Afrika Matematika

, Volume 29, Issue 1–2, pp 81–96 | Cite as

A study on (i-v) prime fuzzy hyperideal of semihypergroups

  • Paltu Sarkar
  • Sukhendu Kar


In this paper, our main objective is to introduce and investigate the interval-valued (in short, (i-v)) prime fuzzy hyperideal in semihypergroups in detail. We notice that every (i-v) semiprime fuzzy hyperideal may not be an (i-v) prime fuzzy hyperideal and we produce a counter example to illustrate this result. Moreover, we define (i-v) fuzzy hyper radical of an (i-v) fuzzy hyperideal of a semihypergroup. Finally, we study some interesting properties regarding this radical.


Semihypergroup (i-v)  Prime fuzzy hyperideal (i-v)  Fuzzy hyper radical 

Mathematics Subject Classification




The authors would like to express their sincere thanks to the reviewers for their esteemed comments to improve the presentation of our manuscript.


  1. 1.
    Aslam, M., Abdullah, S., Davvaz, B., Yaqoob, N.: Rough M-hypersystems and fuzzy M-hypersystems in \(\Gamma \)-semihypergroups. Neural Comput. Appl. 21, 281–287 (2012)CrossRefGoogle Scholar
  2. 2.
    Aslam, M., Aroob, T., Yaqoob, N.: On cubic \(\Gamma \)-hyperideals in left almost \(\Gamma \)-semihypergroups. Ann. Fuzzy Math. Inform. 5(1), 169–182 (2013)MathSciNetMATHGoogle Scholar
  3. 3.
    Biswas, R.: Rosenfeld’s fuzzy subgroups with interval-valued membership functions. Fuzzy Sets Syst. 63(1), 87–90 (1994)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Corsini, P.: Prolegomena of Hypergroup Theory, 2nd edn. Aviani Editore, Tricesimo (1993)MATHGoogle Scholar
  5. 5.
    Corsini, P., Leoreanu, V.: Applications of hyperstructure theory, Advances in Mathematics. Kluwer Academic Publishers, Dordrecht (2003)CrossRefMATHGoogle Scholar
  6. 6.
    Corsini, P., Shabir, M., Mahmood, T.: Semisimple semihypergroups in terms of hyperideals and fuzzy hyperideals. Iran. J. Fuzzy Syst. 8, 95–111 (2011)MathSciNetMATHGoogle Scholar
  7. 7.
    Davvaz, B., Leoreanu-Fotea, V.: Hyperring theory and applications. International Academic Press, Palm Harbor (2007)MATHGoogle Scholar
  8. 8.
    Davvaz, B., Leoreanu-Fotea, V.: Structures of fuzzy \(Gamma\)-hyperideals in \(Gamma\)-semihypergroups. Multiple Valued Logic Soft Comput. 19, 519–535 (2012)MathSciNetGoogle Scholar
  9. 9.
    Davvaz, B., Leoreanu-Fotea, V.: Triangular fuzzy sub \(\Gamma \)-semihypergroups in \(\Gamma \)-semihypergroups. Kuwait J. Sci. 40(1), 93–106 (2014)Google Scholar
  10. 10.
    Dutta, T.K., Kar, S., Purkait, S.: Interval-valued fuzzy prime and semiprime ideals of a hypersemiring. Annu. Fuzzy Math. Inform. 9(2), 261–278 (2015)MathSciNetMATHGoogle Scholar
  11. 11.
    Ersoya, B.A., Davvaz, B.: Atanassov’s intuitionistic fuzzy \(\Gamma \)-hyperideals of \(\Gamma \)-semihypergroups. J. Intell. Fuzzy Syst. 25(2), 463–470 (2013)MathSciNetMATHGoogle Scholar
  12. 12.
    Ersoy, B.A., Saricaoglu, Y., Yenigun, M., Davvaz, B.: On fuzzy interior \(\Gamma \)-hyperideals of \(\Gamma \)-semihypergroups. Utilitas Mathematica 88, 157–170 (2012)MathSciNetMATHGoogle Scholar
  13. 13.
    Freni, D.: Strongly transitive geometric spaces: applications to hypergroups and semigroups theory. Commun. Algebra 32, 969–988 (2004)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Gorzalczany, M.B.: A method of inference in approximate reasoning based on interval-valued fuzzy sets. Fuzzy Sets Syst. 21, 1–17 (1987)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Hasankhani, A.: Ideals in a semihypergroup and Green’s relations. Ratio Mathematica 13, 29–36 (1999)MathSciNetMATHGoogle Scholar
  16. 16.
    Hedayati, H., Azizpour, S., Davvaz, B.: Prime (semiprime) bi-hyperideals of semihypergroups based on intuitionistic fuzzy points. UPB Sci. Bull. Ser. A Appl. Math. Phys. 75(3), 45–58 (2013)MathSciNetMATHGoogle Scholar
  17. 17.
    Hila, K., Davvaz, B., Dine, J.: Study on the structure of \(\Gamma \)-semihypergroups. Commun. Algebra 40, 2932–2948 (2012)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Hila, K., Dine, J.: On hyperideals in left almost semihypergroups. ISRN Algebra. 2011, (2011). doi: 10.5402/2011/953124
  19. 19.
    Hila, K., Davvaz, B., Naka, K.: On quasi-hyperideals in semihypergroups. Commun. Algebra 39(11), 4183–4194 (2011)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Kar, S., Purkait, S.: On \(k\)-regularities in Fuzzy semihyperrings. Int. J. Appl. Comput. Math. (2016). doi: 10.1007/s40819-016-0166-7 (online published)
  21. 21.
    Kar, S., Shum, K.P., Sarkar, P.: Interval-valued prime fuzzy ideals of semigroups. Lobachevskii J. Math. 34(1), 11–19 (2013)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Kar, S., Sarkar, P., Hila, K.: Interval-valued semiprime fuzzy ideals of semigroups. Adv. Fuzzy Syst. 2014, 10 (2014)MathSciNetMATHGoogle Scholar
  23. 23.
    Kar, S., Sarkar, P.: Interval-valued fuzzy completely regular subsemigroups of semigroups. Ann. Fuzzy Math. Inform. 5(3), 583–595 (2013)MathSciNetMATHGoogle Scholar
  24. 24.
    Kar, S., Sarkar, P., Leoreanu-Fotea, V.: On some interval-valued fuzzy hyperideals of semihypergroups. Afrika Matematika 26, 1171–1186 (2015)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Khan, M., Jun, Y.B., Gulistan, M., Yaqoob, N.: The generalized version of Jun’s cubic sets in semigroups. J. Intell. Fuzzy Syst. 28, 947–960 (2015)MathSciNetMATHGoogle Scholar
  26. 26.
    Marty, F.: Sur une generalization de la notion de group. In: Proceedings of the 8th Congres des Mathematiciens Scandinave, Stockholm, pp. 45–49 (1934)Google Scholar
  27. 27.
    Mendel, J.M.: Uncertain rule-based fuzzy logic systems, Introduction and new directions. Prentice-Hall, Upper Saddle River (2001)MATHGoogle Scholar
  28. 28.
    Pibaljommee, B., Davvaz, B.: On fuzzy bi-hyperideals in ordered semihypergroups. J. Intell. Fuzzy Syst. 28, 2141–2148 (2015)CrossRefMATHGoogle Scholar
  29. 29.
    Roy, M.K., Biswas, R.: I–V fuzzy relations and Sanchezs approach for medical diagnosis. Fuzzy Sets Syst. 47, 35–38 (1992)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Vougiouklis, T.: Hyperstructures and their representations. Hadronic Press, Palm Harbor (1994)MATHGoogle Scholar
  31. 31.
    Yaqoob, N., Aslam, M., Ansari, M.A.: Structures of N-G-hyperideals in left almost G-semihypergroups. World Appl. Sci. J. 17(12), 1611–1617 (2012)Google Scholar
  32. 32.
    Yaqoob, N., Aslam, M., Davvaz, B., Ghareeb, A.: Structures of bipolar fuzzy G-hyperideals in G-semihypergroups. J. Intell. Fuzzy Syst. 27(6), 3015–3032 (2014)MathSciNetMATHGoogle Scholar
  33. 33.
    Yaqoob, N., Corsini, P., Yousafzai, F.: On intra-regular left almost semihypergroups with pure left identity. J. Math. 2013, (2013). doi: 110.1155/2013/510790
  34. 34.
    Zadeh, L.A.: The concept of a linguistic variable and its application to approximate reasoning. Inf. Sci. 8, 199–249 (1975)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© African Mathematical Union and Springer-Verlag GmbH Deutschland 2017

Authors and Affiliations

  1. 1.Department of MathematicsAnanda Chandra CollegeJalpaiguriIndia
  2. 2.Department of MathematicsJadavpur UniversityKolkataIndia

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