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On the Intuitionistic Fuzzy Topological Structures of Rough Intuitionistic Fuzzy Sets

  • You-Hong Xu
  • Wei-Zhi WuEmail author
  • Guoyin Wang
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8449)

Abstract

A rough intuitionistic fuzzy set is the result of approximation of an intuitionistic fuzzy set with respect to a crisp approximation space. In this paper, we investigate topological structures of rough intuitionistic fuzzy sets. We first show that a reflexive crisp rough approximation space can induce an intuitionistic fuzzy Alexandrov space. It is proved that the lower and upper rough intuitionistic fuzzy approximation operators are, respectively, an intuitionistic fuzzy interior operator and an intuitionistic fuzzy closure operator if and only if the binary relation in the crisp approximation space is reflexive and transitive. We then verify that a similarity crisp approximation space can produce an intuitionistic fuzzy clopen topological space. We further examine sufficient and necessary conditions that an intuitionistic fuzzy interior (closure, respectively) operator derived from an intuitionistic fuzzy topological space can associate with a reflexive and transitive crisp relation such that the induced lower (upper, respectively) rough intuitionistic fuzzy approximation operator is exactly the intuitionistic fuzzy interior (closure, respectively) operator.

Keywords

Approximation operators Binary relations Intuitionistic fuzzy sets Intuitionistic fuzzy topologies Rough intuitionistic fuzzy sets Rough sets 

Notes

Acknowledgments

This work was supported by grants from the National Natural Science Foundation of China (Nos. 61272021, 61075120, 11071284, 61202206, and 61173181), the Zhejiang Provincial Natural Science Foundation of China (No. LZ12F03002), and Chongqing Key Laboratory of Computational Intelligence (No. CQ-LCI-2013-01).

References

  1. 1.
    Albizuri, F.X., Danjou, A., Grana, M., Torrealdea, J., Hernandezet, M.C.: The high-order Boltzmann machine: learned distribution and topology. IEEE Trans. Neural Netw. 6, 767–770 (1995)CrossRefGoogle Scholar
  2. 2.
    Arenas, F.G.: Alexandroff Spaces. Acta Math. Univ. Comenianae. 68, 17–25 (1999)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Atanassov, K.: Intuitionistic Fuzzy Sets. Physica-Verlag, Heidelberg (1999)CrossRefGoogle Scholar
  4. 4.
    Boixader, D., Jacas, J., Recasens, J.: Upper and lower approximations of fuzzy sets. Int. J. Gen. Syst. 29, 555–568 (2000)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Chakrabarty, K., Gedeon, T., Koczy, L.: Intuitionistic fuzzy rough set. In: Proceedings of 4th Joint Conference on Information Sciences (JCIS), Durham, NC, pp. 211–214 (1998)Google Scholar
  6. 6.
    Chang, C.L.: Fuzzy topological spaces. J. Math. Anal. Appl. 24, 182–189 (1968)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Choudhury, M.A., Zaman, S.I.: Learning sets and topologies. Kybernetes. 35, 1567–1578 (2006)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Chuchro, M.: On rough sets in topological boolean algebras. In: Ziarko, W. (ed.) Rough Sets, Fuzzy Sets and Knowledge Discovery, pp. 157–160. Springer, Berlin (1994)CrossRefGoogle Scholar
  9. 9.
    Chuchro, M.: A certain conception of rough sets in topological boolean algebras. Bull. Sect. Logic. 22, 9–12 (1993)zbMATHGoogle Scholar
  10. 10.
    Coker, D.: An introduction of intuitionistic fuzzy topological spaces. Fuzzy Sets Syst. 88, 81–89 (1997)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Coker, D.: Fuzzy rough sets are intuitionistic \(L\)-fuzzy sets. Fuzzy Sets Syst. 96, 381–383 (1998)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Cornelis, C., Cock, M.D., Kerre, E.E.: Intuitionistic fuzzy rough sets: at the crossroads of imperfect knowledge. Expert Syst. 20, 260–270 (2003)CrossRefGoogle Scholar
  13. 13.
    Cornelis, C., Deschrijver, G., Kerre, E.E.: Implication in Intuitionistic fuzzy and interval-valued fuzzy set theory: construction, classification application. Int. J. Approximate Reasoning 35, 55–95 (2004)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Dubois, D., Prade, H.: Rough fuzzy sets and fuzzy rough sets. Int. J. Gen. Syst. 17, 191–209 (1990)CrossRefGoogle Scholar
  15. 15.
    Hao, J., Li, Q.G.: The relationship between L-Fuzzy rough set and L-Topology. Fuzzy Sets Syst. 178, 74–83 (2011)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Jena, S.P., Ghosh, S.K.: Intuitionistic fuzzy rough sets. Notes Intuitionistic Fuzzy Sets 8, 1–18 (2002)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Kall, L., Krogh, A., Sonnhammer, E.L.L.: A combined transmembrane topology and signal peptide prediction method. J. Mol. Biol. 338, 1027–1036 (2004)CrossRefGoogle Scholar
  18. 18.
    Kondo, M.: On the structure of generalized rough sets. Inf. Sci. 176, 589–600 (2006)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Kortelainen, J.: On relationship between modified sets, topological space and rough sets. Fuzzy Sets Syst. 61, 91–95 (1994)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Kortelainen, J.: On the evaluation of compatibility with gradual rules in information systems: a topological approach. Control Cybern. 28, 121–131 (1999)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Kortelainen, J.: Applying modifiers to knowledge acquisition. Inf. Sci. 134, 39–51 (2001)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Lai, H.L., Zhang, D.X.: Fuzzy preorder and fuzzy topology. Fuzzy Sets Syst. 157, 1865–1885 (2006)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Lashin, E.F., Kozae, A.M., Khadra, A.A.A., Medhat, T.: Rough set theory for topological spaces. Int. J. Approximate Reasoning 40, 35–43 (2005)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Li, J.J., Zhang, Y.L.: Reduction of subbases and its applications. Utilitas Math. 82, 179–192 (2010)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Li, Y.L., Li, Z.L., Chen, Y.Q., Li, X.X.: Using raster quasi-topology as a tool to study spatial entities. Kybernetes 32, 1425–1449 (2003)CrossRefGoogle Scholar
  26. 26.
    Li, Z.W., Xie, T.S., Li, Q.G.: Topological structure of generalized rough sets. Comput. Math. Appl. 63, 1066–1071 (2012)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Lowen, R.: Fuzzy topological spaces and fuzzy compactness. J. Math. Anal. Appl. 56, 621–633 (1976)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Mi, J.-S., Leung, Y., Zhao, H.-Y., Feng, T.: Generalized fuzzy rough sets determined by a triangular norm. Inform. Sci. 178, 3203–3213 (2008)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Mi, J.-S., Zhang, W.-X.: An axiomatic characterization of a fuzzy generalization of rough sets. Inf. Sci. 160, 235–249 (2004)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Morsi, N.N., Yakout, M.M.: Axiomatics for fuzzy rough sets. Fuzzy Sets Syst. 100, 327–342 (1998)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Pawlak, Z.: Rough Sets: Theoretical Aspects of Reasoning about Data. Kluwer Academic Publishers, Boston (1991)CrossRefGoogle Scholar
  32. 32.
    Pei, Z., Pei, D.W., Zheng, L.: Topology vs generalized rough sets. Int. J. Approximate Reasoning 52, 231–239 (2011)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Qin, K.Y., Pei, Z.: On the topological properties of fuzzy rough sets. Fuzzy Sets Syst. 151, 601–613 (2005)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Qin, K.Y., Yang, J., Pei, Z.: Generalized rough sets based on reflexive and transitive relations. Inf. Sci. 178, 4138–4141 (2008)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Radzikowska, A.M., Kerre, E.E.: A comparative study of fuzzy rough sets. Fuzzy Sets Syst. 126, 137–155 (2002)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Rizvi, S., Naqvi, H.J., Nadeem, D.: Rough intuitionistic fuzzy set. In: Proceedings of the 6th Joint Conference on Information Sciences (JCIS), Durham, NC, pp. 101–104 (2002)Google Scholar
  37. 37.
    Samanta, S.K., Mondal, T.K.: Intuitionistic fuzzy rough sets and rough intuitionistic fuzzy sets. J. Fuzzy Math. 9, 561–582 (2001)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Thiele, H.: On axiomatic characterisation of fuzzy approximation operators II, the rough fuzzy set based case. In: Proceedings of the 31st IEEE International Symposium on Multiple-Valued Logic, pp. 330–335 (2001)Google Scholar
  39. 39.
    Tiwari, S.P., Srivastava, A.K.: Fuzzy rough sets, fuzzy preorders and fuzzy topologies. Fuzzy Sets Syst. 210, 63–68 (2013)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Wiweger, R.: On topological rough sets. Bull. Pol. Acad. Sci. Math. 37, 89–93 (1989)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Wu, Q.E., Wang, T., Huang, Y.X., Li, J.S.: Topology theory on rough sets. IEEE Trans. Syst. Man Cybern. Part B-Cybern 38, 68–77 (2008)CrossRefGoogle Scholar
  42. 42.
    Wu, W.-Z.: A study on relationship between fuzzy rough approximation operators and fuzzy topological spaces. In: Wang, L., Jin, Y. (eds.) FSKD 2005. LNCS (LNAI), vol. 3613, pp. 167–174. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  43. 43.
    Wu, W.-Z.: On some mathematical structures of T-fuzzy rough set algebras in infinite universes of discourse. Fundam. Informatica 108, 337–369 (2011)MathSciNetzbMATHGoogle Scholar
  44. 44.
    Wu, W.-Z., Leung, Y., Mi, J.-S.: On characterizations of (\({\cal I, T}\))-fuzzy rough approximation operators. Fuzzy Sets Syst. 154, 76–102 (2005)MathSciNetCrossRefGoogle Scholar
  45. 45.
    Wu, W.-Z., Leung, Y., Shao, M.-W.: Generalized fuzzy rough approximation operators determined by fuzzy implicators. Int. J. Approximate Reasoning 54, 1388–1409 (2013)MathSciNetCrossRefGoogle Scholar
  46. 46.
    Wu, W.-Z., Leung, Y., Zhang, W.-X.: On generalized rough fuzzy approximation operators. In: Peters, J.F., Skowron, A. (eds.) Transactions on Rough Sets V. LNCS, vol. 4100, pp. 263–284. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  47. 47.
    Wu, W.-Z., Xu, Y.-H.: Rough approximations of intuitionistic fuzzy sets in crisp approximation spaces. In: Proceedings of Seventh International Conference on Fuzzy Systems and Knowledge Discovery (FSKD 2010), vol. 1, pp. 309–313 (2010)Google Scholar
  48. 48.
    Wu, W.-Z., Xu, Y.-H.: On Fuzzy Topological Structures of Rough Fuzzy Sets. In: Peters, J.F., Skowron, A., Ramanna, S., Suraj, Z., Wang, X. (eds.) Transactions on Rough Sets XVI. LNCS, vol. 7736, pp. 125–143. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  49. 49.
    Wu, W.-Z., Zhang, W.-X.: Constructive and axiomatic approaches of fuzzy approximation operators. Inf. Sci. 159, 233–254 (2004)MathSciNetCrossRefGoogle Scholar
  50. 50.
    Wu, W.-Z., Zhou, L.: On intuitionistic fuzzy topologies based on intuitionistic fuzzy reflexive and transitive relations. Soft Comput. 15, 1183–1194 (2011)CrossRefGoogle Scholar
  51. 51.
    Yang, L.Y., Xu, L.S.: Topological properties of generalized approximation spaces. Inf. Sci. 181, 3570–3580 (2011)MathSciNetCrossRefGoogle Scholar
  52. 52.
    Yao, Y.Y.: Constructive and algebraic methods of the theory of rough sets. J. Inf. Sci. 109, 21–47 (1998)MathSciNetCrossRefGoogle Scholar
  53. 53.
    Zhang, H.-P., Yao, O.Y., Wang, Z.D.: Note on “Generlaized Rough Sets Based on Reflexive and Transitive Relations”. Inf. Sci. 179, 471–473 (2009)CrossRefGoogle Scholar
  54. 54.
    Zhou, L., Wu, W.-Z.: On generalized intuitionistic fuzzy approximation operators. Inf. Sci. 178, 2448–2465 (2008)MathSciNetzbMATHGoogle Scholar
  55. 55.
    Zhou, L., Wu, W.-Z., Zhang, W.-X.: On intuitionistic fuzzy rough sets and their topological structures. Int. J. Gen. Syst. 38, 589–616 (2009)MathSciNetCrossRefGoogle Scholar
  56. 56.
    Zhou, L., Wu, W.-Z., Zhang, W.-X.: On characterization of intuitionistic fuzzy rough sets based on intuitionistic fuzzy implicators. Inf. Sci. 179, 883–898 (2009)MathSciNetCrossRefGoogle Scholar
  57. 57.
    Zhu, W.: Topological approaches to covering rough sets. Inf. Sci. 177, 1499–1508 (2007)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.School of Mathematics, Physics and Information ScienceZhejiang Ocean UniversityZhoushanPeople’s Republic of China
  2. 2.Chongqing Key Laboratory of Computational IntelligenceChongqing University of Posts and TelecommunicationsChongqingPeople’s Republic of China

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