Approximations of bipolar fuzzy \(\Gamma \)-hyperideals of \(\Gamma \)-semihypergroups

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Abstract

This paper studies relationship between bipolar fuzzy sets, rough sets and \(\Gamma \)-semihypergroups. We study the notion of rough bipolar fuzzy \(\Gamma \)-hyperideals in \(\Gamma \)-semihypergroups. Then we prove that the lower and upper approximation of a bipolar fuzzy \(\Gamma \)-hyperideal is a bipolar fuzzy \(\Gamma \)-hyperideal.

Keywords

Rough sets Bipolar fuzzy sets \(\Gamma \)-Semihypergroups Rough bipolar fuzzy \(\Gamma \)-hyperideals 

Mathematics Subject Classification

20N20 03E72 

Notes

Acknowledgements

The authors are highly grateful to the referees for their valuable comments and suggestions which were helpful in improving this paper.

Compliance with ethical standards

Conflict of interest

The authors of this paper Naveed Yaqoob and Muhammad Aslam declare that they have no conflict of interest.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.

References

  1. 1.
    Pawlak, Z.: Rough sets. Int. J. Comput. Inf. Sci. 11, 341–356 (1982)CrossRefMATHGoogle Scholar
  2. 2.
    Biswas, R., Nanda, S.: Rough groups and rough subgroups. Bull. Polish Acad. Sci. Math. 42, 251–254 (1994)MathSciNetMATHGoogle Scholar
  3. 3.
    Kuroki, N.: Rough ideals in semigroups. Inf. Sci. 100, 139–163 (1997)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Jun, Y.B.: Roughness of ideals in BCK-algebras. Sci. Math. Jpn. 57(1), 165–169 (2003)MathSciNetMATHGoogle Scholar
  5. 5.
    Chinram, R.: Rough prime ideals and rough fuzzy prime ideals in gamma-semigroups. Commun. Korean Math. Soc. 24(3), 341–351 (2009)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Zadeh, L.A.: Fuzzy sets. Inf. Control 8, 338–353 (1965)CrossRefMATHGoogle Scholar
  7. 7.
    Chang, C.L.: Fuzzy topological spaces. J. Math. Anal. Appl. 24, 182–190 (1968)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Rosenfeld, A.: Fuzzy groups. J. Math. Anal. Appl. 35, 512–517 (1971)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Lee, K.M.: Bi-polar-valued fuzzy sets and their operations. Proceedings of the International Conference on Intelligent Technologies, Bangkok, Thailand, pp. 307–312 (2000)Google Scholar
  10. 10.
    Jun, Y.B., Park, C.H.: Filters of BCH-algebras based on bi-polar-valued fuzzy sets. Int. Math. Forum 13, 631–643 (2009)MATHGoogle Scholar
  11. 11.
    Lee, K.J.: Bi-polar fuzzy subalgebras and bi-polar fuzzy ideals of BCK/BCI-algebras. Bull. Malaysian Math. Sci. Soc. 32(3), 361–373 (2009)MathSciNetGoogle Scholar
  12. 12.
    Akram, M., Chen, W., Lin, Y.: Bipolar fuzzy Lie superalgebras. Quasigroups Relat. Syst. 20, 139–156 (2012)MathSciNetMATHGoogle Scholar
  13. 13.
    Akram, M.: Bipolar fuzzy \(\cal{L}\)-Lie algebras. World Appl. Sci. J. 14(12), 1908–1913 (2011)Google Scholar
  14. 14.
    Akram, M., Al-Shehrie, N.O.: Bipolar fuzzy Lie ideals. Utilit. Math. 87, 265–278 (2012)MathSciNetMATHGoogle Scholar
  15. 15.
    Marty, F.: Sur une generalization de la notion de group, 8th Congres Math, pp. 45–49. Stockholm, Scandinaves (1934)Google Scholar
  16. 16.
    Corsini, P., Leoreanu-Fotea, V.: Applications of Hyperstructure Theory. Kluwer Academic Publications, Dordrecht (2003)CrossRefMATHGoogle Scholar
  17. 17.
    Zhan, J., Cristea, I.: \(\Gamma \)-hypermodules: isomorphism theorems and regular relations. UPB Sci. Bull. Ser. A 73, 71–78 (2011)MATHGoogle Scholar
  18. 18.
    Zhan, J., Davvaz, B., Shum, K.P.: Probability n-ary hypergroups. Inf. Sci. 180, 1159–1166 (2010)CrossRefMATHGoogle Scholar
  19. 19.
    Ma, X., Zhan, J., Leoreanu-Fotea, V.: On (fuzzy) isomorphism theorems of gama-hyperrings. Comput. Math. Appl. 60, 2594–2600 (2010)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Anvariyeh, S.M., Mirvakili, S., Davvaz, B.: On \(\Gamma \) -hyperideals in \(\Gamma \)-semihypergroups. Carpath. J. Math. 26(1), 11–23 (2010)MathSciNetMATHGoogle Scholar
  21. 21.
    Heidari, D., Dehkordi, S.O., Davvaz, B.: \(\Gamma \)-semihypergroups and their properties. UPB Sci. Bull. Ser. A 72, 197–210 (2010)MathSciNetMATHGoogle Scholar
  22. 22.
    Mirvakili, S., Anvariyeh, S.M., Davvaz, B.: \(\Gamma \)-semihypergroups and regular relations. J. Math. 2013, 915250 (2013).  https://doi.org/10.1155/2013/915250 MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Hila, K., Davvaz, B., Dine, J.: Study on the structure of \( \Gamma \)-semihypergroups. Commun. Algebra 40, 2932–2948 (2012)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Aslam, M., Abdullah, S., Davvaz, B., Yaqoob, N.: Rough M-hypersystems and fuzzy M-hypersystems in \(\Gamma \)-semihypergroups. Neural Comput. Appl. 21(1), 281–287 (2012)CrossRefGoogle Scholar
  25. 25.
    Davvaz, B.: A survey of fuzzy algebraic hyperstructures. Algebra Groups Geometr. 27(1), 37–62 (2010)MathSciNetMATHGoogle Scholar
  26. 26.
    Davvaz, B., Leoreanu-Fotea, V.: Structures of fuzzy \(\Gamma \) -hyperideals of \(\Gamma \)-semihypergroups. J. Multiple Valued Logic Soft Comput. 19, 519–535 (2012)MathSciNetGoogle Scholar
  27. 27.
    Yaqoob, N., Aslam, M., Davvaz, B., Saeid, A.B.: On rough \( (m, n) \) bi-\(\Gamma \)-hyperideals in \(\Gamma \)-semihypergroups. UPB Sci. Bull. Ser. A 75(1), 119–128 (2013)MathSciNetMATHGoogle Scholar
  28. 28.
    Yaqoob, N., Aslam, M.: Prime \((m, n)\) bi-\(\Gamma \) -hyperideals in \(\Gamma \)-semihypergroups. Appl. Math. Inf. Sci. 8(5), 2243–2249 (2014)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Anvariyeh, S.M., Mirvakili, S., Davvaz, B.: Pawlak’s approximations in \(\Gamma \)-semihypergroups. Comput. Math. Appl. 60, 45–53 (2010)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Yaqoob, N., Aslam, M., Davvaz, B., Ghareeb, A.: Structures of bipolar fuzzy \(\Gamma \)-hyperideals in \(\Gamma \)-semihypergroups. J. Intell. Fuzzy Syst. 27(6), 3015–3032 (2014)MathSciNetMATHGoogle Scholar
  31. 31.
    Ersoy, B.A., Davvaz, B.: Atanassov’s intuitionistic fuzzy \( \Gamma \)-hyperideals of \(\Gamma \)-semihypergroups. J. Intell. Fuzzy Syst. 25, 463–470 (2013)MathSciNetMATHGoogle Scholar
  32. 32.
    Hila, K., Abdullah, S., Dine, J.: On intuitionistic fuzzy \( \Gamma \)-hyperideals in \(\Gamma \)-semihypergroups through left operator semihypergroup. Utilitas Math. 100, 277–297 (2016)MathSciNetMATHGoogle Scholar
  33. 33.
    Hila, K., Abdullah, S.: A study on intuitionistic fuzzy sets in \(\Gamma \)-semihypergroups. J. Intell. Fuzzy Syst. 26(4), 1695–1710 (2014)MathSciNetMATHGoogle Scholar
  34. 34.
    Abd-Allah, M.A., El-Saady, K., Ghareeb, A.: Rough intuitionistic fuzzy subgroup. Chaos Solitons Fractal. 42, 2145–2153 (2009)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© African Mathematical Union and Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics, College of Science Al-ZulfiMajmaah UniversityAl-ZulfiSaudi Arabia
  2. 2.Department of Mathematics and Science, College of Arts and Applied SciencesDhofar UniversitySalalahOman
  3. 3.Department of MathematicsQuaid-i-Azam UniversityIslamabadPakistan

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