A NOTE ON (∈ , ∈ ∨q)-FUZZY EQUIVALENCE RELATIONS AND INDISTINGUISHABILITY OPERATORS

Abstract

A Note on (Î, Î Ú q)-Fuzzy Equivalence Relations and Indistinguishability Operators  ABSTRACT  |  FULL TEXT

Abstract

In this paper (∈ , ∈∨q)-fuzzy equivalence relations are defined and some of their properties are studied. The partition of an (∈ , ∈∨q)-fuzzy equivalence relation is studied. It is shown that under a semibalanced mapping the preimage of an (∈ , ∈∨q)-fuzzy equivalence relation is an (∈ , ∈∨q)-fuzzy equivalence relation, whereas the image of an (∈ , ∈∨q)-fuzzy equivalence relation under a balanced mapping is an (∈, ∈∨q)-fuzzy equivalence relation. To conclude, (∈ , ∈∨q)-fuzzy indistinguishability relations are defined and some of their properties are studied.

Keywords

• Fuzzy set,Fuzzy equivalence relation,Fuzzy G-equivalence relation,Fuzzy partition,Fuzzy indistinguishability operator,2000 AMS Classification: 03 B 52

References

1. Beg, I. and Ashraf, S. Fuzzy dissimilarity and distance function, Fuzzy Inf. Eng. 2, 205–217, 2009. [2] Beg, I. and Ashraf, S. Fuzzy equivalence relations, Kuwait J. Sci. and Eng. 35 (1A), 191–206, 2008. [3] Behounek, L. Extenionality in graded properties of fuzzy relations, Proc. IPMU Eleventh 10th Int. Conf. Vol II, 1604–1611, 2006.
2. Behounek, L., Bodenhofer, U. and Cintula, P. Relations in fuzzy class theory: Initial steps, Fuzzy Sets and Systems 159, 1729–1772, 2008.
3. Bhakat, S. K. (∈ , ∈ ∨q)-fuzzy normal, quasinormal and maximal subgroups, Fuzzy Sets and Systems 112, 299–312, 2000.
4. Bhakat, S. K. (∈ , ∈ ∨q)-level subsets, Fuzzy Sets and Systems 103, 529–533, 1999.
5. Bhakat, S. K. and Das, P. (∈ , ∈ ∨q)-fuzzy subgroups, Fuzzy Sets and Systems 80, 359–368, 1996. [8] Bhakat, S. K. and Das, P. Fuzzy subrings and ideals redefined, Fuzzy Sets and Systems 81, 383–393, 1996.
6. Boixader, D. and Recasens, J. Approximate fuzzy preorder and equivalences, Fuzzy IEEE Korea, 20–24, 2009.
7. Chakraborty, M. and Das, S. On fuzzy equivalence 1, Fuzzy Sets and Systems 11, 185–193, 1983. [11] Chakraborty, M. and Das, S. On fuzzy equivalence 2, Fuzzy Sets and Systems 12, 299–307, 1983. [12] Davvaz, B. (∈ , ∈ ∨q)-fuzzy subnearrings and ideals, Soft Comput. 10, 206–211, 2006.
8. Demirci, M. and Recasens, J. Fuzzy groups, fuzzy functions and fuzzy equivalence relations, Fuzzy Sets and Systems 144, 441–458, 2004.

9. Dudek, W. A., Shabir, M. and Ali, M. I. (α, β)-fuzzy ideals of hemirings, Computers and Mathematics with Applications 58, 310–321, 2009.
10. Gottwald, S. Fuzzified fuzzy relation, Proc. 4th IFSA Conf. Vol. Mathematics, Brussels, 82–86, 1991.
11. Gupta, K. C. and Gupta, R. K. Fuzzy equivalence relation redefined, Fuzzy Sets and Systems 79, 227–233, 1996.
12. Gupta, K. C. and Singh, T. P. Fuzzy G-equivalences and G-congruences on a groupoid under semibalanced maps, Fuzzy Sets and Systems 108, 111–116, 1999.
13. Jun, Y. B. and Song, S. Z. Generalized fuzzy interior ideals in semigroups, Information Sciences 176, 3079–3093, 2006.
14. Jacas, J. and Recasens, J. Fuzzified properties of fuzzy relations, Proc. IPMU 10th Int. Conf. 157–161, 2002.
15. Kazanci, O. and Yamak, S. Generalized fuzzy bi-ideals of semigroups, Soft computing 12, 1119–1124, 2008.
16. Kim, K. H. On fuzzy points in semigroups, Inter. J. Math. and Math. Sci. 26 (11), 707–712, 2001. [22] Kim, Y. H. and Kim, K. H. The semigroups of fuzzy points, The J. of Fuzzy Mathematics 12, 561–572, 2004.
17. Kuroki, N. On Fuzzy semigroups, Information Sciences 53, 203–236, 1991.
18. Kuroki, N.Rough ideals in Semigroups, Information Sciences 100, 139–163, 1997.
19. Makamaba, B. B. and Murali, V. Normality and congruence in fuzzy subgroups, Information Sciences 59, 121–129, 1992.
20. Ming, P. P. and Ming, L. Y. Fuzzy topology 1: Neighbourhood structure of a fuzzy point and Moore-Smith convergence, J. Math. Anal. Appl. 76, 571–599, 1980.
21. Murali, V. Fuzzy equivalence relations, Fuzzy Sets and Systems 30, 155–163, 1989.
22. Nemitz, W. C. Fuzzy relations and fuzzy functions, Fuzzy Sets and Systems 19, 177–1191, 1986. [29] Narayanan, Al. and Manikantan, T. (∈ , ∈ ∨q)-fuzzy subnear-rings (∈ , ∈ ∨q)-fuzzy ideals of near-rings, Fuzzy J. Appl. Math. Computing 18, 419–430, 2005.
23. Rosenfeld, A. Fuzzy groups, J. Math. Anal. Appl. 35, 512–517, 1971.
24. Shabir, M. Fully fuzzy prime semigroups, Int. J. Math. and Math. Sciences 2005 (1), 163– 168, 2005.
25. Shabir, M., Ali, M. I. and Khan, A. Rough S-Acts, Lobachevskii J. Math. 29, 96–109, 2008. [33] Shabir, M. and Ali, M. I. Soft ideals and generalized fuzzy ideals in semigroups, New Math. & Natural Computation 5 (3), 599–615, 2009.
26. Valverde, L. On the structure of F -indistinguishability operators, Fuzzy Sets and Systems 17, 313–328, 1985.
27. Xiao, Q. -M. and Zhang, Z. -L. Rough prime ideals and rough fuzzy prime ideals in semi- groups, Information Sciences 176, 725–733, 2006.
28. Yeh, R. T. Toward an algebraic theory of fuzzy relational systems, Proc. Int. Cong. Cyber., Namur, Belgium, 205–223, 1973.
29. Yuan, X., Zhang, C. and Ren, Y. Generalized fuzzy groups and many-valued implications, Fuzzy Sets and Systems 138, 205–211, 2003.
30. Zadeh, L. A. Fuzzy sets, Inform. and Control 8, 267–274, 1965.
31. Zadeh, L. A. Similarity relation and fuzzy ordering, Information Sciences 3, 177–200, 1971.