation spaces. Int. J. Fuzzy Syst. 19(2), 285–291 (2017)

    Article  MathSciNet  Google Scholar 

  76. Zadeh, L.A.: Fuzzy sets. Inform. Control 8, 338–352 (1965)

    Article  Google Scholar 

  77. Zadeh, L.A.: Similarity relations and fuzzy orderings. Inform. Sci. 3, 177–200 (1971)

    Article  MathSciNet  Google Scholar 

  78. Zhao, H., Zhang, H.-Y.: On hesitant neutrosophic rough set over two universes and its application. Artif. Intell. Rev. 53, 4387–4406 (2020)

    Article  Google Scholar 

  79. Zhou, L., Wu, W.-Z.: On generalized intuitionistic fuzzy rough approximation operators. Inform. Sci. 178, 2448–2465 (2008)

    MathSciNet  Google Scholar 

  80. Zhou, L., Wu, W.-Z., Zhang, W.-X.: On intuitionistic fuzzy rough sets and their topological structures. Int. J. Gen. Syst. 38(6), 589–616 (2009)

    Article  MathSciNet  Google Scholar 

  81. Zhou, L., Wu, W.-Z., Zhang, W.-X.: On characterization of intuitionistic fuzzy rough sets based on intuitionistic fuzzy implicators. Inform. Sci. 179(7), 883–898 (2009)

    Article  MathSciNet  Google Scholar 

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Kumar, V., Tiwari, S. A survey on topological structures on fuzzy rough sets. Afr. Mat. 35, 42 (2024). https://doi.org/10.1007/s13370-024-01181-w

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