Soft Computing

, Volume 23, Issue 3, pp 907–920 | Cite as

Interval valued L-fuzzy prime ideals, triangular norms and partially ordered groups

  • Babushri Srinivas Kedukodi
  • Syam Prasad Kuncham
  • B. JagadeeshaEmail author
Methodologies and Application


We introduce interval valued equiprime, 3-prime and c-prime L-fuzzy ideals of a nearring N by using interval valued t-norms and interval valued t-conorms. We characterize interval valued prime L-fuzzy ideals in terms of their level subsets. We define interval valued equisemiprime, 3-semiprime and c-semiprime L-fuzzy ideals of nearrings and study their properties. We find interrelations among different interval valued prime L-fuzzy ideals. We study these concepts further in a partially ordered group and define implications based on interval valued L-fuzzy ideals.


Lattice t-norm t-conorm Ideal Equiprime 3-prime c-prime 



We thank the anonymous referees and the editor for their constructive comments and suggestions which improved this paper. All authors acknowledge Manipal University for their encouragement. The third author acknowledges St Joseph Engineering College, Mangaluru, India, for their encouragement.

Compliance with ethical standards

Conflicts of interest

The authors declare that they have no conflict of interest.

Ethical approval

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


  1. Akram M, Dudek WA (2008) Intuitionistic fuzzy left k-ideals of semirings. Soft Comput 12:881–890CrossRefzbMATHGoogle Scholar
  2. Bedregal BRC, Takahashi A (2006) The best interval representations of t-norms and automorphisms. Fuzzy Sets Syst 157:3220–3230MathSciNetCrossRefzbMATHGoogle Scholar
  3. Bertoluzza C, Doldi V (2006) On the distributivity between t-norms and t-conorms. Fuzzy Sets Syst 142:85–104MathSciNetCrossRefzbMATHGoogle Scholar
  4. Bhavanari S, Kuncham SP (2013) Nearrings, fuzzy ideals and graph theory. Chapman and Hall/ CRC Press, LondonzbMATHGoogle Scholar
  5. Bhavanari S, Kuncham SP, Kedukodi BS (2010) Graph of a nearring with respect to an ideal. Commun Algebra 38:1957–1962MathSciNetCrossRefzbMATHGoogle Scholar
  6. Blyth TS (2005) Lattices and ordered algebraic structures. Springer, LondonzbMATHGoogle Scholar
  7. Booth GL, Groenewald NJ, Veldsman S (1990) A Kurosh–Amitsur prime radical for near-rings. Commun Algebra 18:3111–3122MathSciNetCrossRefzbMATHGoogle Scholar
  8. Davvaz B (2001) Fuzzy ideals of nearring with interval valued membership functions. J Sci Islam Repub Iran 12:171–175MathSciNetGoogle Scholar
  9. Davvaz B (2006) (\(\epsilon, \epsilon \vee q\))-fuzzy subnear-rings and ideals. Soft Comput 10:206–211CrossRefzbMATHGoogle Scholar
  10. Davvaz B (2008) Fuzzy R-subgroups with thresholds of nearrings and implication operators. Soft Comput 12:875–879CrossRefzbMATHGoogle Scholar
  11. Deschrijver G (2008) A reprentation of t-norms in interval valued L-fuzzy set theroy. Fuzzy Sets Syst 159:1597–1618CrossRefzbMATHGoogle Scholar
  12. Dymek G (2008) Fuzzy prime ideals of pseudo-MV algebras. Soft Comput 12:365–372CrossRefzbMATHGoogle Scholar
  13. Goguen JA (1967) L-fuzzy sets. J Math Anal Appl 18(1):145–174Google Scholar
  14. Gratzer G (2011) Lattice theory: foundation. Birkhauser verlag, BaselCrossRefzbMATHGoogle Scholar
  15. Groenewald NJ (1991) Different prime ideals in near-rings. Commun Algebra 19:2667–2675MathSciNetCrossRefzbMATHGoogle Scholar
  16. Gu WX, Li SY, Chen DG, Lu YH (1995) The generalized t-norms and TLPF-groups. Fuzzy Sets Syst 72:357–364MathSciNetCrossRefzbMATHGoogle Scholar
  17. Jagadeesha B, Kedukodi BS, Kuncham SP (2015) Interval valued L-fuzzy ideals based on t-norms and t-conorms. J Intell Fuzzy Syst 28(6):2631–2641MathSciNetCrossRefzbMATHGoogle Scholar
  18. Jagadeesha B, Kuncham SP, Kedukodi BS (2016) Implications on a lattice. Fuzzy Inf Eng 8(4):411–425MathSciNetCrossRefGoogle Scholar
  19. Kazanc O, Yamak S (2008) Generalized fuzzy bi-ideals of semigroups. Soft Comput 12:1119–1124Google Scholar
  20. Kedukodi BS, Kuncham SP, Bhavanari S (2007) C-prime fuzzy ideals of nearrings. Soochow J Math 33:891–901MathSciNetzbMATHGoogle Scholar
  21. Kedukodi BS, Kuncham SP, Bhavanari S (2009) Equiprime, 3-prime and c-prime fuzzy ideals of nearrings. Soft Comput 13:933–944CrossRefzbMATHGoogle Scholar
  22. Kedukodi BS, Jagadeesha B, Kuncham SP (2016) Automorphisms, t-norms and t-conorms on a lattice, CommunicatedGoogle Scholar
  23. Kedukodi BS, Jagadeesha B, Kuncham SP, Juglal S (2017) Different prime graphs of a nearring with respect to an ideal. In: Nearrings, nearfields and related topics. World Scientific, Singapore, pp 185 -203. doi: 10.1142/9789813207363_0018
  24. Klement EP, Mesiar R, Pap E (2000) Triangular norms. Kluwer Academic Publishers, DordrechtCrossRefzbMATHGoogle Scholar
  25. Kondo M, Dudek WA (2015) On the transfer principle in fuzzy theory. Math Soft Comput 12:41–55MathSciNetzbMATHGoogle Scholar
  26. Kuncham SP, Kedukodi BS, Jagadeesha B (2016) Interval valued L-fuzzy cosets and isomorphism theorems. Afr Mat 27:393–408MathSciNetCrossRefzbMATHGoogle Scholar
  27. Ma J (2014) Lecture notes on algebraic structure of lattice ordered rings. World Scientific, SingaporeCrossRefzbMATHGoogle Scholar
  28. Ma X, Zhan J, Davvaz B, Jun YB (2008) Some kinds of \((\epsilon,\epsilon \vee q)\)-interval-valued fuzzy ideals of BCI-algebras. Inf Sci 178:3738–3754MathSciNetCrossRefzbMATHGoogle Scholar
  29. Pan X, Xu Y (2007) On the algebraic structure of binary lattice-valued fuzzy relations. Soft Comput 11:1053–1057CrossRefGoogle Scholar
  30. Pilz G (1983) Near-rings. Revised edition. North Hollond, AmsterdamGoogle Scholar
  31. Suzuki M (1951) On the lattice of subgroups of finite group. Trans Am Math Soc 70(2):345–371CrossRefzbMATHGoogle Scholar
  32. Veldsman S (1992) On equiprime near-rings. Commun Algebra 20(9):2569–2587MathSciNetCrossRefzbMATHGoogle Scholar
  33. Wang GJ, Li XP (1996) TH interval valued fuzzy subgroups. J Lanzhou Univ 32:96–99Google Scholar
  34. Zhan J, Davvaz B, Shum KP (2007) On fuzzy isomorphism theorems of hypermodules. Soft Comput 11:1053–1057CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  • Babushri Srinivas Kedukodi
    • 1
  • Syam Prasad Kuncham
    • 1
  • B. Jagadeesha
    • 2
    Email author
  1. 1.Department of Mathematics, Manipal Institute of TechnologyManipal UniversityManipalIndia
  2. 2.Department of MathematicsSt Joseph Engineering CollegeVamanjoor, MangaluruIndia

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