Skip to main content
SpringerLink
Log in

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

  • Published:
Afrika Matematika Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Pawlak, Z.: Rough sets. Int. J. Comput. Inf. Sci. 11, 341–356 (1982)

    Article  MATH  Google Scholar 

  2. Biswas, R., Nanda, S.: Rough groups and rough subgroups. Bull. Polish Acad. Sci. Math. 42, 251–254 (1994)

    MathSciNet  MATH  Google Scholar 

  3. Kuroki, N.: Rough ideals in semigroups. Inf. Sci. 100, 139–163 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  4. Jun, Y.B.: Roughness of ideals in BCK-algebras. Sci. Math. Jpn. 57(1), 165–169 (2003)

    MathSciNet  MATH  Google Scholar 

  5. Chinram, R.: Rough prime ideals and rough fuzzy prime ideals in gamma-semigroups. Commun. Korean Math. Soc. 24(3), 341–351 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Zadeh, L.A.: Fuzzy sets. Inf. Control 8, 338–353 (1965)

    Article  MATH  Google Scholar 

  7. Chang, C.L.: Fuzzy topological spaces. J. Math. Anal. Appl. 24, 182–190 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  8. Rosenfeld, A.: Fuzzy groups. J. Math. Anal. Appl. 35, 512–517 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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)

    MATH  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  12. Akram, M., Chen, W., Lin, Y.: Bipolar fuzzy Lie superalgebras. Quasigroups Relat. Syst. 20, 139–156 (2012)

    MathSciNet  MATH  Google Scholar 

  13. Akram, M.: Bipolar fuzzy \(\cal{L}\)-Lie algebras. World Appl. Sci. J. 14(12), 1908–1913 (2011)

    Google Scholar 

  14. Akram, M., Al-Shehrie, N.O.: Bipolar fuzzy Lie ideals. Utilit. Math. 87, 265–278 (2012)

    MathSciNet  MATH  Google Scholar 

  15. Marty, F.: Sur une generalization de la notion de group, 8th Congres Math, pp. 45–49. Stockholm, Scandinaves (1934)

  16. Corsini, P., Leoreanu-Fotea, V.: Applications of Hyperstructure Theory. Kluwer Academic Publications, Dordrecht (2003)

    Book  MATH  Google Scholar 

  17. Zhan, J., Cristea, I.: \(\Gamma \)-hypermodules: isomorphism theorems and regular relations. UPB Sci. Bull. Ser. A 73, 71–78 (2011)

    MATH  Google Scholar 

  18. Zhan, J., Davvaz, B., Shum, K.P.: Probability n-ary hypergroups. Inf. Sci. 180, 1159–1166 (2010)

    Article  MATH  Google Scholar 

  19. Ma, X., Zhan, J., Leoreanu-Fotea, V.: On (fuzzy) isomorphism theorems of gama-hyperrings. Comput. Math. Appl. 60, 2594–2600 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  20. Anvariyeh, S.M., Mirvakili, S., Davvaz, B.: On \(\Gamma \) -hyperideals in \(\Gamma \)-semihypergroups. Carpath. J. Math. 26(1), 11–23 (2010)

    MathSciNet  MATH  Google Scholar 

  21. Heidari, D., Dehkordi, S.O., Davvaz, B.: \(\Gamma \)-semihypergroups and their properties. UPB Sci. Bull. Ser. A 72, 197–210 (2010)

    MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  23. Hila, K., Davvaz, B., Dine, J.: Study on the structure of \( \Gamma \)-semihypergroups. Commun. Algebra 40, 2932–2948 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  25. Davvaz, B.: A survey of fuzzy algebraic hyperstructures. Algebra Groups Geometr. 27(1), 37–62 (2010)

    MATH  Google Scholar 

  26. Davvaz, B., Leoreanu-Fotea, V.: Structures of fuzzy \(\Gamma \) -hyperideals of \(\Gamma \)-semihypergroups. J. Multiple Valued Logic Soft Comput. 19, 519–535 (2012)

    MathSciNet  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  28. Yaqoob, N., Aslam, M.: Prime \((m, n)\) bi-\(\Gamma \) -hyperideals in \(\Gamma \)-semihypergroups. Appl. Math. Inf. Sci. 8(5), 2243–2249 (2014)

    Article  MathSciNet  Google Scholar 

  29. Anvariyeh, S.M., Mirvakili, S., Davvaz, B.: Pawlak’s approximations in \(\Gamma \)-semihypergroups. Comput. Math. Appl. 60, 45–53 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  31. Ersoy, B.A., Davvaz, B.: Atanassov’s intuitionistic fuzzy \( \Gamma \)-hyperideals of \(\Gamma \)-semihypergroups. J. Intell. Fuzzy Syst. 25, 463–470 (2013)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  33. Hila, K., Abdullah, S.: A study on intuitionistic fuzzy sets in \(\Gamma \)-semihypergroups. J. Intell. Fuzzy Syst. 26(4), 1695–1710 (2014)

    MathSciNet  MATH  Google Scholar 

  34. Abd-Allah, M.A., El-Saady, K., Ghareeb, A.: Rough intuitionistic fuzzy subgroup. Chaos Solitons Fractal. 42, 2145–2153 (2009)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

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

Author information

Authors and Affiliations

  1. Department of Mathematics, College of Science Al-Zulfi, Majmaah University, Al-Zulfi, Saudi Arabia

    Naveed Yaqoob

  2. Department of Mathematics and Science, College of Arts and Applied Sciences, Dhofar University, Salalah, Oman

    Inayatur Rehman

  3. Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan

    Muhammad Aslam

Authors
  1. Naveed Yaqoob
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Inayatur Rehman
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Muhammad Aslam
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Naveed Yaqoob.

Ethics declarations

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.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yaqoob, N., Rehman, I. & Aslam, M. Approximations of bipolar fuzzy \(\Gamma \)-hyperideals of \(\Gamma \)-semihypergroups. Afr. Mat. 29, 869–886 (2018). https://doi.org/10.1007/s13370-018-0585-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13370-018-0585-1

Keywords

Mathematics Subject Classification

Search

Navigation