On k-Regularities in Fuzzy Semihyperrings

Original Paper

Abstract

We introduce the notion of interval-valued (i.v.) fuzzy k-quasi ideals and i.v. fuzzy k-bi-ideals of a semihyperring and deduce certain characterizations for k-regularities in semihyperring by using these i.v. fuzzy k-hyperideals. Finally, our goal is to establish a relation between different types of k-hyperideals of a semihyperring and i.v. fuzzy k-hyperideals of the associated i.v. fuzzy semihyperring and also to find out the relation between k-regularity, k-intra-regularity of semihyperring and the associated i.v. fuzzy semihyperring.

Keywords

Semihyperring k-Regular semihyperring i.v. fuzzy semihyperring k-Quasi-hyperideal k-Bi-hyperideal i.v. fuzzy k-quasi-hyperideal i.v. fuzzy k-bi-hyperideal 

Mathematics Subject Classification

08A72 

Notes

Acknowledgments

The authors are extremely grateful to the anonymous referee(s) for their constructive comments to improve the presentation of this paper.

References

  1. 1.
    Ameri, R., Motameni, M.: Fuzzy hyperideals of fuzzy hyperrings. World Appl. Sci. J. 16(11), 1604–1614 (2012)MATHGoogle Scholar
  2. 2.
    Ameri, R., Hedayati, H.: On \(k\)-hyperideals of hypersemirings. J. Discrete Math. Sci. Cryptogr. 10(1), 41–54 (2007)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Corsini, P.: Join spaces, power sets, fuzzy sets. In: Proceedings of Fifth International Congress of Algebraic Hyperstructures and Applications (1993), Iasi, Romania. Hadronic Press, Palm Harbor, pp. 45–52 (1994)Google Scholar
  4. 4.
    Corsini, P.: Fuzzy sets, join spaces and factor spaces. Pure Math. Appl. 11(3), 439–446 (2000)MathSciNetMATHGoogle Scholar
  5. 5.
    Corsini, P., Leoreanu, V.: Applications of Hyperstructure Theory; Advances in Mathematics, vol. 5. Kluwer, Dordrecht (2003)CrossRefMATHGoogle Scholar
  6. 6.
    Corsini, P., Leoreanu, V.: Fuzzy sets and join spaces associated with rough sets. Rend. Circ. Mat. Palermo (2) 51, 527–536 (2002)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Corsini, P., Leoreanu, V.: Join spaces associated with fuzzy sets. J. Comb. Inf. Syst. Sci. 20(1–4), 293–303 (1995)MathSciNetMATHGoogle Scholar
  8. 8.
    Corsini, P., Tofan, I.: On fuzzy hypergroups. Pure Math. Appl. 8, 29–37 (1997)MathSciNetMATHGoogle Scholar
  9. 9.
    Davvaz, B.: Interval-valued fuzzy subhypergroups. Korean J. Comput. Appl. Math. 6(1), 197–202 (1999)MathSciNetMATHGoogle Scholar
  10. 10.
    Davvaz, B.: Interval-valued fuzzy ideals in a hyperring. Ital. J. Pure Appl. Math. 10, 117–124 (2001)MathSciNetMATHGoogle Scholar
  11. 11.
    Davvaz, B.: Rings derived from semihyperrings. Algebras Groups Geom. 20, 245–252 (2003)MathSciNetMATHGoogle Scholar
  12. 12.
    Davvaz, B.: Isomorphism theorems of hyperrings. Indian J. Pure Appl. Math. 35(3), 321–331 (2004)MathSciNetMATHGoogle Scholar
  13. 13.
    Davvaz, B., Cristea, I.: Fuzzy Algebraic Hyperstructures; Studies Fuzziness and Soft Computing 321. Springer, Berlin (2015)CrossRefGoogle Scholar
  14. 14.
    De Salvo, M.: Hyperrings and hyperfields. Ann. Sci. Univ. Clermont-Ferrand II(22), 89–107 (1984)MathSciNetGoogle Scholar
  15. 15.
    Dutta, T.K., Kar, S., Purkait, S.: Interval-valued fuzzy \(k\)-ideals and \(k\)-regularity of semirings. Fuzzy Inf. Eng. 5(2), 235–251 (2013)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Dutta, T.K., Kar, S., Purkait, S.: Interval-valued fuzzy prime and semiprime ideals of a hypersemiring. Ann. Fuzzy Math. Inform. 9(2), 261–278 (2015)MathSciNetMATHGoogle Scholar
  17. 17.
    Grattan-Guiness, I.: Fuzzy membership mapped onto interval and many-valued quantities. Z. Math. Logik Grundladen Math. 22, 149–160 (1975)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Jahn, K.U.: Intervall-wertige Mengen. Math. Nachr. 68, 115–132 (1975)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Jun, Y., Kim, K.: Interval-valued fuzzy R-subgroups of near-rings. Indian J. Pure Appl. Math. 33(1), 71–80 (2002)MathSciNetMATHGoogle Scholar
  20. 20.
    Kar, S., Purkait, S.: Characterization of some k-regularities of semirings in terms of fuzzy ideals of semirings. J. Intell. Fuzzy Syst. 27, 3089–3101 (2014). doi: 10.3233/IFS-141266 MathSciNetMATHGoogle Scholar
  21. 21.
    Kar, S., Purkait, S., Shum, K.P.: Interval-valued fuzzy k-quasi-ideals and k-regularity of semirings. Afr. Mat. 26, 1413–1425 (2015). doi: 10.1007/s13370-014-0296-1 MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Krasner, M.: A class of hyperrings and hyperfields. Int. J. Math. Math. Sci. 6(2), 307–311 (1983)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Leoreanu-Fotea, V., Davvaz, B.: \(n\)-hypergroups and binary relations. Eur. J. Comb. 29, 1207–1218 (2008)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Leoreanu-Fotea, V., Davvaz, B.: Fuzzy hyperrings. Fuzzy Sets Syst. 160(16), 2366–2378 (2009)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Marty, F.: Sur une generalisation de la notion de group. In: 8th Congress of Mathematics, pp. 45–49. Stockholm, Scandinaves (1934)Google Scholar
  26. 26.
    Mirvakili, S., Davvaz, B.: Relationship between the rings and hyperrings by using the notion of fundamental relations. Commun. Algebra 41, 70–82 (2013)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Rota, R.: Strongly distributive multiplicative hyperrings. J. Geom. 39(1990), 130–138 (1990)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Sambuc, R.: Fonctions \(\phi \)-floues; Application l’aide au diagnostic en pathologie thyroidienne. Ph. D. thesis, Univ. Marseille, Marseille (1975)Google Scholar
  29. 29.
    Sen, M.K., Ameri, R., Chowdhury, G.: Fuzzy hypersemigroups. Soft Comput. (2007). doi: 10.1007/s00500-007-0257-9 MATHGoogle Scholar
  30. 30.
    Vougiouklis, T.: On some representations of hypergroups. Ann. Sci. Univ. Clermont-Ferrand II Math. 26, 21–29 (1990)MathSciNetMATHGoogle Scholar
  31. 31.
    Vougiouklis, T.: Hyperstructures and Their Representations. Hadronic Press, Nonantum (1994)MATHGoogle Scholar
  32. 32.
    Zadeh, L.: The concept of a linguistic variable and its application to approximate reasoning. Inf. Sci. 8, 199–249 (1975)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Zahedi, M.M., Ameri, R.: On the prime, primary and maximal subhypermodules. Ital. J. Pure Appl. Math. 5, 61–80 (1999)MathSciNetMATHGoogle Scholar
  34. 34.
    Zahedi, M.M., Bolurian, M., Hasankhani, A.: On polygroups and fuzzy subpolygroups. J. Fuzzy Math. 3, 1–15 (1995)MathSciNetMATHGoogle Scholar
  35. 35.
    Zhan, J., Davvaz, B., Shum, K.P.: Generalized fuzzy hyperideals of hyperrings. Comput. Math. Appl. 56(7), 1732–1740 (2008)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer India Pvt. Ltd. 2016

Authors and Affiliations

  1. 1.Department of MathematicsJadavpur UniversityKolkataIndia

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