# FUZZY SUBGROUPS BASED ON FUZZY POINTS

• Jun, Young-Bae (Department of Mathematics Education (and RINS) Gyeongsang National University) ;
• Kang, Min-Su (Department of Mathematics Hanyang University) ;
• Park, Chul-Hwan (School of Digital, Mechanics Ulsan College)
• Published : 2011.07.31
• 75 11

#### Abstract

Using the "belongs to" relation and "quasi-coincident with" relation between a fuzzy point and a fuzzy subgroup, Bhakat and Das, in 1992 and 1996, initiated general types of fuzzy subgroups which are a generalization of Rosenfeld's fuzzy subgroups. In this paper, more general notions of "belongs to" and "quasi-coincident with" relation between a fuzzy point and a fuzzy set are provided, and more general formulations of general types of fuzzy (normal) subgroups by Bhakat and Das are discussed. Furthermore, general type of coset is introduced, and related fundamental properties are investigated.

#### Keywords

(${\in}$, ${\in}$)-fuzzy subgroup;(strong) (${\in}$, ${\in}{\vee}q_{\kappa}$)-fuzzy subgroup;(${\in}$, ${\in}{\vee}q_{\kappa}$)-fuzzy subgroup generated by a fuzzy subset;(${\in}$, ${\in}{\vee}q_{\kappa}$)-fuzzy normal subgroup;(${\in}$, ${\in}{\vee}q_{\kappa}$)-fuzzy left (resp. right) coset;(${\in}$, ${\in}{\vee}q_{\kappa}$)-level subgroup

#### References

1. S. K. Bhakat, (${\in} {\vee}q$)-level subset, Fuzzy Sets and Systems 103 (1999), no. 3, 529-533. https://doi.org/10.1016/S0165-0114(97)00158-9
2. S. K. Bhakat, (${\in}, {\in} {\vee}q$)-fuzzy normal, quasinormal and maximal subgroups, Fuzzy Sets and Systems 112 (2000), no. 2, 299-312. https://doi.org/10.1016/S0165-0114(98)00029-3
3. S. K. Bhakat and P. Das, On the definition of a fuzzy subgroup, Fuzzy Sets and Systems 51 (1992), no. 2, 235-241. https://doi.org/10.1016/0165-0114(92)90196-B
4. S. K. Bhakat and P. Das, (${\in}, {\in} {\vee} q$)-fuzzy subgroup, Fuzzy Sets and Systems 80 (1996), no. 3, 359-368. https://doi.org/10.1016/0165-0114(95)00157-3
5. V. N. Dixit, R. Kumar, and N. Ajmal, Level subgroups and union of fuzzy subgroups, Fuzzy Sets and Systems 37 (1990), no. 3, 359-371. https://doi.org/10.1016/0165-0114(90)90032-2
6. D. Dubois and H. Prade, Gradual elements in a fuzzy set, Soft Comput. 12 (2008), 165-175.
7. T. Head, A metatheorem for deriving fuzzy theorems from crisp versions, Fuzzy Sets and Systems 73 (1995), no. 3, 349-358. https://doi.org/10.1016/0165-0114(94)00321-W
8. W.-J. Liu, Fuzzy invariant subgroups and fuzzy ideals, Fuzzy Sets and Systems 8 (1982), no. 2, 133-139. https://doi.org/10.1016/0165-0114(82)90003-3
9. P. K. Maji, A. R. Roy, and R. Biswas, An application of soft sets in a decision making problem, Comput. Math. Appl. 44 (2002), no. 8-9, 1077-1083. https://doi.org/10.1016/S0898-1221(02)00216-X
10. D. Molodtsov, Soft set theory-first results, Comput. Math. Appl. 37 (1999), no. 4-5, 19-31. https://doi.org/10.1016/S0898-1221(99)00056-5
11. N. P. Mukherjee and P. Bhattacharya, Fuzzy normal subgroups and fuzzy cosets, Inform. Sci. 34 (1984), no. 3, 225-239. https://doi.org/10.1016/0020-0255(84)90050-1
12. N. P. Mukherjee and P. Bhattacharya, Fuzzy groups: some group-theoretic analogs, Inform. Sci. 39 (1986), no. 3, 247-268. https://doi.org/10.1016/0020-0255(86)90039-3
13. V. Murali, Fuzzy points of equivalent fuzzy subsets, Inform. Sci. 158 (2004), 277-288. https://doi.org/10.1016/j.ins.2003.07.008
14. P. M. Pu and Y. M. Liu, Fuzzy topology. I. Neighborhood structure of a fuzzy point and Moore-Smith convergence, J. Math. Anal. Appl. 76 (1980), no. 2, 571-599. https://doi.org/10.1016/0022-247X(80)90048-7
15. A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl. 35 (1971), 512-517. https://doi.org/10.1016/0022-247X(71)90199-5
16. X. Yuan, C. Zhang, and Y. Ren, Generalized fuzzy groups and many-valued implications, Fuzzy Sets and Systems 138 (2003), no. 1, 205-211. https://doi.org/10.1016/S0165-0114(02)00443-8
17. L. A. Zadeh, Fuzzy sets, Information and Control 8 (1965), 338-353. https://doi.org/10.1016/S0019-9958(65)90241-X

#### Cited by

1. Product of the GeneralizedL-Subgroups vol.2016, 2016, https://doi.org/10.1155/2016/4918948