Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
권호 표지
DOI QR코드

DOI QR Code

A STUDY ON (∈, ∈ ∨ q)-FUZZY CONGRUENCE ON RING

  • N. PRADIPKUMAR (Department of Mathematics, Manipur University) ;
  • O. RATNABALA DEVI (Department of Mathematics, Manipur University)
  • Received : 2023.09.13
  • Accepted : 2024.07.21
  • Published : 2024.07.30
Download PDF
⟨ Previous Next ⟩

Abstract

The purpose of this paper is to introduce the concept of (∈, ∈ ∨q)-fuzzy congruence relation over ring and discuss some properties of the (∈, ∈ ∨q)-fuzzy congruence relation. We also establish a brief relation between (∈, ∈ ∨q)-fuzzy ideal and (∈, ∈ ∨q)-fuzzy congruence relation. The image and preimage of (∈, ∈ ∨q)-fuzzy congruence are also studied under the so called semibalanced map.

Keywords

Acknowledgement

The authors wish to thank the anonymous reviewers for their valuable suggestions.

References

  1. S. Abou-Zaid, On Fuzzy subnear-rings and ideals, Fuzzy Sets and Systems 44 (1991), 139-146.
  2. M.I. Ali, F. Feng and M. Shabir, A note on (∈, ∈ ∨q)-fuzzy equivalence relations and indistinguishability operations, Hacettape Journal of Mathematics and Statistics 40 (2011), 383-400.
  3. S.K. Bhakat, (∈ ∨q)-level subset, Fuzzy Sets and Systems 103 (1999), 529-533.
  4. S.K. Bhakat and P. Das, On the definition of a fuzzy subgroup, Fuzzy Sets and Systems 51 (1992), 235-241.
  5. S.K. Bhakat and P. Das, (∈, ∈ ∨q)-fuzzy subgroup, Fuzzy Sets and Systems 80 (1996), 359-368.
  6. S.K. Bhakat and P. Das, Fuzzy subrings and ideals redefined, Fuzzy Sets and Systems 81 (1996), 383-393.
  7. P. Bhattacharya and N.P Mukherjee, Fuzzy relations and fuzzy groups, Information Sciences 36 (1985), 267-282.
  8. M. Chakraborty and S. Das, On fuzzy equivalence I, Fuzzy sets and Systems 11 (1983), 185-193.
  9. M. Chakraborty and S. Das, On fuzzy equivalence II, Fuzzy sets and Systems 12 (1983), 299-307.
  10. P.S. Das, Fuzzy groups and level subgroups, Journal of Mathematical Analysis and Applications 84(1) (1981), 264-269.
  11. K.C. Gupta and R.K. Gupta, Fuzzy equivalent relation redefined, Fuzzy Sets and Systems 79 (1996), 227-23.
  12. K.C. Gupta and T.P. Singh, Fuzzy G-congruences on a groupoid, Fuzzy Sets and Systems 93 (1998), 211-222.
  13. K.C. Gupta and T.P. Singh, Fuzzy G-equivalences and G-congruences on a groupoid under semibalanced maps, Fuzzy Sets and Systems 108 (1999), 111-116.
  14. J.P. Kim and D.R. Bae, Fuzzy congruence in groups, Fuzzy Sets System 85 (1997), 115-120.
  15. N. Kuroki, Fuzzy Congruences and Fuzzy Normal Subgroups, Information Sciences 60 (1992), 247-259.
  16. W.J. Liu, Fuzzy invariant subgroups and fuzzy ideals, Fuzzy sets and Systems 8 (1982), 133-139.
  17. Liu, Yong Lin, Jie Meng, and Xiao Long Xin, Quotient rings induced via fuzzy ideals, Korean Journal of Computational and Applied Mathematics 8 (2001), 631-643.
  18. B.B. Makamba and V. Murali, Normality and Congruence in Fuzzy Subgroups, Information Sciences 59 (1992), 121-129.
  19. P.P. Ming and L.Y. Ming, Fuzzy topology I, Neighborhood structure of a fuzzy point and Moore-Smith convergence, Journal of Mathematical Analysis and Applications 76 (1980), 571-599.
  20. N.P. Mukherjee, and P. Bhattacharya, Fuzzy normal sub-groups and fuzzy cosets, Information Sciences 34 (1984), 225-239.
  21. V. Murali, Fuzzy Equivalence Relations, Fuzzy Sets and Systems 30 (1989), 155-163.
  22. V. Murali, Fuzzy Congruence Relations, Fuzzy Sets and Systems 41 (1991), 359-369.
  23. Nemitz and C. William, Fuzzy relations and fuzzy functions, Fuzzy Sets and Systems 19 (1986), 177-191.
  24. N. Pradipkumar and O. Ratanabala Devi, (∈, ∈ ∨q)-fuzzy congruences on Groups, Bulletin Calcutta Mathematical Society 114 (2022), 597-616.
  25. A. Rosenfeld, Fuzzy groups, Journal of Mathematical Analysis and Applications 35 (1971), 512-517.
  26. M.A. Samhan, Fuzzy congruence on semigroups, Information Sciences 74 (1993), 165-175.
  27. M.A. Samhan and T.G.M. Ahsanullah, Fuzzy congruence on groups and rings, International Journal of Mathematics and Mathematical Sciences 17 (1994), 469-474.
  28. Y.O. Sung and B.G. Son, Notes of fuzzy congruences, Far East J. Math. Sci. 100 (2016), 385-396.
  29. L.A. Zadeh, Fuzzy sets, Information and Control 8 (1965), 338-353.
  30. L.A. Zadeh, Similarity relations and fuzzy ordering, Information Sciences 3 (1971), 177-200.
  31. Zhou, Xiaowu, Dajing Xiang, and Jianming Zhan, Quotient rings via fuzzy congruence relations, Italian Journal of Pure and Applied Mathematics 33 (2014), 411-424.