Skip to main content

Advertisement

Springer Nature Link
Log in

On \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy ideals of BCI-algebras

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

Abstract

The concepts of \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy (p-, q- and a-) ideals of BCI-algebras are introduced and some related properties are investigated. In particular, we describe the relationships among ordinary fuzzy (p-, q- and a-) ideals, (∈, ∈ ∨ q)-fuzzy (p-, q- and a-) ideals and \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy (p-,q- and a-) ideals of BCI-algebras. Moreover, we prove that a fuzzy set μ of a BCI-algebra X is an \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy a-ideal of X if and only if it is both an \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy p-ideal and an \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy q-ideal. Finally, we give some characterizations of three particular cases of BCI-algebras by these generalized fuzzy ideals.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Explore related subjects

Discover the latest articles and news from researchers in related subjects, suggested using machine learning.

References

  1. Bhakat SK, Das P (1996) (∈, ∈ ∨ q)-fuzzy subgroups. Fuzzy Sets Syst 80:359–368

    Article  MATH  MathSciNet  Google Scholar 

  2. Davvaz B (2006) (∈,  ∈ ∨ q)-fuzzy subnear-rings and ideals. Soft Comput 10:206–211

    Article  MATH  Google Scholar 

  3. Dudek WA (1998) On group-like BCI-algebras. Demonstratio Math 21:369–376

    MathSciNet  Google Scholar 

  4. Dudek WA, Thomys J (1990) On decompositions of BCH-algebras. Math Japon 35:1131–1138

    MATH  MathSciNet  Google Scholar 

  5. Esteva F, Godo L (2001) Monoidal t-norm based logic: towards a logic for left-continuous t-norms. Fuzzy Sets Syst 124:271–288

    Article  MATH  MathSciNet  Google Scholar 

  6. Hájek P (1998) Metamathematics of fuzzy logic. Kluwer, Dordrecht

    MATH  Google Scholar 

  7. Imai Y, Iséki K (1966) On axiom system of propositional calculus. Proc Japan Acad 42:19–22

    Article  MATH  MathSciNet  Google Scholar 

  8. Iséki K (1980) On BCI-algebras. Math Seminar Notes(now Kobe Math J) 8:125–130

    MATH  Google Scholar 

  9. Jun YB (2004) On (α, β)-fuzzy ideals of BCK/BCI-algebras. Sci Math Japon 60:613–617

    MATH  Google Scholar 

  10. Jun YB, Meng J (1994) Fuzzy p-ideals in BCI-algebras. Math Japon 40:271–282

    MATH  MathSciNet  Google Scholar 

  11. Liu YL, Meng J (2000) Fuzzy q-ideals in BCI-algebras. J Fuzzy Math 8:873–881

    MATH  MathSciNet  Google Scholar 

  12. Liu YL, Meng J (2001) Fuzzy ideals in BCI-algebras. Fuzzy Sets Syst 123:227–237

    Article  MATH  MathSciNet  Google Scholar 

  13. Liu YL, Meng J, Zhang XH and Yue ZC (2000) q-ideals and a-ideals in BCI-algebras. SEA Bull Math 24:243–253

    Article  MATH  MathSciNet  Google Scholar 

  14. Liu YL, Zhang XH (2002) Fuzzy a-ideals in BCI-algebras. Adv Math (China) 31:65–73

    MathSciNet  Google Scholar 

  15. Ma X, Zhan J, Davvaz B, Jun YB (2008) Some kinds of (∈, ∈ ∨ q)-interval-valued fuzzy ideals of BCI-algebras. Inform Sci 178:3738–3754

    Article  MATH  MathSciNet  Google Scholar 

  16. Meng J, Guo X (2005) On fuzzy ideals in BCK-algebras. Fuzzy Sets Syst 149:509–525

    Article  MATH  MathSciNet  Google Scholar 

  17. Pu PM, Liu YM (1980) Fuzzy topology I: Neighbourhood structure of a fuzzy point and Moore-Smith convergence. J Math Anal Appl 76:571–599

    Article  MATH  MathSciNet  Google Scholar 

  18. Xu Y (1993) Lattice implication algebras. J Southeast Jiaotong Univ 1:20–27

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  20. Zhan J, Jun YB, Davvaz B (2009) On (∈, ∈ ∨ q)-fuzzy ideals of BCI-algebras. Iran J Fuzzy Syst 6(1):81–94

    MATH  MathSciNet  Google Scholar 

  21. Zhan J, Tan Z (2004) Intuitionistic fuzzy a-ideals in BCI-algebras. Soochow Math J 30:207–216

    MATH  MathSciNet  Google Scholar 

  22. Zhang XH, Jiang H, Bhatti SA (1994) On p-ideals of BCI-algebras. Punjab Univ J Math 27:121–128

    MATH  MathSciNet  Google Scholar 

Download references

Acknowledgements

This research is partially supported by a grant of the National Natural Science Foundation of China (60875034); a grant of the Natural Science Foundation of Education Committee of Hubei Province, China (D20092901) and also the support of the Natural Science Foundation of Hubei Province, China (2009CDB340).

Author information

Authors and Affiliations

  1. Department of Mathematics, Hubei Institute for Nationalities, 445000, Enshi, Hubei Province, People’s Republic of China

    Jianming Zhan

  2. Department of Mathematics Education, Gyeongsang National University, Chinju, 660-701, Korea

    Young Bae Jun

Authors
  1. Jianming Zhan
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Young Bae Jun
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Jianming Zhan.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zhan, J., Jun, Y.B. On \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy ideals of BCI-algebras. Neural Comput & Applic 20, 319–328 (2011). https://doi.org/10.1007/s00521-010-0376-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00521-010-0376-6

Keywords