A new type of interval valued fuzzy normal subgroups of groups

Saleem Abdullah; Muhammad Naeem

A New Type of Interval Valued Fuzzy Normal Subgroups of Groups

Year 2015, Volume: 3 Issue: 1, 62 - 77, 22.12.2014

Abstract

In this paper we are using the notions of not belonging (∈) and non quasi-k-coincidenceqk ( ) of an interval valued fuzzypoint with an interval valued fuzzy set, we define the concepts of interval valued (∈,∈ ∨ qk)-fuzzy normal subgroups and interval valued(∈,∈ ∨ qk)-fuzzy cosets which is a generalization of fuzzy normal subgroups, fuzzy coset, interval valued fuzzy normal subgroups,interval valued fuzzy coset, interval valued (∈,∈ ∨ q)-fuzzy normal subgroups and interval valued (∈,∈ ∨ q)-fuzzy cosets. We givesome characterizations of an interval valued (∈,∈ ∨ qk)-fuzzy normal subgroup and interval valued (∈,∈ ∨ qk)-fuzzy coset, and dealwith several related properties. The important achievement of the study with an interval valued (∈,∈ ∨ qk)-fuzzy normal subgroupand interval valued (∈,∈ ∨ qk)-fuzzy cosets is the generalization of that the notions of fuzzy normal subgroups, fuzzy coset, intervalvalued fuzzy normal subgroups, interval valued fuzzy coset, interval valued (∈,∈ ∨ q)-fuzzy normal subgroups and interval valued(∈,∈ ∨ q)-fuzzy cosets. We prove that the set of all interval valued (∈,∈ ∨ qk)-fuzzy cosets of G is a group, where the multiplicationis defined byλx·λy=λxyfor all x, y∈ G. Ifµ : F → D[0,1] is defined by µ←−←−(←−)λx =λ (x) for all x ∈ G. Then µ is an interval valued( )-fuzzy normal subgroup of F

Keywords

Group, normal subgroup, coset, interval valued (∈, ∈ ∨ qk)-fuzzy normal subgroup, interval valued (∈, ∈ ∨ qk)-fuzzy coset

References

S. Abdullah, A new type of interval valued fuzzy subgroups of groups, Submitted. S. Abdullah, M.Aslam, T. Ahmad and M. Naeem, A new typ fuzzy normal subgroups and fuzzy cosets, Journal of Intelligent and Fuzzy Systems, 2012, 10.3233/IFS-2012-0612.
M. Aslam, S. Abdullah and N. Amin, Characterizations of gamma LA-semigroups by generalized fuzzy gamma ideals, International J. of Mathematics and Statistics, 12 (2012): 29-50.
M. Akram, K.H. Dar, , K.P. Shum, Interval-valued (α,β)-fuzzy K-algebras, Applied Soft Computing, 11, 1, (2011), 1213-1222.
S. K. Bhakat, (∈,∈ ∨q)-Level subset, Fuzzy Sets and System 103 (1999) 529-533.
S. K. Bhakat, (∈,∈ ∨q)-fuzzy normal, quasinormal and maximal subgroups, Fuzzy Sets and System 112 (2000) 299-312.
S. K. Bhakat, P. Das, (∈,∈ ∨q)-fuzzy subgroup, Fuzzy Sets and System 80 (1996) 359-368.
S. K. Bhakat, P. Das, On the definition of a fuzzy subgroup, Fuzzy Sets and System 51 (1992) 235-241.
R. Biswas, Rosenfeld’s fuzzy subgroups with interval-valued membership functions, Fuzzy Sets and Systems, 63, 1, (1994), 87-90.
P. S. Das, Fuzzy groups and level subgroups, J. Math. Anal. Appl. 85 (1981) 264-269.
B. Davvaz, Interval-valued fuzzy subhypergroups, Korean Journal of Computational and Applied Mathematics, 6, 1, (1999), 197- 20
Y. B. Jun, Generalizations of (∈,∈ ∨q)-fuzzy subalgebras in BCK/BCI-algebras, Comput. Math. Appl. , 58 (2009): 1383 1390
Y. B. Jun, M. S. Kang, and C. H. Park, Fuzzy subgroups based on fuzzy points, Commun. Korean Math. Soc. 26 (2011), No. 3, pp. 349-3
K.B. Latha, D.R. P. Williams and E.Chandrasekar, Interval valued (α,β)-fuzzy subgroups, Mamthematica, Tome 52, (75), 2, (2010), 177-184
W.-J. Liu, Fuzzy invariant subgroups and fuzzy ideals, Fuzzy Sets and Systems 8 (1982), no. 2, 133–139.
Mukherjee and P. Bhattacharya, Fuzzy normal subgroups and fuzzy cosets, Inform. Sci. 34 (1984), no. 3, 225–239.
V. Murali, Fuzzy points of equivalent fuzzy subsets, Inform. Sci. 158 (2004) 277-288.
P. M. Pu, Y.M. Liu, Fuzzy topology I, neighbourhood structure of a fuzzy point and Moore-Smith convergence, J. Math. Anal. Appl. 76 (1980) 571-599.
A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl. 35 (1971) 512-517.
J. S. Rose, A Course on Group Theory, Cambridge University Press, 1978.
M. Shabir, Y.B. Jun, Y. Nawaz, Characterizations of regular semigroups by (α,β)-fuzzy ideals, Comput. Math. Appl. , 59 (2010) 161-1
M. Shabir, Y.B. Jun and Y. Nawaz, Semigroups characterized by (∈,∈ ∨q)-fuzzy ideals, Comput. Math. Appl. , 60 (2010) 1473- 14
X. Yuan, C. Zhang, and Y. Ren, Generalized fuzzy groups and many-valued implications, Fuzzy Sets and Systems 138 (2003), no. 1, 205–211.
L.A. Zadeh, Fuzzy sets, Inform. and Control 8 (1965) 338-353.
L.A.Zadeh, The concept of a linguistic variable and its application to approximate reasoning-I, Information Sciences, 8, (1975), 199-2
J. Zhan, B. Davvaz and K.P. Shum, A new view of fuzzy hypernear-rings, Information Sciences, 178, (2008), 425-438.
J. Zhan, B. Davvaz and K.P. Shum, A new view of fuzzy hyperquasigroups, Journal of Intelligent & Fuzzy Systems: Applications in Engineering and Technology, 20(4, 5), 147-157.