INTRODUCTION
After the introduction of fuzzy sets by Zadeh (1965),
there have been a number of generalizations of this fundamental concept. Rosenfeld
(1971) introduced the concept of fuzzy group. The concept was discussed
further by many researchers, as Naraghi (2009) and Sulaiman
and Ahmad (2010). On the other hand many scholars have studied the antihomomorphism
in fuzzy groups and obtained many results. The concept of antihomomorphism
was discussed further by many researchers and obtained many results, such that
Abdullah and Jeyaraman (2010). The aim of this study
is to define the fuzzy subgroups with operators "mfuzzy subgroups", mnormal
fuzzy subgroups and study some of its properties.
PRELIMINARY DEFINITIONS
Definition (Pandiammal et al., 2010):
Let G be a group, M be a set, if:
• 
mx∈G ∀x∈G, m∈M 
• 
m (xy) = (mx)y = x (my) ∀x, y∈G, m∈M 
Then m is said to be a left operator of G, M is said to be a left operator set of G. G is said to be a group with operators. We use phrase " G is an Mgroup" in stead of a group with operators. If a subgroup of Mgroup G is also Mgroup, then it is said to be an Msubgroup of G.
Definition (Zadeh, 1965): Let X be a non empty
set, a fuzzy set μ is just a function from X onto [0, 1].
Definition (Rosenfeld, 1971): Let G be a group
and μ be a fuzzy set on G. μ is said to be a fuzzy subgroup of G,
if for x, y ∈ G:
μ (xy)≥min {μ (x), μ (y)}
μ (x ^{ 1}) = μ (x) 
Definition (Wu, 1981): A fuzzy subgroup μ
of a group G is called normal fuzzy subgroup if μ (x^{1}yx)≥μ
(y) ∀ x, y ∈ G.
Definition (Kim, 2003): If Φ is a homomorphism
from a group G_{1} to the group G_{2} and let μ be a fuzzy
subgroup of G_{2}, then the inverse image of μ under:
• 
Φ is (Φ^{1}(μ) (x) = μ (Φ
(x)) ∀x∈G. 
Mfuzzy subgroups
Definition (Muthuraj et al., 2010): Let G
be a group and μ be a fuzzy group of G, if μ (mx)≥μ (x) hold
for any x ∈ G, m ∈ M then μ is said to be a fuzzy group with
operator of G. We use the phrase "μ is an mfuzzy subgroup of G" instead
of a fuzzy subgroup with operator of G.
Example: Let μ be a fuzzy subset of an mgroup G and μ defined by:
μ is an mfuzzy subgroup of G.
Definition (Muthuraj et al., 2010): Let
G_{1}, G_{2} be an mgroup, Φ be a homomorphism from G_{1}
into G_{2}. If Φ (m x) = m. Φ (x) ∀ x∈G, m∈M,
then Φ is called mhomomorphism .
Proposition: Let G be an mgroup and μ, δ both be mfuzzy subgroups of G, then, μ∩δ is an mfuzzy subgroup of G.
Proof: Its clear that μ∩δ is a fuzzy subgroup of G. For any x∈G, m∈M.
(μ∩δ) (mx) = min {μ (mx), δ
(mx)}≥min {μ (x), δ (x)} = (μ∩δ) (x) 
Hence μ∩δ is an mfuzzy subgroup of G.
Corollary: The intersection of any family of mfuzzy subgroups is an mfuzzy subgroup.
Definition (Pandiammal et al., 2010):
Let G be an mgroup, μ is said to be an mnormal fuzzy subgroup of G, if
μ is not only an mfuzzy subgroup of G but also a normal fuzzy subgroup
of G.
Proposition: Let G be an mgroup, μ, δ both be mnormal fuzzy subgroups of G, then μ∩δ is an mnormal fuzzy subgroup of G.
Proof: It easy to know by above proposition μ∩δ is an mfuzzy subgroup of G, also we know μ∩δ is a normal fuzzy subgroup of G, hence μ∩δ is an mnormal fuzzy subgroup of G.
Corollary: The intersection of any family of mnormal fuzzy subgroups is an mnormal fuzzy subgroup.
Theorem: Let G_{1}, G_{2} be an mgroup and μ, δ be two mfuzzy subgroup of G_{2} and μ be mnormal fuzzy subgroup of δ . Let Φ be an mhomomorphism G_{1} into G_{2}, then Φ^{1 }(μ) is an mnormal fuzzy subgroup of Φ^{1 }(δ).
Proof: Clearly Φ^{1 }(δ ), Φ^{1 }(μ)
is an mfuzzy subgroups of G. It follows easily that:
Now:
Φ^{1 }(μ) (xyx^{¯1})
= μ (Φ (xyx^{¯1}) = μ (Φ(x) Φ (y)
(Φ (x)^{1})≥min {μ (Φ (y)), δ(Φ (x))}
= min {Φ^{1 }(μ) (y), Φ^{1 }(δ)
(x)};
∀x, y∈G 
Hence Φ^{1} (μ) is normal fuzzy subgroup of Φ^{1
}(δ).
Thus Φ^{1} (μ) is an mnormal fuzzy subgroup of Φ^{1
}(δ).
Theorem: Let Φ be an mhomomorphism from the mgroup G_{1} to the mgroup G_{2}, then:
• 
If μ is an mfuzzy subgroup of G_{2}, then Φ^{1}(μ)
is an mfuzzy subgroup of G_{1} 
• 
If μ is an mnormal fuzzy subgroup of G_{2}, then Φ^{1}(μ)
is an mnormal fuzzy subgroup of G_{1} 
Proof:
• 
Since Φ^{1}(μ) is an fuzzy subgroup of
G_{1}: 

(Φ^{¯1}(μ)) (mx) = μ (Φ(mx))
= μ(mΦ(x))≥μ(Φ(x)) = (Φ^{1}(μ))
(x) 
• 
Since we know Φ^{1}(μ) is normal fuzzy subgroup of
G_{1} and by above Φ^{1}(μ) is an mfuzzy subgroup,
hence Φ^{1}(μ) is an mnormal fuzzy subgroup of G_{1} 
Theorem: Let Φ be a homomorphism from the mgroup G_{1} into mgroup G_{2} and λ is an mfuzzy subgroup of G_{2}. Then λ o Φ is an mfuzzy subgroup of G_{1}.
Proof: Since λ is an m –fuzzy subgroup of G_{2}. Thus:
λ(m Φ(x))≥λ(Φ(x)) = (λoΦ)
(x) 
Therefore λoΦ is an mfuzzy subgroup.
Corollary: If μ is an mnormal fuzzy subgroup of G_{2},
then μoΦ is an mnormal fuzzy subgroup of G_{1}.
CONCLUSION
In this study, we studied the concept of fuzzy subgroup with operator and we used it to study some properties on the subgroup with operator like to introduce the concept normal fuzzy subgroup with operator and discuss the intersection operation. After that, we have studied on this topic the m homomorphism and the effect on the image and inverse image of fuzzy and normal fuzzy subgroups with operator.
In the next studies we will formulize the concept of fuzzy subgroups with two operators and applied it to some properties such as fuzzy cosets, conjugate, etc.