Acta Mathematica Sinica, English Series

, Volume 27, Issue 1, pp 97–118 | Cite as

L-fuzzy roughness of n-ary polygroups

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Abstract

In this paper, we consider the relations among L-fuzzy sets, rough sets and n-ary polygroup theory. Some properties of (normal) TL-fuzzy n-ary subpolygroups of an n-ary polygroup are first obtained. Using the concept of L-fuzzy sets, the notion of ϑ-lower and T-upper L-fuzzy rough approximation operators with respect to an L-fuzzy set is introduced and some related properties are presented. Then a new algebraic structure called (normal) TL-fuzzy rough n-ary polygroup is defined and investigated. Also, the (strong) homomorphism of ϑ-lower and T-upper L-fuzzy rough approximation operators is studied.

Keywords

n-Ary polygroups (normal) TL-fuzzy n-ary polygroups ϑ-lower L-fuzzy rough approximation T-upper L-fuzzy rough approximation (strong) homomorphism 

MR(2000) Subject Classification

20N20 20N15 20N25 
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References

  1. [1]
    Marty, F.: Sur une generalization de la notion de groupe. In: 8th Congress Math. Scandinaves, Stockholm, 1934, 45–49Google Scholar
  2. [2]
    Corsini, P.: Prolegomena of Hypergroup Theory, Aviani Editore, Italy, 1993MATHGoogle Scholar
  3. [3]
    Vougiouklis, T.: Hyperstructures and Their Representations, Hadronic Press Inc., Palm Harbor, USA, 1994MATHGoogle Scholar
  4. [4]
    Corsini, P., Leoreanu, V.: Applications of Hyperstructure Theory, Advances in Mathematics (Dordrecht), Kluwer Academic Publishers, Dordrecht, 2003Google Scholar
  5. [5]
    Zadeh, L. A.: Fuzzy sets. Inform. Control, 8, 338–358 (1965)MATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    Rosenfeld, A.: Fuzzy groups. J. Math. Anal. Appl., 35, 512–517 (1971)MATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    Mordeson, J. N., Malik, M. S.: Fuzzy Commutative Algebra, World Publishing, Singapore, 1998MATHGoogle Scholar
  8. [8]
    Corsini, P., Leoreanu, V.: Join spaces associated with fuzzy sets. J. Comb., Inf. Syst. Sci., 20, 293–303 (1995)MATHMathSciNetGoogle Scholar
  9. [9]
    Corsini, P., Leoreanu, V.: Fuzzy sets and join spaces associated with rough sets. Rend. Circ. Mat. Palermo, 51, 527–536 (2002)MATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    Cristea, I.: Hyperstructures and fuzzy sets endowed with two membership functions. Fuzzy Sets Syst., 160, 1114–1124 (2009)MATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    Davvaz, B.: Fuzzy Hv-submodules. Fuzzy Sets Syst., 117, 477–484 (2001)MATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    Kehagias, A., Serafimidis, K.: The L-fuzzy Nakano hypergroup. Inform. Sci., 169, 305–327 (2005)MATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    Leoreanu, V.: Fuzzy hypermodules. Comput. Math. Appl., 57, 466–475 (2009)MATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    Sen, M. K., Ameri, R., Chowdhury, G.: Fuzzy hypersemigroups. Soft Comput., 12, 891–900 (2008)MATHCrossRefGoogle Scholar
  15. [15]
    Stefanescu, M., Cristea, I.: On the fuzzy grade of hypergroups. Fuzzy Sets Syst., 159, 1097–1106 (2008)MATHCrossRefMathSciNetGoogle Scholar
  16. [16]
    Yin, Y., Zhan, J., Xu, D., Wang, J.: The L-fuzzy hypermodules. Comput. Math. Appl., 59, 953–963 (2010)MATHCrossRefMathSciNetGoogle Scholar
  17. [17]
    Zhan, J., Davvaz, B., Shum, K. P.: Isomorphism theorems of hypermodules. Acta Mathematica Sinica, Chinese Series, 50(4), 909–914 (2007)MATHMathSciNetGoogle Scholar
  18. [18]
    Zhan, J., Davvaz, B., Shum, K. P.: A new view of fuzzy hypermodules. Acta Mathematica Sinica, English Series, 23(8), 1345–1356 (2007)MATHCrossRefMathSciNetGoogle Scholar
  19. [19]
    Zhan, J., Dudek, W. A.: Interval valued intuitionistic (S, T)-fuzzy Hv-submodules. Acta Mathematica Sinica, English Series, 22, 963–970 (2006)MATHCrossRefMathSciNetGoogle Scholar
  20. [20]
    Davvaz, B., Vougiouklis, T.: n-Ary hypergroups. Iran. J. Sci. Technol. Trans. A, 30, 165–174 (2006)MathSciNetGoogle Scholar
  21. [21]
    Leoreanu, V., Davvaz, B.: Roughness in n-ary hypergroups. Inform. Sci., 178, 4114–4124 (2008)MATHCrossRefMathSciNetGoogle Scholar
  22. [22]
    Leoreanu, V., Davvaz, B.: n-hypergroups and binary relations. European J. Combin., 29(5), 1207–1218 (2008)MATHCrossRefMathSciNetGoogle Scholar
  23. [23]
    Davvaz, B., Corsini, P.: Fuzzy n-ary hypergroups. J. Intell. Fuzzy Systems, 18(4), 377–382 (2007)MATHGoogle Scholar
  24. [24]
    Davvaz, B., Corsini, P., Leoreanu, V.: Atanassov’s intuitionistic (S, T)-fuzzy n-ary subhypergroups and their properties. Inform. Sci., 179, 654–666 (2009)MATHCrossRefMathSciNetGoogle Scholar
  25. [25]
    Ghadiri, M., Waphare, B. N.: n-ary polygroups. Iran J. Sci. Technol. Trans. A, in pressGoogle Scholar
  26. [26]
    Davvaz, B., Corsini, P., Leoreanu, V.: Fuzzy n-ary subpolygroups. Comput. Math. Appl., 57, 141–152 (2009)MATHCrossRefMathSciNetGoogle Scholar
  27. [27]
    Davvaz, B., Leoreanu, V.: Applications of interval valued fuzzy n-ary polygroups with respect to t-norms (t-conorms). Comput. Math. Appl., 57, 1413–1424 (2009)MATHCrossRefMathSciNetGoogle Scholar
  28. [28]
    Pawlak, Z.: Rough sets. Int. J. Comput. Inform. Sci., 11, 341–356 (1982)MATHCrossRefMathSciNetGoogle Scholar
  29. [29]
    Biswas, R., Nanda, S.: Rough groups and rough subgroups. Bull. Polishe Acad. Sci. Math., 42, 251–254 (1994)MATHMathSciNetGoogle Scholar
  30. [30]
    Davvaz, B.: Roughness in rings. Inform. Sci., 164, 147–163 (2004)MATHCrossRefMathSciNetGoogle Scholar
  31. [31]
    Davvaz, B., Mahdavipour, M.: Roughness in modules. Inform. Sci., 176, 3658–3674 (2006)MATHCrossRefMathSciNetGoogle Scholar
  32. [32]
    Jun, Y. B.: Roughness of ideals in BCK-algebras. Sci. Math. Japon., 57(1), 165–169 (2003)Google Scholar
  33. [33]
    Kuroki, N.: Rough ideals in semigroups. Inform. Sci., 100, 139–163 (1997)MATHCrossRefMathSciNetGoogle Scholar
  34. [34]
    Kuroki, N., Mordeson, J. N.: Structure of rough sets and rough groups. J. Fuzzy Math., 5(1), 183–191 (1997)MATHMathSciNetGoogle Scholar
  35. [35]
    Kuroki, N., Wang, P. P.: The lower and upper approximations in a fuzzy group. Inform. Sci., 90, 203–220 (1996)MATHCrossRefMathSciNetGoogle Scholar
  36. [36]
    Leoreanu, V.: The lower and upper approximations in a hypergroup. Inform. Sci., 178, 3605–3615 (2008)MATHCrossRefMathSciNetGoogle Scholar
  37. [37]
    Davvaz, B.: Approximations in Hv-modules. Taiwanese J. Math., 6(4), 499–505 (2002)MATHMathSciNetGoogle Scholar
  38. [38]
    Davvaz, B.: A new view of approximations in Hv-groups. Soft Comput., 10(11), 1043–1046 (2006)MATHCrossRefGoogle Scholar
  39. [39]
    Dubois, D., Prade, H.: Editorial. Fuzzy Sets Syst., 122, 1–3 (2001)CrossRefMathSciNetGoogle Scholar
  40. [40]
    Jiang, J. S., Wu, C. X., Chen, D. G.: The product structure of fuzzy rough sets on a group and the rough T-fuzzy group. Inform. Sci., 175, 2–11 (2004)MathSciNetGoogle Scholar
  41. [41]
    Li, F., Yin, Y., Lu, L.: (ϑ, T)-fuzzy rough approximation operators and the TL-fuzzy rough ideals on a ring. Inform. Sci., 177, 4711–4726 (2007)MATHCrossRefMathSciNetGoogle Scholar
  42. [42]
    Kazancı, O., Davvaz, B.: On the structure of rough prime (primary) ideals and rough fuzzy prime (primary) ideals in commutative rings. Inform. Sci., 178, 1343–354 (2008)MATHCrossRefMathSciNetGoogle Scholar
  43. [43]
    Kazancı, O., Yamak, S., Davvaz, B.: The lower and upper approximations in a quotient hypermodule with respect to fuzzy sets. Inform. Sci., 178, 2349–2359 (2008)MATHMathSciNetGoogle Scholar
  44. [44]
    Morsi, N. N., Yakout, M. M.: Axiomatics for fuzzy rough sets. Fuzzy Sets Syst., 100, 327–342 (1998)MATHCrossRefMathSciNetGoogle Scholar
  45. [45]
    Goguen, J. A.: L-fuzzy sets. J. Math. Anal. Appl., 18, 145–174 (1967)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  1. 1.School of Mathematics and Information SciencesEast China Institute of TechnologyFuzhouP. R. China
  2. 2.Department of MathematicsHubei University for NationalitiesEnshiP. R. China
  3. 3.Department of Biology and Agro-Industrial EconomyUdineItaly

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