Soft Computing

, Volume 23, Issue 24, pp 13025–13033 | Cite as

Fundamental relation on fuzzy hypermodules

  • B. DavvazEmail author
  • N. Firouzkouhi


The fundamental relation \(\theta ^{*}\) on M can be defined as the smallest equivalence relation such that the quotient \(M/\theta ^{*}\) be a module over the corresponding fundamental ring such that \(M/\theta ^{*}\) as a group is not abelian. Moreover, the fundamental ring is not commutative with respect to both sum and product. Now, we would like the fundamental module as a group to be abelian and the fundamental ring to be commutative with respect to both sum and product. Also, we assign fundamental functor between the category of fuzzy hypermodules and the category of abelian groups to convey its features and related commutative diagram. Finally, we find necessary and sufficient conditions such that \(\theta \) is transitive.


Fuzzy hypermodule Fuzzy hyperring Fuzzy strongly regular equivalence relation Fundamental functor 


Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

Ethical approval

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

Informed consent

Informed consent was obtained from all individual participants included in the study.


  1. Ameri A, Nozari T (2010) Complete parts and fundamental relation of fuzzy hypersemigroup. J Mult-valued Logic Soft Comput 19:451–460MathSciNetGoogle Scholar
  2. Anvariyeh SM, Mirvakili S, Davvaz B (2008) \(\theta ^{\ast }\)-Relation on hypermodules and fundamental modules over commutative fundamental rings. Commun Algebra 36(2):622–631Google Scholar
  3. Corsini P (1991) Prolegomena of hypergroup theory. Aviani Editore, ItalyzbMATHGoogle Scholar
  4. Corsini P (1993) Join spaces, power sets, fuzzy sets. In: Proceedings of fifth international congress of algebraic hyperstructures and application, Iasi, Romania, Hadronic Press, Palm Harbor, p 4552Google Scholar
  5. Corsini P (2000) Fuzzy sets, join spaces and factor spaces. Pure Math Appl 11(3):439–446MathSciNetzbMATHGoogle Scholar
  6. Corsini P, Leoreanu V (eds) (2013) Applications of hyperstructure theory. Kluwer Academic, DordrechtzbMATHGoogle Scholar
  7. Corsini P, Tofan I (1997) On fuzzy hypergroups. Pure Math Appl 8:29–37MathSciNetzbMATHGoogle Scholar
  8. Davvaz B, Cristea I (eds) (2015) Studies in fuzziness and soft computing, vol 321. Fuzzy algebraic hyperstructures—an introduction. Springer, ChamGoogle Scholar
  9. Davvaz B, Leoreanu-Fotea V (2007) Hyperring theory and applications. International Academic Press, Palm HarborzbMATHGoogle Scholar
  10. Davvaz B, Vougiouklis T (2007) Commutative rings obtained from hyperrings (\(H_{v}\)-rings) with \(\alpha ^{\ast }\)-relations. Commun Algebra 35(11):3307–3320Google Scholar
  11. Leoreanu-Fotea V (2009) Fuzzy hypermodules. Comput Math Appl 57(3):466–475MathSciNetCrossRefGoogle Scholar
  12. Leoreanu-Fotea V, Davvaz B (2009) Fuzzy hyperrings. Fuzzy Sets Syst 160(16):2366–2378MathSciNetCrossRefGoogle Scholar
  13. Mirvakili S, Anvariyeh SM, Davvaz B (2008) Transitivity of \(\Gamma \)-relation on hyperfields. Bull Math Soc Sci Math Roumanie 51(99):233–243Google Scholar
  14. Mirvakili S, Davvaz B (2013) Relationship between rings and hyperrings by using the notion of fundamental relations. Commun Algebra 41(1):70–82MathSciNetCrossRefGoogle Scholar
  15. Mirvakili S, Davvaz B (2012) Strongly transitive geometric spaces: applications to hyperrings. Rev Un Mat Argent 53(1):43–53MathSciNetzbMATHGoogle Scholar
  16. Mirvakili S, Anvariyeh SM, Davvaz B (2008) On \(\alpha \)-relation and transitivity conditions of \(\alpha \). Commun Algebra 36(5):1695–1703Google Scholar
  17. Sen MK, Ameri R, Chowdhury G (2008) Fuzzy hypersemigroups. Soft Comput 12(9):891–900CrossRefGoogle Scholar
  18. Vougiouklis T (1990) The fundamental relation in hyperrings. The general hyperfield. In: Proceedings of the 4th international in congress on algebraic hyperstructures and Applications (AHA 1990), Xanthi, pp 203–211Google Scholar
  19. Vougiouklis T (1994) Hyperstructures and their representations. Hadronic Press Inc., FloridazbMATHGoogle Scholar
  20. Zahedi MM, Bolurian M, Hasankhani A (1995) On polygroups and fuzzy subpolygroups. J Fuzzy Math 3:1–15MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsYazd UniversityYazdIran
  2. 2.SariIran

Personalised recommendations