On the Relationship Between L-fuzzy Closure Spaces and L-fuzzy Rough Sets

  • Vijay K. Yadav
  • Swati Yadav
  • S. P. Tiwari
Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 834)


This work is towards the establishment of bijective correspondence between the family of all L-fuzzy reflexive/tolerance approximation spaces and the family of all quasi-discrete L-fuzzy closure spaces satisfying a certain condition.


L-fuzzy closure space L-fuzzy reflexive approximation space L-fuzzy tolerance approximation space 


  1. 1.
    Blount, K., Tsinakis, C.: The structure of residuated lattices. Int. J. Algebra Comput. 13, 437–461 (2003)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Boixader, D., Jacas, J., Recasens, J.: Upper and lower approximations of fuzzy sets. Int. J. Gen. Syst. 29, 555–568 (2000)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Dubois, D., Prade, H.: Rough fuzzy set and fuzzy rough set. Int. J. Gen. Syst. 17, 191–209 (1990)CrossRefGoogle Scholar
  4. 4.
    Gautam, V., Yadav, V.K., Singh, A.K., Tiwari, S.P.: On the topological structure of rough soft sets. In: RSKT 2014, LNAI, vol. 8818, pp. 39-48 (2014)Google Scholar
  5. 5.
    Goguen, J.A.: L-fuzzy sets. J. Math. Anal. Appl. 18, 145–174 (1967)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Hao, J., Li, Q.: The relationship between L-fuzzy rough set and L-topology. Fuzzy Sets Syst. 178, 74–83 (2011)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Kondo, M.: On the structure of generalized rough sets. Inf. Sci. 176, 586–600 (2006)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Lowen, R.: Fuzzy topological space and fuzzy compactness. J. Math. Anal. Appl. 56, 621–633 (1976)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Mashhour, A.S., Ghanim, M.H.: Fuzzy closure spaces. J. Math. Anal. Appl. 106, 154–170 (1985)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Ma, Z.M., Hu, B.Q.: Topological and lattice structures of L-fuzzy rough sets determined by lower and upper sets. Inf. Sci. 218, 194–204 (2013)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Pawlak, Z.: Rough sets. Int. J. Comput. Inf. Sci. 11, 341–356 (1982)CrossRefGoogle Scholar
  12. 12.
    Qin, K., Pei, Z.: On the topological properties of fuzzy rough sets. Fuzzy Sets Syst. 151, 601–613 (2005)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Qin, K., Yang, J., Pei, Z.: Generalized rough sets based on reflexive and transitive relations. Inf. Sci. 178, 4138–4141 (2008)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Ramadan, A.A., Elkordy, E.H., El-Dardery, M.: L-fuzzy approximation space and L-fuzzy topological spaces. Iran. J. Fuzzy Syst. 13(1), 115–129 (2016)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Radzikowska, A.M., Kerre, E.E.: Fuzzy rough sets based on residuated lattices. In: Peters, J.F., Skowron, A., Dubois, D., Grzymała-Busse, J.W., Inuiguchi, M., Polkowski, L. (eds.) Transactions on Rough Sets II. LNCS, vol. 3135, pp. 278–296. Springer, Heidelberg (2004). Scholar
  16. 16.
    Sharan, S., Tiwari, S.P., Yadav, V.K.: Interval type-2 fuzzy rough sets and interval type-2 fuzzy closure spaces. Iran. J. Fuzzy Syst. 12, 127–135 (2015)MathSciNetzbMATHGoogle Scholar
  17. 17.
    She, Y.H., Wang, G.J.: An axiomatic approach of fuzzy rough sets based on residuated lattices. Comput. Math. Appl. 58, 189–201 (2009)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Srivastava, R., Srivastava, M.: On \(T_0\)- and \(T_1\)-fuzzy closure spaces. Fuzzy sets Syst. 109, 263–269 (2000)CrossRefGoogle Scholar
  19. 19.
    Tiwari, S.P., Sharan, S., Yadav, V.K.: Fuzzy closure spaces vs. fuzzy rough sets. Fuzzy Inf. Eng. 6, 93–100 (2014)MathSciNetCrossRefGoogle Scholar
  20. 20.
    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). Scholar
  21. 21.
    Yao, Y.Y.: Constructive and algebraic methods of the theory of rough sets. Inf. Sci. 109, 21–47 (1998)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Zhu, W.: Generalized rough sets based on relations. Inf. Sci. 177(22), 4997–5011 (2007)MathSciNetCrossRefGoogle Scholar

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© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.Department of Mathematics, School of Mathematics, Statistics and Computational SciencesCentral University of RajasthanAjmerIndia
  2. 2.Department of Applied MathematicsIndian Institute of Technology (Indian School of Mines)DhanbadIndia

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