International Journal of Soft Computing, Mathematics and Control (IJSCMC)

International Journal of Soft Computing, Mathematics and Control (IJSCMC), Vol. 6, No.1, February 2017
DOI: 10.14810/ijscmc.2017.6101 1
A NEW APPROACH TO M(G)-GROUP SOFT UNION
ACTION AND ITS APPLICATIONS TO M(G)-GROUP
THEORY
P.Jeyaraman 1
& R.Nagarajan2
1
Assistant Professor, Department of Mathematics, Bharathiar University,Coimbatore-046.
2
Associate Professor, Department of Mathematics ,J J College of Engineering &
Technology Tiruchirappalli- 620009, Tamilnadu, India
ABSTRACT:
In this paper, we define a new type of M(G)-group action , called M(G)-group soft union(SU) action and
M(G)-ideal soft union(SU) action on a soft set. This new concept illustrates how a soft set effects on an
M(G)-group in the mean of union and inclusion of sets and its function as bridge among soft set theory, set
theory and M(G)-group theory. We also obtain some analog of classical M(G)- group theoretic concepts
for M(G)-group SU-action. Finally, we give the application of SU-actions on M(G)-group to M(G)-group
theory.
KEYWORDS:
soft set, M(G)-group, M(G)-group SU-action, M(G)--ideal SU-action, soft pre-image, soft anti-image, -
inclusion.
AMS MATHEMATICS SUBJECT CLASSIFICATION: 03E70,08E40,
1.INTRODUCTION:
Soft set theory as in [1, 2, 11, 14, 15, 16, 18, 25, 28] was introduced in 1999 by Molodtsov [22]
for dealing with uncertainties and it has gone through remarkably rapid strides in the mean of
algebraic structures. Maji et al. [19] presented some definitions on soft sets and based on the
analysis of several operations on soft sets Ali et al. [3] introduced several operations Moreover,
Atagun and Sezgin [4] defined the concepts of soft sub rings and ideals of a ring, soft subfields of
a field and soft sub modules of a module and studied their related properties with respect to soft
set operations. Furthermore, soft set relations and functions [5] and soft mappings [21] with many
related concepts were discussed. The theory of soft set has also a wide-ranging applications
especially in soft decision making as in the following studies: [6, 7, 23, 29] Sezgin et.al [25]
introduced a new concept to the literature of N-group called N-group soft intersection action.
Operations of soft sets have been studied by some authors, too. of soft sets and Sezgin and
Atagun [26] studied on soft set operations as well. In this paper, we define a new type of M(G)-
group action on a soft set, which we call M(G)- group soft union action and abbreviate as “M(G)-
group SU action “which is based on the inclusion relation and union of sets. Since M(G)- group

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SU-action gathers soft set theory and set theory and M(G) –group theory, it is useful in improving
the soft set theory with respect to M(G)- group structures. Based on this new notion, we then
introduce the concepts of M(G)-ideal SU-action and show that if M(G)-group SU-action over U.
Moreover, we investigate these notions with respect to soft image, soft pre-image and give their
applications to M(G)- group theory.
2.PRELIMINARIES:
In this section, we recall some basic notions relevant to M(G)- groups and soft sets. By a near-
ring, we shall mean an algebraic system (M(G),+, • ),
where
(N ) (M(G), +) forms a group (not necessarily abelian)
(N ) (M(G), •) forms a semi group and
(N ) (x + y)z = xz + yz for all x,y,z ∈ G.
Throughout this paper, M(G) will always denote right near-ring. A normal subgroup H of M(G) is
called a left ideal of M(G) if g(f+i)-gf ∈ H for all g,f ∈ M(G) and i ∈ I and denoted by
H⊲ℓM(G). For a near-ring M(G), the zero-symmetric part of M(G) denoted by M (G) is defined
by M (G) ={g∈ S / g0=0}.
Let G be a group, written additively but not necessarily abelian, and let M(G) be the set { f / f :
G→ G } of all functions from G to G. An addition operation can be defined on M(G) ; given f,g
in M(G), then the mapping f+g from G to G is given by (f+g)x = f(x) +g(x) for x in G.Then
(M(G), + ) is also group, which is abelian if and only if G is abelian. Taking the composition of
mappings as the product, M(G) becomes a near-ring.
Let G be a group. Then, under the operation below;
µ : M(G) x G → G
(f,a ) → fa
(G, µ ) is called M(G)-group. Let M(G) be a near-ring, G1 and G2 two M(G)-groups. Then ϕ : G1
→ G2 is called M(G)- homomorphism if for all x,y ε G1, for all g ε M(G),
(i) Φ (x+y) = ϕ(x) + ϕ(y)
(ii) Φ (gx) = g ϕ(x). It is denoted by G. Clearly M(G) itself is an M(G)-group by natural
operations.
For all undefined concepts and notions we refer to (24). From now on, U refers to on initial
universe, E is a set of parameters P(U) is the power set of U and A,B,C⊆ E.
2.1.Definition[22]: A pair (F,A) is called a soft set over U, where F is a mapping given by
F : A→P(U).
In other words, a soft set over U is a parameterized family of subsets of the universe U.
Note that a soft set (F, A) can be denoted by FA. In this case, when we define more than one soft
set in some subsets A, B, C of parameters E, the soft sets will be denoted by FA, FB, FC,
respectively. On the other case, when we define more than one soft set in a subset A of the set of

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parameters E, the soft sets will be denoted by FA,GA, HA, respectively. For more details, we refer
to [11,17,18,26,29,7].
2.2.Definition[6] :The relative complement of the soft set FA over U is denoted by Fr
A, where
Fr
A : A → P(U) is a mapping given as Fr
A(a) =U FA(a), for all a ∈ A.
2.3.Definition[6]: Let FA and GB be two soft sets over U such that A∩B ≠ ∅,. The restricted
intersection of FA and GB is denoted by FA ⋓ GB, and is defined as FA ⋓ GB =(H,C),
where C = A∩B and for all c ∈ C, H(c) = F(c)∩G(c).
2.4.Definition[6]: Let FA and GB be two soft sets over U such that A∩B ≠ ∅,. The restricted
union of FA and GB is denoted by FA∪R GB, and is defined as FA∪R GB = (H,C),where C = A∩B
and for all c ∈ C, H(c) = F(c)∪G(c).
2.5 Definition[12]: Let FA and GB be soft sets over the common universe U and be a function
from A to B. Then we can define the soft set (FA) over U, where (FA) : B→P(U) is a set
valued function defined by (FA)(b) =∪{F(a) | a ∈ A and (a) = b},
if −1
(b) ≠ ∅, = 0 otherwise for all b ∈ B. Here, (FA) is called the soft image of FA under .
Moreover we can define a soft set −1
(GB) over U, where −1
(GB) : A → P(U) is a set-valued
function defined by −1
(GB)(a) = G( (a)) for all a ∈ A. Then, −1
(GB) is called the soft pre
image (or inverse image) of GB under .
2.6.Definition[13]: Let FA and GB be soft sets over the common universe U and be a function
from A to B. Then we can define the soft set ⋆(FA) over U, where ⋆(FA) : B→P(U) is a set-valued
function defined by ⋆(FA)(b)=∩{F(a) | a ∈ A and (a) = b}, if −1
(b) ≠ ∅,
=0 otherwise for all b ∈ B. Here, ⋆(FA) is called the soft anti image of FA under .
2.7 Definition [8]: Let fA be a soft set over U and α be a subset of U. Then, lower α-inclusion of
a soft set fA, denoted by f α
A, is defined as f α
A = {x ∈A : fA(x) ⊆ α}
3. M(G) –GROUP SU-ACTION
In this section, we first define M(G)-group soft union action, abbreviated as M(G)-group SU-
action with illustrative examples. We then study their basic results with respect to soft set
operation.
3.1Definition: Let S be an M(G)- group and be a fuzzy soft set over U, then is called fuzzy
SU-action on M(G)- group over U if it satisfies the following conditions;
(FSUN-1) (x+y) ⊆ (x) U (y)
(FSUN-2) (-x) ⊆ (x)
(FSUN-3) (gx) ⊆ (x)
For all x,y ∈ S and g ∈M(G).

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3.1Example: Consider the near-ring module M(G) ={e,f,g,h},be the near-ring under the
operation defined by the following table:
Let G=M(G) be the set of functions and
and U=
f e
f e
/ f, e ∈ M(G)!,2⨉2 matrices with four terms, is the universal set .we construct a
soft set.
(e)=
# 0
# 0
, !, (f)= (g)= (h)= %&
0
0
' , &
( 0
( 0
' ,
3 0
0 0
*,
Then one can easily show that the soft set is a M(G)-group SU-action over U.
3.1 Proposition: Let be a fuzzy SU-action on M(G)- group over U. Then, (0)⊆ (x)
for all x ∈ +.
Proof: Assume that is fuzzy SU-action over U.Then, for all x∈S ,
(0)= (x-x)⊆ (x)∪ (-x)= (x)∪ (x)= (x).
3.1 Theorem: Let S be a fuzzy SU-action on M(G)- group and be a fuzzy soft set over U.
Then is SU-action of M(G)- group over U if and only if
(i) (x-y) ⊆ (x) ∪ (y)
(ii) (gx) ⊆ (x) for all x,y ∈ S and g ∈ M(G).
Proof: Suppose is a fuzzy SU-action on M(G)- group over U. Then, by definition-3.1,
(xy) ⊆ (y) and (x-y) ⊆ (x) ∪ (-y) = (x) ∪ (y) for all x,y ∈ S
Conversely, assume that (xy) ⊆ (y) and (x-y) ⊆ (x) ∪ (y) for all x,y ∈ S.
If we choose x=0,then (0-y)= (-y) ⊆ (0) ∪ (y) = (y) by proposition-3.1. Similarly
(y)= (−(-y)) ⊆ (-y),thus (-y) = (y) for all y ∈ S. Also, by assumption
(x-y) ⊆ (x) ∪ (-y) = (x) ∪ (y). This complete the proof.
3.2Theorem: Let be a fuzzy SU-action on M(G)- group over U.
(i) If (x-y)= (0) for any x,y ∈ S ,then (x) = (y).
(ii) (x-y)= (0) for any x,y ∈ S ,then (x) = (y).
+ e f g h
e
f
g
h
e f g h
f e f+g f+h
g g+h e g+h
h h+f h+g e
. e f g h
0
f
g
h
e f g h
f e f•g f •h
g g •h e g •h
h h •f h •g e

5
Proof: Assume that (x-y)= (0) for any x,y ∈ S ,then
(x) = (x-y+y) ⊆ (x-y) ∪ (y)
= (0) ∪ (y) = (y)
and similarly,
(y) = ((y-x)+x) ⊆ (y-x) ∪ (x)
= (−(y-x)) ∪ (x)
= (0) ∪ (x) = (x)
Thus, (x) = (y) which completes the proof .Similarly, we can show the result (ii).
It is known that if S is an M(G)- group, then (S,+)is a group but not necessarily abelian.
That is, for any x,y ∈ S, x+y needs not be equal to y+x. However, we have the following:
3.3Theorem: Let be fuzzy SU-action on M(G)- group over U and x ∈ S. Then,
(x) = (0)⇔ (x+y) = (y+x) = (y) for all y ∈ S .
Proof: Suppose that (x+y)= (y+x) = (y) for all y ∈ S. Then, by choosing y =0,
we obtain that (x) = (0).
Conversely, assume that (x) = (0). Then by proposition-3.1,we have
(0) = (x) ⊂ (y) , ∀ y ∈ S…………… (1)
Since is fuzzy SU-action on N-module over U, then
(x+y) ⊆ (x) ∪ (y) = (y), ∀ y ∈ S. Moreover, for all y ∈ S
(y) = ((-x)+x)+y) = (-x+(x+y) ) ⊆ (-x) ∪ (x+y)
= (x) ∪ (x+y)= (x+y)
Since by equation (1), (x) ⊆ (y) for all y ∈ S and x,y ∈ S ,implies that x+y ∈ S .Thus, it
follows that (x) ⊆ (x+y). So (x+y) = (y) for all y ∈ S.
Now, let x ∈ S. Then, for all x, y ∈ S
(y+x) = (y+x+(y-y))
= (y+(x+y)-y)
⊆ (y) ∪ (x+y) ∪ (y)
= (y) ∪ (x+y)= (y)
Since (x+y) = (y). Furthermore, for all y ∈ S
(y) = (y+(x-x))
= ((y+x)-x )
⊆ (y+x) ∪ (x)
= (y+x) by equation(1).
It follows that (y+x) = (y) and so (x+y)= (y+x)= (y) , for all y ∈ S ,which completes
the proof.
3.4Theorem: Let S be a near-field and be a fuzzy soft set over U. If (0)⊆ (1)= (x)
for all 0≠ x ∈ S, then it is fuzzy SU-action on M(G)- group over U.

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Proof: Suppose that (0)⊆ (1) = (x) for all 0≠ x ∈ S. In order to prove that it is
fuzzy SU-action on M(G)- group over U,it is enough to prove that (x-y)⊆ (x)∪ (y)
and (gx)⊆ (x).
Let x, y ∈ S. Then we have the following cases:
Case-1: Suppose that x≠ 0 and y=0 or x=0 and y≠ 0. Since S is a near-field, so it follows that
gx=0 and (gx) = (0).since (0)⊆ (x),for all x ∈ S ,so (nx) = (0) ⊆ (x),and
(nx) = (0) ⊆ (y). This imply (gx) ⊆ (x).
Case-2: Suppose that x≠ 0 and y≠0. It follows that nx≠0.Then, (nx) = (1) = (x) and
(gx) = (1) = (y),which implies that (gx)⊆ (x).
Case-3: suppose that x= 0 and y=0,then clearly (gx)⊆ (x).Hence (gx)⊆ (x),
for all x,y ∈ S.
Now, let x,y ∈ S. Then x-y=0 or x-y≠ 0. If x-y=0,then either x=y=0 or x≠ 0,y≠ 0 and x=y.
But, since (x-y) = (0) ⊆ (x) , for all x ∈ S, it follows that (x-y) = (0) ⊆ (x) ∪ (y).
If x-y≠0,then either x≠ 0,y≠ 0 and x≠y or x≠ 0 and y=0 or x=0 and y≠ 0.
Assume that x≠ 0,y≠ 0 and x≠y.This follows that
(x-y) = (1) = (x) ⊆ (x) ∪ (y).
Now, let x≠ 0 and y=0. Then (x-y) ⊆ (x) ∪ (y). Finally, let x=0 and y≠ 0.
Then, (x-y) ⊆ (x) ∪ (y). Hence (x-y) ⊆ (x) ∪ (y), for all x, y ∈ S.
Thus, is fuzzy SU-action on M(G)- group over U.
3.5Theorem: Let and 1 be two fuzzy SU-action on M(G)- group over U. Then ˄ 1 is fuzzy
soft SU-action on M(G)- group over U.
Proof: let (2 , 3 ), (2 , 3 ) ∈ S×T.Then
5˄1 672 ,3 8 − 72 ,3 89 = 5˄1 (2 :2 , 3 :3 )
= 5 (2 :2 ) ∩ 1 (3 :3 )
⊆ ( 5 (2 ) ∪ 5 (2 )) ∩ ( 1 (3 ) ∪ 1 (3 ))
=( 5 (2 ) ∪ 1 (3 )) ∩ ( 5 (2 ) ∪ 1 (3 ))
= 5˄172 ,3 8 ∩ 5˄172 ,3 8
and
5˄1 67( ,( 8, 72 ,3 89 = 5˄17; 2 ,; 3 8
= 5(( 2 ) ∩ 1 (( 3 )
⊆ 5(2 ) ∩ 1 (3 )
= 5˄172 ,3 8
Thus ˄ 1 is fuzzy SU-action on M(G)- group over U.
Note that ˅ 1 is not fuzzy SU-action on M(G)- group over U.

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3.2Example: Assume U = < is the universal set. Let S = = and H =
> >
? ?
/ >, ? ∈ = !
2× 2 matrices with = terms, be set of parameters. We define fuzzy SU-action on M(G)- group
5 over U= < by
5(0) = <
5(1) = @(1), (1 2), (1 3 2)B
5(2) =@(1), (1 2), (1 2 3), (1 3 2)B
We define fuzzy SU-action on N-module Cover U= < by
C
0 0
0 0
! = <
C
0 0
1 1
! =@(1), (1 2), (1 3 2)B
Then ˅ 1 is not fuzzy SU-action on M(G)- group over U.
3.2Definition: Let 5, (1 be fuzzy SU-action on M(G)- group over U. Then product of fuzzy
SU-action on M(G)- group 5 and (1 is defined as 5 × (1 = ℎI×J, where
ℎI×J(2, 3) = 5 (2) × (1(y) for all (x,y) ∈ S × T.
3.6Theorem: If 5 >;M (1 are fuzzy SU-action on M(G)- group over U. Then so is 5 × (1
over U×U.
Proof: By definition-3.2, let 5 × (1 =ℎI×J, where ℎI×J(2, 3) = 5 (2) × (1(y)
for all (x,y) ∈ S × T. Then for all (2 , 3 ), (2 , 3 ) ∈ S × T and 7; ,; 8=N×N.
ℎI×J 672 ,3 8 − 72 ,3 89 = ℎI×J (2 :2 , 3 :3 )
= 5 (2 :2 ) × (1 (3 :3 )
⊆ ( 5 (2 ) ∪ 5 (2 )) × ((1 (3 ) ∪ (1 (3 ))
=( 5 (2 ) × (1 (3 )) ∪ ( 5 (2 ) × (1 (3 ))
= ℎI×J72 ,3 8 ∪ ℎI×J72 ,3 8
ℎI×J 67( ,( 872 ,3 89 = ℎI×J7; 2 ,; 3 8
= 5(( 2 ) × (1 (( 3 )
⊆ 5(2 ) × (1 (3 )
= ℎI×J72 ,3 8
Hence 5 × (1 =ℎI×J is fuzzy SU-action on M(G)- group over U.
3.7Theorem: If 5 and ℎI are fuzzy SU-action on M(G)-group over U,then so is 5 ∩
N ℎI over U.
Let x, y ∈ s and n ∈ N then
( 5 ∩
N ℎI) (x-y) = 5 (x-y) ∩ ℎI (x-y)
⊆ ( 5 (2 ) ∪ 5 (3 )) ∩ (ℎ5 (2 ) ∪ ℎ5 (3 ))
= ( 5 (2 ) ∩ ℎ5(2 )) ∪ ( 5 (3 ) ∩ ℎ5 (3 ))
= ( 5 ∩
N ℎI) (2) ∪ ( 5 ∩
N ℎI)(3)

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( 5 ∩
N ℎI) (;x) = 5 (;2) ∩ ℎI(;2)⊆ 5 (2) ∩ ℎI(2) = ( 5 ∩
N ℎI) (x)
Therefore, ( 5 ∩
N ℎI) is fuzzy SU-action on M(G)- group over U.
4.SU-ACTION ON M(G)-IDEAL STRUCTURES
4.1 Definition : Let S be an M(G)- group and 5 be a fuzzy soft set over U. Then 5 is
called fuzzy SU-action on M(G)-ideal of S over U if the following conditions are satisfied:
(i) (2 + 3) ⊆ (2) ∪ (3)
(ii) (−2) = (2)
(iii) (2 + 3 − 2) ⊆ (3)
(iv) (((2 + 3) − (2) ⊆ (3) for all x, y ∈ S and g ∈ M(G).
Here, note that
(2 + 3) ⊆ (2) ∪ (3) and (−2) = (2) imply (2 − 3) ⊆ (2) ∪ (3)
4.1Example: Consider M(G) =@0, x, y, zB with the following tables
Let S=M(G) be the parameters and U= U ,dihedral group, be the universal set. We define a
fuzzy soft set over U by (0) = U , (2) = @#, ?, ?>B , (3) = @>, ?B , (V) = @?B.
Then, one can show that is fuzzy SU-action on M(G)-ideal of S over U.
4.2 Example: Consider the near –ring N=@0,1,2,3B with the following tables
Let S=M(G) be the set of parameters and U= =W
be the universal set. We define a fuzzy soft set
over U by (0) = @ 1, 2, 3, 5, 6, 7, 9, 10, 11, 17 B
(1) = (3) = @ 1, 3, 5, 7, 9, 11 B , (2) = @ 1, 5, 7, 9, 11 B
. 0 x y z
0
x
y
z
0 0 0 0
0 0 0 x
0 x y y
0 x y z
+ 0 x y z
0
x
y
z
0 x y z
x 0 z y
y z 0 x
z y x 0
+ 0 1 2 3
0
1
2
3
0 1 2 3
1 2 3 0
2 3 0 1
3 0 1 2
. 0 x y z
0
x
y
z
0 0 0 0
0 1 0 1
0 3 0 3
0 2 0 2

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Since (2. (3 + 1) − 2.3) = (2.1 − 2.3) = (3 − 3) = (0) ⊈ (1)
Therefore, is not fuzzy SU-action on M(G)-ideal over U. It is known that if M(G) is a zero-
symmetric near-ring, then every M(G)-ideal of S is also M(G)- group of S. Here, we have an
analog for this case.
4.1Theorem: Let M(G) be a zero- symmetric near-ring. Then, every fuzzy SU-action on M(G)-
ideal is fuzzy SU-action on M(G)- group over U.
Proof: Let be an fuzzy SU-action on M(G)-ideal on S over U. Since (g(x+y)-gx) ⊆ (y), for
all x,y ∈ S, and g ∈ M(G), in particular for x=0, it follows that (g(0+y)-g.0) = (ny-
0)= (y)⊆ (y). Since the other condition is satisfied by definition-4.1, is fuzzy SU-action
on M(G)-ideals of S over U.
4.2Theorem: Let be fuzzy SU-action on M(G)-ideal of S and 1 be fuzzy SU-action on M(G)-
ideal of T over U. Then ˄ 1 is fuzzy SU-action on M(G)-ideal of S×T over U.
4.3 Theorem : If is fuzzy SU-action on M(G)-ideal of S and 1 be fuzzy SU-action on M(G)-
ideal of T over U, then × 1 is fuzzy SU-action on M(G)-ideal over U×U.
4.4 Theorem : If and ℎ are two fuzzy SU-action on M(G)- group of S over U, then 5 ∩
N ℎI is
Fuzzy SU-action on M(G)-ideal over U.
5.APPLICATION OF FUZZY SU-ACTION ON M(G)- GROUP
In this section, we give the applications of fuzzy soft image, soft pre-image, lower
-inclusion of fuzzy soft sets and M(G)- group homomorphism with respect to fuzzy SU-action
on M(G)- group and M(G)-ideal.
5.1Theorem: If is fuzzy SU-action on M(G)-ideal of S over U, then +i
={x ∈S / (x)= (0)}
is a M(G)-ideal of S.
Proof: It is obvious that 0 ∈ +i
we need to show that (i) x-y ∈ +i
, (ii) s+x-s ∈ +i
and
(iii) g(s+x)-gs ∈ +i
for all x,y ∈ +i
and g∈M(G) and s∈S.
If x,y∈ +i
, then (x)= (y) = (0). By proposition-3.1,
(0) ⊆ (x-y), (0) ⊆ (s+x-s), and (0) ⊆ (n(s+x)-ns) for all x,y ∈ +i
Since is fuzzy SU-action on M(G)-ideal of S over U, then for all x,y ∈ +i
and g∈M(G) and
s∈S.
(i) (x-y) ⊆ (x)∪ (y) = (0).
(ii) (s+x-s)⊆ (x) = (0).
(iii) (g(s+x)-gs) ⊆ (x) = (0).
Hence (x-y) = (0), (s+x-s)= (0) and (g(s+x)-gs) = (0), for all x,y ∈ +i
and g∈M(G) and
s∈S. Therefore +i
is M(G)-ideal of S.

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5.2Theorem: Let be fuzzy soft set over U and be a subset of U such that ∅ ⊇ ⊇ (0). If
is fuzzy SU-action on M(G)-ideal over U, then ⊆l
is an M(G)-ideal of S.
Proof: Since (0) ⊆ , then 0 ∈ ⊆l
and ∅ ≠ ⊆l
⊇ S. Let x,y ∈ ⊆l
, then (x) ⊆ and
(y) ⊆ . We need to show that
(i) x-y ∈ ⊆l
(ii) s+x-s ∈ ⊆l
(iii) g(s+x)-ns ∈ ⊆l
for all x,y ∈ ⊆l
Since is fuzzy SU-action on M(G)-ideal over U, it follows that
(i) (x-y) ⊆ (x)∪ (y) ⊆ ∪ = ,
(ii) (s+x-s)⊆ (x) ⊆ and
(iii) (g(s+x)-gs) ⊆ (x) ⊆ . Thus, the proof is completed.
5.3 Theorem : Let and 1 be fuzzy soft sets over U and m be an M(G) -isomorphism from S to
T.If is fuzzy SU-action on M(G)-ideal of S over U, then m( ) is fuzzy SU-action on M(G)-
ideal of T over U.
Proof: Let n , n and g∈ M(G). Since χ is surjective, there exists o , o ∈ S such that χ(o ) = n
and χ(o )= n . Then
(χ ) (n - n ) = ∪ { (s) / s∈ S , χ(s) = n - n }
= ∪ { (s) / s∈ S , s = m:
(n - n ) }
= ∪ { (s) / s∈ S , s = m:
(m(o - o ))= o - o }
= ∪ { (o - o ) / op ∈ S , χ(op) = np , q =1,2,…}
⊆ ∪ { (o ) ∪ (o )) / op ∈ S , χ(op) = np , q =1,2,…}
=7r∪ @ (o )/o ∈ S , χ(o ) = n Bt ∪ r∪ @ (o )/o ∈ S , χ(o ) = n Bt8
= (χ( )) (n ) ∪ (χ( )) (n )
Also (χ ) (n + n -n ) = ∪ { (s) / s∈ S , χ(s) = n + n -n }
= ∪ { (s) / s∈ S , s = m:
(n + n -n ) }
= ∪ { (s) / s∈ S , s = m:
(m(o + o -o ))= o + o -o }
= ∪ { (o + o -o ) / op ∈ S , χ(op) = np , q =1,2,…}
⊆ ∪ @ (o )/o ∈ S , χ(o ) = n B
= (χ( )) (n )
Furthermore, (χ ) (g(n + n )-gn ) = ∪ { (s) / s∈ S , χ(s) = g(n + n )-gn }
= ∪ { (s) / s∈ S , s = m:
(g(n + n )-gn ) }
= ∪ { (s) / s∈ S , s = g(o + o )-go }
= ∪ { ((+ o )-go ) / op ∈ S , χ(op) = np , q =1,2,…}
⊆ ∪ @ (o )/o ∈ S , χ(o ) = n B
= (χ( )) (n ).
Hence χ( ) is fuzzy SU-action on M(G)-ideal of T over U.

11
5.4Theorem: Let and 1 be fuzzy soft sets over U and m be an M(G)--isomorphism from S to
T. If 1 is fuzzy SU-action on M(G)-ideal of T over U, then m:
( 1) is fuzzy SU-action on
M(G)-ideal of S over U.
Proof: Let o , o ∈S and g∈ M(G). Then
(m:
( 1)) (o - o ) = 1(χ (o - o ))
= 1(χ (o )-χ(o ))
⊆ 1(χ (o )) ∪ 1(χ (o ))
= (m:
( 1))( o ) ∪ (m:
( 1))( o ).
Also (m:
( 1)) (o +o − o ) = 1(χ (o +o − o ))
= 1(χ (o )+χ(o ) − m(o ))
⊆ 1(χ (o )) = (m:
( 1)) (o )
Furthermore, (m:
( 1)) (g(o +o ) − go ) = 1(χ (g(o +o ) − go ))
= 1(g(χ (o )+χ(o )) − gm(o ))
⊆ 1(χ (o )) = (m:
( 1)) (o )
Hence, (m:
( 1)) is fuzzy SU-action on M(G)-ideal of S over U.
CONCLUSION:
In this paper, we have defined a new type of N-module action on a fuzzy soft set, called fuzzy
SU-action on M(G)- group by using the soft sets. This new concept picks up the soft set theory,
fuzzy theory and M(G)- group theory together and therefore, it is very functional for obtaining
results in the mean of M(G)- group structure. Based on this definition, we have introduced the
concept of fuzzy SU-action on M(G)-ideal. We have investigated these notions with respect to
soft image, soft pre-image and lower -inclusion of soft sets. Finally, we give some application
of fuzzy SU-action on M(G)-ideal to M(G)- group theory. To extend this study, one can further
study the other algebraic structures such as different algebra in view of their SU-actions.
ACKNOWLEDGEMENT:
The authors are highly grateful to the referees for their valuable comments and suggestions for
improving papers.
REFERENCES
[1] Acar U., Koyuncu F., Tanay B., Soft sets and soft rings, Comput. Math. Appl., 59(2010),
3458-3463.
[2] Aktas. H., C. agman N., ˘ Soft sets and soft groups, Inform. Sci., 177(2007), 2726-2735.
[3] Ali M.I.,Feng F., Liu X., Min W.K., Shabir M., On some new operations in soft set theory,
Comput. Math. Appl., 57(2009), 1547-1553.
[4] Atagun A.O., Sezgin A., Soft substructures of rings, fields and modules, Comput. Math.
Appl., 61(3)(2011), 592-601.
[5] Babitha K.V., Sunil J.J., Soft set relations and functions, Comput. Math. Appl., 60(7)(2010),
1840-1849.

12
[6] Cagman N., Enginoglu S.,Soft matrix theory and its decision making, Comput. Math. Appl.,
59(2010), 3308-3314.
[7] Cagman N, Engino ˘ glu S., ˘ Soft set theory and uni-int decision making, Eur. J. Oper. Res.,
207(2010), 848-855.
[8] Cagman N., C¸ ıtak F., Aktas¸ H., ˘ Soft int-groups and its applications to group theory,
Neural Comput. Appl., DOI: 10.1007/s00521-011-0752-x.
[9] Cagman N., Sezgin A., Atag ˘ un A.O., ¨ Soft uni-groups and its applications to group theory,
(submitted).
[10] Cagman N., Sezgin A., Atag ˘ un A.O., ¨ α-inclusions and their applications to group theory,
(submitted). ˘
[11] F. Feng, W. Pedrycz, On scalar products and decomposition theorems of fuzzy soft sets,
Journal of Multivalued Logic and Soft Computing, 2015, 25(1), 45-80.
[12] F. Feng, J. Cho, W. Pedrycz, H. Fujita, T. Herawan, Soft set based association rule mining,
Knowledge-Based Systems, 111 (2016), 268-282
[13] Feng F., Li C., Davvaz B., Ali M.I., Soft sets combined with fuzzy sets and rough sets: a
tentative approach, Soft Comput., 14(6)(2010), 899-911.
[14] Jun Y.B., Soft BCK/BCI-algebras, Comput. Math. Appl., 56(2008), 1408 -1413.
[15] Jun Y.B., Park C.H., Applications of soft sets in ideal theory of BCK/ BCI-algebras, Inform.
Sci., 178(2008), 2466-2475.
[16] Jun Y.B., Lee K.J., Zhan J., Soft p-ideals of soft BCI-algebras, Comput. Math. Appl.,
58(2009), 2060-2068.
[17] Jun Y.B., Lee K.J., Park C.H., Soft set theory applied to ideals in d-algebras, Comput. Math.
Appl., 57(3)(2009), 367-378.
[18] Kazancı O., Yılmaz S¸., Yamak S., Soft sets and soft BCH-algebras, Hacet. J. Math. Stat.,
39(2)(2010), 205-217.
[19] Maji P.K., Biswas R., Roy A.R., Soft set theory, Comput. Math. Appl., 45(2003), 555-562.
[20] Maji P.K., Roy A.R., Biswas R., An application of soft sets in a decision making problem,
Comput. Math. Appl., 44(2002), 1077-1083.
[21] Majumdar P., Samanta S.K., On soft mappings, Comput. Math. Appl., 60 (9)(2010), 2666-
2672.
[22] Molodtsov D., Soft set theory-first results, Comput. Math. Appl., 37(1999), 19-31.

13
[23] Molodtsov D.A., Leonov V.Yu., Kovkov D.V., Soft sets technique and its application,
Nechetkie Sistemy i Myagkie Vychisleniya, 1(1)(2006), 8-39.
[24] Pilz G., Near-rings, North Holland Publishing Company, Amsterdam-New York-Oxford,
1983.
[25] Sezgin A., Atagun A.O., Ayg ¨ un E., ¨ A note on soft near-rings and idealistic soft near-
rings, Filomat., 25(1)(2011), 53-68.
[26] Sezgin A., Atagun A.O., ¨ On operations of soft sets, Comput. Math. Appl., 61(5)(2011),
1457-1467.
[27] Wendt G., On Zero Divisors in Near-Rings, Int. J. Algebra, 3(1)(2009), 21-32.
[28] Zhan J., Jun Y.B., Soft BL-algebras based on fuzzy sets, Comput. Math. Appl., 59(6)(2010),
2037-2046.
[29] Zou Y., Xiao Z., Data analysis approaches of soft sets under incomplete information, Knowl-
Based Syst., 21(2008), 941-945.
[30] J. Zhan, B. Yu, V. Fotea, Characterizations of two kinds of hemi rings based on probability
spaces, Soft Comput, 20 (2016), 637-648.

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