Skip to main content
SpringerLink
Log in

Interval valued intuitionistic fuzzy sets in \(\Gamma \)-semihypergroups

  • Original Article
  • Published:
International Journal of Machine Learning and Cybernetics Aims and scope Submit manuscript

Abstract

The notion of interval valued intuitionistic fuzzy sets was introduced by Atanassov and Gargov as a generalization of the notion of intuitionistic fuzzy sets and interval valued fuzzy sets. In this paper, we initiate a study on interval valued intuitionistic fuzzy sets in \(\Gamma \)-semihypergroups. We introduce the notions of interval valued intuitionistic fuzzy left (right, two sided) \(\Gamma \)-hyperideal, interval valued intuitionistic fuzzy bi-\(\Gamma \)-hyperideal and interval valued intuitionistic fuzzy (1,2) \(\Gamma \)-hyperideal in a \(\Gamma \)-semihypergroup and some properties of them are obtained. We use the interval valued intuitionistic fuzzy left, right, two-sided and bi-\(\Gamma \)-hyperideals to characterize some classes of \(\Gamma \)-semihypergroups. We also introduce the notion of an interval valued intuitionistic fuzzy \(M\) (resp. \(N\))-hypersystem of a \(\Gamma \)-semihypergroup and some properties of them are investigated.

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

Abbreviations

\(\mathcal {P}^{*}(S)=\mathcal {P}(S)\backslash \{\emptyset \}\) :

The set of all non-empty subsets of \(S\)

\(\Gamma \) :

The set of all hyperoperations

\(D\) [0,1]:

The collections of all closed subinterval of \( [0,1]\)

\(\widetilde{\lambda }\) :

An interval valued fuzzy set

i.v fuzzy set:

Interval valued fuzzy set

\(A=\langle \widetilde{\mu }_{A},\) \(\widetilde{\gamma }_{A}\rangle \) :

Notion for interval valued intuitionistic fuzzy set

i.v intuitionistic fuzzy set:

Interval valued intuitionistic fuzzy set

References

  1. Abdullah S, Hila K, Aslam M (2012) On bi-\(\Gamma \)-hyperideals of \(\Gamma \)-semihypergroups. UPB Sci Bull Ser A 74(4):79–90

    MathSciNet  MATH  Google Scholar 

  2. Abdullah S, Aslam M, Anwar T (2011) A note on \(M\)-hypersystems and \(N\)-hypersystems in \(\Gamma \)-semihypergroups. Quasigroups Relat Syst 19:169–172

    MathSciNet  MATH  Google Scholar 

  3. Abdullah S (2012) \(N\)-dimensional \(\left( \alpha ,\beta \right) \)-fuzzy \(H\)-ideals in hemirings. Int J Mach Learn Cyber. doi:10.1007/s13042-012-0141-5

  4. Akram M, Dudek WA (2009) Interval valued intuitionistic fuzzy Lie ideals of Lie algebras. World App Sci J 7(7):812–819

    Google Scholar 

  5. Aslam M, Abdullah S, Davvaz B, Yaqoob N (2012) Rough M-hypersystems and fuzzy M-hypersystems in \(\Gamma \)-semihypergroups. Neural Comput Appl 21:S281–S287

    Article  Google Scholar 

  6. Atanassov K (1983) Intuitionistic fuzzy sets, Central Tech. Library, Bulgarian Academy Science, Sofia, Bulgaria, Rep. No. 1697/84

  7. Atanassov K (1986) Intuitionistic fuzzy sets. Fuzzy Sets Systems 20:87–96

    Article  MathSciNet  MATH  Google Scholar 

  8. Atanassov KT, Gargov G (1989) Interval valued intuitionistic fuzzy sets. Fuzzy Sets Syst 31:343–349

    Article  MathSciNet  MATH  Google Scholar 

  9. Atanassov K (1994) New operations defined over the intuitionistic fuzzy sets. Fuzzy Sets Syst 61:137–142

    Article  MathSciNet  MATH  Google Scholar 

  10. Atanassov K (1999) Intuitionistic Fuzzy sets, theory and applications, studies in fuzziness and soft computing, vol 35. Physica, Heidelberg

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  12. Corsini P (1993) Prolegomena of hypergroup theory, Second edition, Aviani editor

  13. Corsini P, Leoreanu V (2003) Applications of hyperstructure theory. Advances in Mathematics. Kluwer Academic Publishers, Dordrecht

    Book  MATH  Google Scholar 

  14. Corsini P, Shabir M, Mahmood T (2011) Semisipmle semihypergroups in terms of hyperideals and fuzzy hyperideals. Iran J Fuzzy Syst 8(1):47

    MathSciNet  MATH  Google Scholar 

  15. Davvaz B, Leoreanu-Fotea V (2007) Hyperring theory and applications. International Academic Press, USA

    MATH  Google Scholar 

  16. Davvaz B (2000) Fuzzy hyperideals in semihypergroups. Italian J Pure Appl Math 8:67–74

    MathSciNet  MATH  Google Scholar 

  17. Davvaz B (2006) Intuitionistic fuzzy hyperideals of semihypergroups. Bull Malays Math Sci Soc (2) 29(1):203–207

    Google Scholar 

  18. Davvaz B, Leoreanu-Fotea V (2012) Structures of fuzzy \( \Gamma \)-hyperideals of \(\Gamma \)-semihypergroups. J Mult Val Log Soft Comput 19:519–535

    MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  22. Hila K, Abdullah S, Dine J. On intuitionistic fuzzy hyperideals in \(\Gamma \)-semihypergroups through left operator semihypergroup, Utilitas Mathematica (accepted)

  23. Hila K, Davvaz B, Dine J (2012) Study on the structure of \(\Gamma \)-semihypergroups. Commun Algebr 40(8):2932–2948

    Article  MathSciNet  MATH  Google Scholar 

  24. Marty F (1934) Sur une generalization de la notion de group, 8th Congres Math. Stockholm, Scandinaves

    Google Scholar 

  25. Mirvakili S, Anvariyeh SM, Davvaz B (2013) \(\Gamma \)-Semihypergroups and regular relations. J Math 2013. doi:10.1155/2013/915250

  26. Mahmood T (2011) On intuitionistic fuzzy bi-hyperideals of semihypergroups. T Res J 01(01):06–13

    Google Scholar 

  27. Mahmood T, Shabir M, Khan A. Generalized fuzzy hyperideals in semihypergroups. Ital J Pure Appl Math (accepted)

  28. Martinez-Soto R, Castillo O (2013) A hybrid optimization method with PSO and GA to automatically design Type-1 and Type-2 fuzzy logic controllers. Int J Mach Learn Cyber. doi:10.1007/s13042-013-0170-8

  29. Narayanam A, Manikantan T (2006) Interval-valued fuzzy ideals generated by an interval valued fuzzy subset in semigroups. J App Math Comput 20:455–464

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  31. Sen MK (1981) On \(\Gamma \)-semigroups, Proceedings of the International Conference on Algebra and its Applications. Dekker Publication, New York, p 301

    Google Scholar 

  32. Sen MK, Saha NK (1986) On \(\Gamma \)-semigroup-I. Bull Cal Math Soc 78:180–186

    MathSciNet  MATH  Google Scholar 

  33. Shabir M, Mahmood T. Regular and intra-regular semihypergroups in terms of bi-hyperideals and fuzzy bi-hyperideals. Ital J Pure Appl Math (accepted)

  34. Shabir M, Khan IA (2008) Interval-valued fuzzy ideals generated by an interval valued fuzzy subset in ordered semigroups. Mathw Soft Comput 15:263–272

    MathSciNet  MATH  Google Scholar 

  35. Vougiouklis T (1994) Hyperstructures and their representations. Hadronic Press, Florida

    MATH  Google Scholar 

  36. Xie X-Y (2001) Fuzzy ideal extension of semigroups. Soochow J Math 27(2):125–138

    MathSciNet  Google Scholar 

  37. Wang X, He Y, Dong L, Zhao H (2011) Particle swarm optimization for determining fuzzy measures from data. Inf Sci 181(19):4230–4252

    Article  MATH  Google Scholar 

  38. Wang X, Dong L, Yan J (2012) Maximum ambiguity based sample selection in fuzzy decision tree induction. IEEE Trans Knowl Data Eng 24(8):1491–1505

    Article  Google Scholar 

  39. Yaqoob N, Aslam M, Davvaz B, Saeid AB (2013) On rough \((m, n)\) bi-\(\Gamma \)-hyperideals in \(\Gamma \)-semihypergroups. UPB Sci Bull Ser A 75(1)

  40. Zadeh LA (1965) Fuzzy sets. Inform Control 8:338–353

    Article  MathSciNet  MATH  Google Scholar 

  41. Zadeh LA (1975) The concept of a lingusistic variable and its application to approximate reasoning. Inf Sci 8:199–249

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

We wish to express our heart thanks to Pakistan Science Foundation. This research work was supported by a grant of Pakistan Science Foundation, Islamabad, Pakistan.

Author information

Authors and Affiliations

  1. Department of Mathematics, Quaid-i-Azam University, Islamabad, 45320, Paksitan

    Saleem Abdullah

  2. Department of Mathematics, King Khalid University, Abha, Saudi Arabia

    Muhammad Aslam

  3. Department of Mathematics and Computer Science, Faculty of Natural Sciences, University of Gjirokastra, Gjirokastra, Albania

    Kostaq Hila

Authors
  1. Saleem Abdullah
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Muhammad Aslam
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Kostaq Hila
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Saleem Abdullah.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Abdullah, S., Aslam, M. & Hila, K. Interval valued intuitionistic fuzzy sets in \(\Gamma \)-semihypergroups. Int. J. Mach. Learn. & Cyber. 7, 217–228 (2016). https://doi.org/10.1007/s13042-014-0250-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13042-014-0250-4

Keywords

Search

Navigation