intuitionistic m-fuzzy groups - Soochow Journal of Mathematics

90 JIANMING ZHAN AND ZHISONG TAN
Theorem 2.10. Let G and G ′ be M-groups and f an M-homomorphism
from G onto G ′ .IfIFS A f =(α f A ,βf A
) is an intuitionistic M-fuzzy subgroup of
G, thenIFS A=(α A ,β A ) is an intuitionistic M-fuzzy subgroup of G ′ .
Proof. By Lemma 2.8, IFS A=(α A ,β A ) is an intuitionistic fuzzy subgroup
of G ′ . For any y ∈ G ′ and m ∈ M,
α f A (my)=
and
β f A (my)=
sup α A (z) ≥ sup α A (mx) (wherex ∈ f −1 (y))
z∈f −1 (my)
mx∈f −1 (my)
= sup α A (mx) =
f(mx)=my
inf β A(z) ≤
z∈f −1 (my)
= inf
f(mx)=my β A(mx) =
sup α A (mx) ≥
mf(x)=my
inf β A(mx)
mx∈f −1 (my)
inf β A(mx) ≤
mf(x)=my
sup α A (x) =α f A (y)
f(x)=y
(where x ∈ f −1 (y))
inf β A(x) =β f A (y).
f(x)=y
Hence IFS A=(α A ,β A ) is an intuitionistic M-fuzzy subgroup of G ′ .
Acknowledgments
The authors are very grateful to the referees for their advice.
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Department of Mathematics, Hubei Institute for Nationalities, Enshi, Hubei Province, 445000,
P.R. China.
E-mail: zhanjianming@hotmail.com