Interval valued L-fuzzy cosets of nearrings and isomorphism theorems

Abstract

In this paper, we study homomorphic images of interval valued L-fuzzy ideals of a nearring. If \(f:N_1\rightarrow N_2\) is an onto nearring homomorphism and \(\hat{\mu }\) is an interval valued L-fuzzy ideal of \(N_2\) then we prove that \(f^{-1}(\hat{\mu })\) is an interval valued L-fuzzy ideal of \(N_1\). If \(\hat{\mu }\) is an interval valued L-fuzzy ideal of \(N_1\) then we show that \(f(\hat{\mu })\) is an interval valued L-fuzzy ideal of \(N_2\) whenever \(\hat{\mu }\) is invariant under \(f\) and interval valued t-norm is idempotent. Finally, we define interval valued L-fuzzy cosets and prove isomorphism theorems.

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References

  1. 1.

    Acar, U.: On L-fuzzy prime submodules. Hacet. J. Math. Stat. 34, 17–25 (2005)

    MathSciNet  MATH  Google Scholar 

  2. 2.

    Aichinger, E., Binder, F., Ecker, J., Mayr, P., Nobauer, C.: SONATA-system of near-rings and their applications. GAP package. 2.6, (2012)

  3. 3.

    Akram, M., Dar, K.H.: Fuzzy left h-ideal in hemirings with respect to a s-norm. Int. J. Comput. Appl. Math. 1, 7–14 (2007)

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Akram, M., Yaqoob, N., Kavikumar, J.: Interval-valued \((\theta,\delta )\)-fuzzy KU-ideals of KU-algebras. Int. J. Pure Appl. Math. 92(3), 335–349 (2014)

    Article  MATH  Google Scholar 

  5. 5.

    Anderson, F.W., Fuller, K.R.: Rings and Categories of Modules, 2nd edn. Springer-Verlag, Berlin (1992)

    Book  MATH  Google Scholar 

  6. 6.

    Aygunoglu, A., Varol, B.P., Cetkin, V., Aygun, H.: Interval-valued intuitionistic fuzzy subgroups based on interval-valued double t-norm. Neural Comput. Appl. 21(1), 207–214 (2012)

    Article  Google Scholar 

  7. 7.

    Baksi, M.: Generalized fuzzy filters in noncommutative residuated lattices. Afr. Mat. 25, 289–305 (2014)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Bertoluzza, C., Doldi, V.: On the distributivity between t-norms and t-conorms. Fuzzy Sets Syst. 142, 85–104 (2006)

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Bhakat, S.K., Das, P.: Fuzzy subrings and ideals redefined. Fuzzy Sets Syst. 81, 383–393 (1992)

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Bhavanari, S.: Nearrings, Fuzzy Ideals and Graph Theory. Chapman and Hall/ CRC Press, New York (2013)

    MATH  Google Scholar 

  11. 11.

    Bhavanari, S., Kuncham, S.P., Kedukodi, B.S.: Graph of a nearring with respect to an ideal. Commun. Algebra. 38, 1957–1962 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Bhowmik, M., Snenpathi, M., Pal, M.: Intuitionistic L-fuzzy ideals of BG-algebras. Afr. Mat. 25, 577–590 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Booth, G.L., Groenewald, N.J., Veldsman, S.: A Kurosh-Amitsur prime radical for near-rings. Commun. Algebra 18(9), 3111–3122 (1990)

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Davvaz, B.: Fuzzy ideals of nearring with interval valued membership functions. J. Sci. Islam. Repub. Iran. 12, 171–175 (2001)

    MathSciNet  Google Scholar 

  15. 15.

    Davvaz, B.: Fuzzy R-subgroups with thresholds of nearring and implication operators. Soft Comput. 12, 875–879 (2008)

    Article  MATH  Google Scholar 

  16. 16.

    Davvaz, B., Fotea, V.L.: Applications of interval valued fuzzy n-ary polygroups with respect to t-norms (t-conorms). Comput. Math. Appl. 57, 1413–1424 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  17. 17.

    Dutta, T.K., Kar, S., Purkait, S.: Interval-valued fuzzy k-ideals and k-regularity of semirings. Fuzzy Inform. Eng. 5(2), 235–251 (2013)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Fisal, Yaqoob, N., Ghareeb, A.: Left regular AG -groupoids in terms of fuzzy interior ideals. Afr. Mat. 24, 577–587 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    Goguen, J.A.: L-fuzzy sets. J. Math. Anal. Appl. 18, 145–174 (1967)

    MathSciNet  Article  MATH  Google Scholar 

  20. 20.

    Gratzer, G.: Lattice Theory: Foundation. Birkhauser verlag, Basel (2011)

    Book  MATH  Google Scholar 

  21. 21.

    Groenewald, N.J.: Completely prime radical in near-rings. Acta. Math. Hung. 51(3), 301–305 (1988)

    MathSciNet  Article  MATH  Google Scholar 

  22. 22.

    Groenewald, N.J.: Different prime ideals in near-rings. Commun. Algebra. 19, 2667–2675 (1991)

    MathSciNet  Article  MATH  Google Scholar 

  23. 23.

    Gu, W.X., Li, S.Y., Chen, D.G., Lu, Y.H.: The generalized t-norms and TLPF-groups. Fuzzy Sets Syst. 72, 357–364 (1995)

    MathSciNet  Article  MATH  Google Scholar 

  24. 24.

    Guijun, W., Xiapping, L.: Interval-valued fuzzy subgroups induced by T-triangular norms. BUSEFAL 65, 80–84 (1996)

    Google Scholar 

  25. 25.

    Jagadeesha, B., Kedukodi, B.S., Kuncham, S.P.: Interval valued L-fuzzy ideals based on t-norms and t-conorms. J. Intell. Fuzzy Syst. (2015). doi:10.3233/IFS-151541

  26. 26.

    Jun, Y.B., Neggers, J., Kim, H.S.: Normal L-fuzzy ideals in semirings. Fuzzy Sets Syst. 82(3), 383–386 (1996)

    MathSciNet  Article  MATH  Google Scholar 

  27. 27.

    Jun, Y.B., Neggers, J., Kim, H.S.: On L-fuzzy ideals in semirings I. Czechoslovak Math. J. 48(4), 669–675 (1998)

    MathSciNet  Article  MATH  Google Scholar 

  28. 28.

    Kedukodi, B.S., Jagadeesha, B., Kuncham, S.P.: Automorphisms, t-norms and t-conorms on a lattice (communicated)

  29. 29.

    Kedukodi, B.S., Jagadeesha, B., Kuncham, S.P.: Interval valued Equiprime, 3-prime and C-prime L-fuzzy ideals of nearrings (communicated)

  30. 30.

    Kedukodi, B.S., Kuncham, S.P., Bhavanari, S.: C-prime fuzzy ideals of nearrings. Soochow J. Math. 33, 891–901 (2007)

    MathSciNet  MATH  Google Scholar 

  31. 31.

    Kedukodi, B.S., Kuncham, S.P., Bhavanari, S.: Equiprime, 3-prime and c-prime fuzzy ideals of nearrings. Soft. Comput. 13, 933–944 (2009)

    Article  MATH  Google Scholar 

  32. 32.

    Kedukodi, B.S., Kuncham, S.P., Bhavanari, S.: Reference points and roughness. Inform. Sci. 180, 3348–3361 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  33. 33.

    Khan, M., Aziz, T., Yaqoob, N.: Generalized interval-valued fuzzy right ideals of Abel Grassmann’s groupoids. Analele Universitatii din Oradea Fascicola Matematica 21(1), 153–160 (2014)

    MathSciNet  MATH  Google Scholar 

  34. 34.

    Khan, A., Jun, Y.B., Shabir, M.: A study of generalized fuzzy ideals in ordered semigroups. Neural Comput. Appl. 21(1), S69–S78 (2012)

    Article  Google Scholar 

  35. 35.

    Klement, E.P., Mesiar, R., Pap, E.: Triangular Norms. Kluwer Academic Publishers, Netherlands (2000)

    Book  MATH  Google Scholar 

  36. 36.

    Koc, A., Balkanay, E.: On \(\theta \)-euclidean L-fuzzy ideals of rings. Turkish J. Math. 28, 137–142 (2004)

    MathSciNet  MATH  Google Scholar 

  37. 37.

    Li, X., Wang, G.: TH-interval-valued fuzzy subgroups. Fuzzy Syst. Math. 12(1), 60–65 (1998)

    MATH  Google Scholar 

  38. 38.

    Pilz, G.: Near-Rings. North Hollond, Amsterdam (1983)

    MATH  Google Scholar 

  39. 39.

    Veldsman, S.: On equiprime near-rings. Commun. Algebra. 20(9), 2569–2587 (1992)

    MathSciNet  Article  MATH  Google Scholar 

  40. 40.

    Xiapping, L., Guijun, W.: The SH interval-valued fuzzy subgroups. Fuzzy Sets Syst. 112, 319–325 (2000)

    Article  Google Scholar 

  41. 41.

    Yaqoob, N.: Interval-valued intuitionistic fuzzy ideals of regular LA-semigroups. Thai J. Math. 11(3), 683–695 (2013)

    MathSciNet  MATH  Google Scholar 

  42. 42.

    Yaqoob, N., Akram, M., Aslam, M.: Intuitionistic fuzzy soft groups induced by (t, s)-norm. Indian J. Sci. Technol. 6(4), 4282–4289 (2013)

    Google Scholar 

  43. 43.

    Yaqoob, N., Chinram, R., Ghareeb, A., Aslam, M.: Left almost semigroups characterized by their interval valued fuzzy ideals. Afr. Mat. 24, 231–245 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  44. 44.

    Yaqoob, N., Khan, M., Akram, M., Khan, A.: Interval valued intuitionistic \((S, T)\)-fuzzy ideals of ternary semigroups. Indian J. Sci. Technol. 6(11), 5418–5428 (2013)

    Google Scholar 

  45. 45.

    Zaid, S.A.: On fuzzy subnear-rings and ideals. Fuzzy Sets Syst. 44, 139–146 (1991)

    MathSciNet  Article  MATH  Google Scholar 

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Acknowledgments

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

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Correspondence to Babushri Srinivas Kedukodi.

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Kuncham, S.P., Jagadeesha, B. & Kedukodi, B.S. Interval valued L-fuzzy cosets of nearrings and isomorphism theorems. Afr. Mat. 27, 393–408 (2016). https://doi.org/10.1007/s13370-015-0348-1

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Keywords

  • Nearring
  • Ideal
  • Coset
  • t-norm
  • t-conorm