Skip to main content
SpringerLink
Log in

A fuzzy view of \(\Upgamma\)-hyperrings

  • Original Article
  • Published:
Neural Computing and Applications Aims and scope Submit manuscript

Abstract

In this paper, by means of a kind of new idea, we consider the ( ∈ ,  ∈ ∨ q)-fuzzy hyperideals of \(\Upgamma\)-hyperrings. The notion of ( ∈ ,  ∈ ∨ q)-fuzzy hyperideals of \(\Upgamma\)-hyperrings is provided and some related properties are investigated. A kind of quotient \(\Upgamma\)-hyperring of a \(\Upgamma\)-hyperring by an ( ∈ ,  ∈ ∨ q)-fuzzy hyperideal is presented and the relationships among ( ∈ ,  ∈ ∨ q)-fuzzy hyperideals, quotient \(\Upgamma\)-hyperrings and homomorphisms are explored. Several homomorphism theorems in the framework of fuzzy setting are provided.

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. Ameri R, Hedayati H, Molaee A (1999) On fuzzy hyperideals of \(\Upgamma\)-hyperrings. Iran J Fuzzy Syst 6:47–59

    MathSciNet  Google Scholar 

  2. Anvariyeh SM, Mirvakili S, Davvaz B (2010) Pawlaks approximations in \(\Upgamma\)-semihypergroups. Comput Math Appl 60:45–53

    Article  MathSciNet  MATH  Google Scholar 

  3. Barnes WE (1966) On the \(\Upgamma\)-rings of Nobusawa. Pac J Math 18:411–22

    MATH  Google Scholar 

  4. Bhakat SK, Das P (1992) On the definition of a fuzzy subgroup. Fuzzy Sets Syst 51:235–241

    Article  MathSciNet  MATH  Google Scholar 

  5. Bhakat SK, Das P (1992) ( ∈ ,  ∈ ∨ q)-fuzzy subgroups. Fuzzy Sets Syst 80:359–368

    Article  MathSciNet  Google Scholar 

  6. Corsini P (1993) Prolegomena of hypergroup theory. Aviani Editore, Italy

    MATH  Google Scholar 

  7. Corsini P, Leoreanu-Fotea V (1995) Join spaces associated with fuzzy sets. J Comb Inf Syst Sci 20:293–303

    MathSciNet  MATH  Google Scholar 

  8. Corsini P, Leoreanu-Fotea V (2002) Fuzzy sets and join spaces associated with rough sets. Rend Circ Mat Palermo 51:527–536

    Article  MathSciNet  MATH  Google Scholar 

  9. Corsini P, Leoreanu-Fotea V (2003) Applications of hyperstructure theory, advances in mathematics (Dordrecht). Kluwer, Dordrecht

    Google Scholar 

  10. Davvaz B (2004) Isomorphism theorems on hyperrings. Indian J Pure Appl Math 35(3):321–331

    MathSciNet  MATH  Google Scholar 

  11. Davvaz B, Corsini P (2006) Generalized fuzzy sub-hyperquasigroups of hyperquasigroups. Soft Comput 10(11):1109–1114

    Article  MATH  Google Scholar 

  12. Davvaz B, Corsini P (2006) Generalized fuzzy hyperideals of hypernear-rings and many valued implications. J Intell Fuzzy Syst 17(3):241–251

    MATH  Google Scholar 

  13. Davvaz B, Corsini P (2007) Fuzzy n-ary hypergroups. J Intell Fuzzy Syst 18((4):377–382

    MATH  Google Scholar 

  14. Davvaz B, Corsini P, Leoreanu-Fotea V (2009) Atanassov’s intuitionistic (S, T)-fuzzy n-ary subhypergroups and their properties. Inform Sci 179:654–666

    Article  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, Salasi A (2006) A realization of hyperrings. Commun Algebra 34(12):4389–4400

    Article  MathSciNet  MATH  Google Scholar 

  17. Davvaz B, Vougiouklis T (2007) Commutative rings obtained from hyperrings (H v -rings) with α*-relations. Commun Algebra 35(11):3307–3320

    Article  MathSciNet  MATH  Google Scholar 

  18. Davvaz B, Zhan J, Kim KH (2010) Fuzzy \(\Upgamma\)-hypernear-rings. Comput Math Appl 59(8):2846–2853

    Article  MathSciNet  MATH  Google Scholar 

  19. Dutta TK, Chanda T (2005) Structures of fuzzy ideals of \(\Upgamma\)-ring. Bull Malays Math Sci Soc 28:9–8

    MathSciNet  MATH  Google Scholar 

  20. Dutta TK, Chanda T (2007) Fuzzy prime ideals in \(\Upgamma\)-rings. Bull Malays Math Sci Soc 30:65–73

    MathSciNet  MATH  Google Scholar 

  21. Jun YB (1995) On fuzzy prime ideals of \(\Upgamma\)-rings. Soochow J Math 21:41–48

    MathSciNet  MATH  Google Scholar 

  22. Jun YB, Lee CY (1992) Fuzzy \(\Upgamma\)-rings. Pusan Kyŏngnam Math J 8:163–170

    Google Scholar 

  23. Jun YB, Lee CY (1993) Fuzzy prime ideals in \(\Upgamma\)-rings. Pusan Kyŏngnam Math J 9(1):105–111

    Google Scholar 

  24. Kehagias A, Serafimidis K (2005) The L-fuzzy Nakano hypergroup. Inform Sci 169:305–327

    Article  MathSciNet  MATH  Google Scholar 

  25. Krasner M (1983) A class of hyperrings and hyperfields. Int J Math Math Sci 2:307–312

    Article  MathSciNet  Google Scholar 

  26. Leoreanu-Fotea V (2000) Direct limit and inverse limit of join spaces associated with fuzzy sets. Pure Math Appl 11:509–516

    MathSciNet  Google Scholar 

  27. Leoreanu-Fotea V (2009) Fuzzy hypermodules. Comput Math Appl 57:466–475

    Article  MathSciNet  MATH  Google Scholar 

  28. Leoreanu-Fotea V, Davvaz B (2009) Fuzzy hyperrings. Fuzzy Sets Syst 160:2366–2378

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  30. Marty F (1934) Sur une generalization de la notion de groupe, in: 8th Congress Math. Scandianaves, Stockholm, pp 45–49

  31. Mirvakili S, Anvariyeh SM, Davvaz B (2008) On α-relation and transitivity conditions of α. Commun Algebra 36:1695–1703

    Article  MathSciNet  MATH  Google Scholar 

  32. Mordeson JN, Malik MS (1998) Fuzzy Commutative Algebra. World Publishing, Singapore

    Book  MATH  Google Scholar 

  33. Nakassis A (1988) Expository and survey article recent results in hyperring and hyperfield theory. Int J Math Math Sci 11:209–220

    Article  MathSciNet  MATH  Google Scholar 

  34. Pu PM, Liu YM (1980) Fuzzy topology I: neighbourhood structure of a fuzzy point and MooreCSmith convergence. J Math Anal Appl 76:571–599

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  36. Sen MK, Ameri R, Chowdhury G (2008) Fuzzy hypersemigroups. Soft Comput 12:891–900

    Article  MATH  Google Scholar 

  37. Vougiouklis T (1991) The fundamental relation in hyperrings. The general hyperfield. In: Proceedings of fourth international congress on algebraic hyperstructures and applications (AHA 1990). World Scientific, pp 203–211

  38. Vougiouklis T (1994) Hyperstructures and their representations. Hadronic Press Inc., Palm Harbor

    MATH  Google Scholar 

  39. Yin Y, Zhan J, Xu D, Wang J (2010) The L-fuzzy hypermodules. Comput Math Appl 59:953–963

    Article  MathSciNet  MATH  Google Scholar 

  40. Yin Y, Zhan J, Corsini P (2010a) Fuzzy roughness of n-ary hypergroups based a complete residuated lattice. Neural Comput Appl. doi:10.1007/s00521-010-0465-6

  41. Yin Y, Zhan J, Corsini P (2010b) L-fuzzy roughness of n-ary polygroups. Acta Math Sin. doi:10.1007/s10114-011-9465-4

  42. Zadeh LA (1965) Fuzzy sets. Inform Control 8:338–358

    Article  MathSciNet  MATH  Google Scholar 

  43. Zahedi MM, Bolurian M, Hasankhani A (1995) On polygroups and fuzzy subpolygroups. J Fuzzy Math 3:1–15

    MathSciNet  MATH  Google Scholar 

  44. Zhan J (2006) On properties of fuzzy hyperideals in hypernear-rings with t-norms. J Appl Math Comput 20:255–277

    Article  MathSciNet  MATH  Google Scholar 

  45. Zhan J, Davvaz B, Shum KP (2008) Generalized fuzzy hyperideals of hyperrings. Comput Math Appl 56:1732–1740

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We wish to express our heart thanks to the referees for their interests in our work and spending their valuable times in reading this manuscript carefully and giving their valuable comments for improving the paper.

This research was supported by National Natural Science Foundation of China (60774049; 60875034); the Natural Science Foundation of Education Committee of Hubei Province, China (D20092901; Q20092907; D20082903; B200529001) and the Natural Science Foundation of Hubei Province, China (2008CDB341).

Author information

Authors and Affiliations

  1. School of Mathematics and Information Sciences, East China Institute of Technology, Fuzhou, Jiangxi, 344000, China

    Yunqiang Yin

  2. Department of Mathematics, Yazd University, Yazd, Iran

    B. Davvaz

  3. Department of Mathematics, Hubei University for Nationalities, Enshi, Hubei Province, 445000, China

    Jianming Zhan

Authors
  1. Yunqiang Yin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. B. Davvaz
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jianming Zhan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yunqiang Yin.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Yin, Y., Davvaz, B. & Zhan, J. A fuzzy view of \(\Upgamma\)-hyperrings. Neural Comput & Applic 21, 979–992 (2012). https://doi.org/10.1007/s00521-010-0509-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00521-010-0509-y

Keywords

AMS Mathematics Subject Classification (2000)

Search

Navigation