EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 11, No. 2, 2018, 517-536
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Soft Uni-Abel-Grassmann’s Groups

Aman Ullah1,∗, Faruk Karaaslan2, Imtiaz Ahmad1

1 Department of Mathematics, Faculty of Sciences, University of Malakand, Chakdara Dir
Lower, Pakistan
2 Department of Mathematics, Faculty of Sciences, Çankırı Karatekin University, 18100,
Çankırı, Türkiye

Abstract. In this paper, we define soft union AG-group (abbreviated as soft uni-AG-group). We
also define e-set and α-inclusion of soft uni-AG-groups, normal soft uni-AG-subgroups, conjugate
of soft uni-AG-groups and commutators of AG-groups. We investigate various properties of these
notions and provide a variety of relevant examples that are produced by a computer package GAP
to illustrate these notations.

2010 Mathematics Subject Classifications: 03Exx, 20Bxx, 03E20

Key Words and Phrases: Soft set, soft uni-AG-group, α-inclusion, conjugate normal soft uni-
AG-group, normal soft uni-AG-subgroup.

1. Introduction

In 1999, soft set theory was proposed by Molodtsov [1] as an alternative approach
to fuzzy set and intuitionistic fuzzy set theories. This study of Molodtsov provided a
general skeleton to researchers that naturally requires a study on algebraic structures.
After that, Maji et al. [2] defined set theoretical operations of soft sets. In 2010, Çağman
and Enginoğlu [3] redefined soft set operations in decision making problems. Ali et al.
[4], Sezgin and Atagün [5] studied some new operations on soft sets. The first study on
algebraic structures of soft sets was made by Aktaş and Çağman [6] in 2007. They defined
concept of soft groups. After Aktaş and Çağman’s study, studies related to algebraic
structures of soft sets increased rapidly. In 2010, Çağman et al. [7] defined concept of
soft int-groups with a similar approach to fuzzy group definition of Rosenfeld [8], and
obtained some properties of soft int-groups. Kaygısız [9] derived some new results on
soft int-groups. Sezgin [10] made some corrections for some problematic case related to
soft groups defined by Aktaş and Çağman, and they defined concepts of normalistic soft
group and normalistic soft group homomorphism. Sezgin et al. [11] gave definitions of

∗Corresponding author.

Email addresses: amanswt@uom.edu.pk (A. Ullah∗), fkaraaslan@karatekin.edu.tr (F. Karaaslan),
iahmad@uom.edu.pk (I. Ahmad)

http://www.ejpam.com 517 c© 2018 EJPAM All rights reserved.



A. Ullah, F. Karaaslan, I. Ahmad / Eur. J. Pure Appl. Math, 11 (2) (2018), 517-536 518

soft uni-groups and uni-soft normal subgroup of a group, and investigated their related
properties especially with respect to anti-image, α−inclusion of a soft set. Sezgin [12]
introduced the concept of soft intersection LA-semigroups (Abel-Grassman’s groupoids)
and studied various ideals in LA-semigroups such as left (right) ideals, bi-ideals, interior
ideals and quasi ideals by defining soft intersection product operations. The concept of
soft intersection Abel-Grassmann’s groups was defined by Ullah et al. [13] and related
properties were investigated. There are many studies on algebraic structures of soft sets,
some of them are as in following [14–18].

In this study, we introduce the notion of soft uni-AG-groups, and define some new con-
cepts related to soft uni-AG-groups such as e-set and α-inclusion of soft uni-AG-groups,
normal soft uni-AG-subgroups, conjugate of soft uni-AG-groups and commutators of AG-
groups. We support our definitions with examples to be more understandable. Further-
more, we obtain interrelations of these concepts.

Definition 1. [1] Let U be the universal set, E be the set of parameters and P (U) be the
power set of U . Then a soft set, A is a set of ordered pairs

A = {(ε, fA(ε)) : ε ∈ E } ,

where fA is a set valued function from E to P (U) i.e. fA : E → P (U). Note that if
fA(ε) = ∅, where ε ∈ E. Then (ε, fA(ε)) is not appeared in the set A. The set of all soft
sets over U is denoted by S(U).

Definition 2. [3] Let A,B ∈ S(U). Then

1. If fA(ε) = ∅ for all ε ∈ E, A is said to be a null soft set, denoted by φ.

2. If fA(ε) = U for all ε ∈ E, A is said to be absolute soft set, denoted by Û .

3. A is soft subset of B, denoted by A⊆̃B, if fA(ε) ⊆ fB(ε) for all ε ∈ E.

4. A=̃B, if A⊆̃B and B⊆̃A.

5. Soft union of A and B, denoted by A∪̃B, is a soft set over U and defined by

A∪̃B = {(ε, (fA∪̃fB) (ε)) : ε ∈ E }
= {(ε, (fA(ε) ∪ fB(ε))) : ε ∈ E } .

6. Soft intersection of A and B, denoted by A∩̃B, is a soft set over U and defined by

A∩̃B = {(ε, (fA∩̃fB) (ε)) : ε ∈ E }
= {(ε, (fA(ε) ∩ fB(ε))) : ε ∈ E } .



A. Ullah, F. Karaaslan, I. Ahmad / Eur. J. Pure Appl. Math, 11 (2) (2018), 517-536 519

Definition 3. [3] Let A,B ∈ S(U). Then, AND and OR operators of A and B is repre-
sented by A ∧B and A ∨B respectively, defined by

A ∧B =
{(

(ε, ε′), fA∧B(ε, ε′)
)

: ε, ε′ ∈ E
}

=
{(

(ε, ε′), fA(ε) ∩ fB(ε′)
)

: ε, ε′ ∈ E
}
,

and

A ∨B =
{(

(ε, ε′), fA∨B(ε, ε′)
)

: ε, ε′ ∈ E
}

=
{(

(ε, ε′), fA(ε) ∪ fB(ε′)
)

: ε, ε′ ∈ E
}
.

In the rest of this paper, G denotes an AG-group and e denotes the left identity of G unless
otherwise stated. An AG-group is a non-associative structure, in which commutativity and
associativity imply each other and thus AG-group become an abelian group if any one of
the property is allowed in AG-group. AG-group is a generalization of abelian group and
a special case of quasi-group. An AG-groupoid (or LA-semigroup) is a non-associative
groupoid in general, in which the left invertive law: (ab)c = (cb)a holds for all a, b, c ∈ G.
An AG-groupoid G is called an AG-group or left almost group (LA-group), if there exists
a unique left identity e in G (i.e. ea = a for all a ∈ G), and for all a ∈ G there exists
a−1 ∈ G such that aa−1 = a−1a = e. Nowadays, many researchers take keen interest to
fuzzify AG-groupoids and AG-groups; also they develop soft theory of AG-groupoids and
AG-groups [19–23]. An AG-group (G, ∗) can be easily obtained from an abelian group
(G1, ·) by:

a ∗ b = a−1 · b or a ∗ b = b · a−1 ∀ a, b ∈ G1.

It is easy to prove that in an AG-group G the right identity become the two sided identity,
and thus G with right identity become an abelian group. AG-group posses the property
of cancellativity like groups. A nonempty subset H of G is called an AG-subgroup of G,
if H itself is an AG-group under the same binary operation defined on G.

Various comparative properties of AG-groups and groups are explored in [24–26].
The following identities can be easily proved in an AG-group G.

Lemma 1. [24] Let e ∈ G, and a, b, c, d ∈ G, then

1. (ab)(cd) = (ac)(bd) (medial law).

2. a(bc) = b(ac).

3. (ab)(cd) = (db)(ca) (paramedial law).

4. (ab)(cd) = (dc)(ba).

5. (ab)−1 = a−1b−1.



A. Ullah, F. Karaaslan, I. Ahmad / Eur. J. Pure Appl. Math, 11 (2) (2018), 517-536 520

2. Soft Uni-AG-groups

In this section the basic definition of soft union AG-group (soft uni-AG-group) is given,
some of the basic results along with suitable examples are provided.

Definition 4. Let G be an AG-group and A ∈ S(U) be a soft set. Then, A is called soft
uni-AG-group over U if

1. fA(ab) ⊆ fA(a) ∪ fA(b) ∀ a, b ∈ G,

2. fA(a−1) = fA(a) ∀ a ∈ G.

The set of all soft uni-AG-group over U is symbolically represented by S∪AG(U).

Example 1. Consider a non-associative AG-group G = {0, 1, 2} of order 3 with left
identity 0, defined in the following table:

. 0 1 2

0 0 1 2
1 2 0 1
2 1 2 0

Let A be a soft set over U = {u1, u2, . . . , u10}, defined by

A = {(0, fA(0)) , (1, fA(1)) , (2, fA(2))}
= {(0, {u1, u3}) , (1, {u1, u3, u5}) , (2, {u1, u3, u5})} .

Then, A is a soft uni-AG-group.

Example 2. Consider a non-associative AG-group G = {0, 1, 2, 3} of order 4 with left
identity 0 defined by:

. 0 1 2 3

0 0 1 2 3
1 3 0 1 2
2 2 3 0 1
3 1 2 3 0

Let A be a soft set over U = Z, defined by

A = {(0, fA(0)) , (1, fA(1)) , (3, fA(3)) , (4, fA(4))}
= {(0, {1, 3}) , (1, {1, 3, 5, 7}) , (2, {1, 3, 5, 7}) , (3, {1, 3, 5, 7})} .

Then, one can easily show that A ∈ S∪AG(U).

Lemma 2. Let A ∈ S∪AG(U). Then, fA(e) ⊆ fA(a) for all a ∈ G.



A. Ullah, F. Karaaslan, I. Ahmad / Eur. J. Pure Appl. Math, 11 (2) (2018), 517-536 521

Proof. Since A ∈ S∪AG(U). Then, for all a ∈ G,

fA(e) = fA(aa−1)

⊆ fA(a) ∪ fA(a−1)

= fA(a) ∪ fA(a)

= fA(a).

Hence, fA(e) ⊆ fA(a) for all a ∈ G.

Lemma 3. Let A ∈ S∪AG(U). Then fA(ab) = fA(ba) for all a, b ∈ G.

Proof. Let A ∈ S∪AG(U). Then for all a, b ∈ G,

fA(ab) = fA((ea)b)

= fA ((ba)e) (by the left invertive law)

⊆ fA(ba) ∪ fA(e)

= fA(ba) (by Lemma 2)

⇒ fA(ab) ⊆ fA(ba).

Similarly, it can be shown that fA(ba) ⊆ fA(ab). Hence, fA(ab) = fA(ba) for all a, b ∈ G.

Theorem 1. A soft set A over U is a soft uni-AG-group over U if and only if fA(ab−1) ⊆
fA(a) ∪ fA(b) for all a, b ∈ G.

Proof. Suppose A ∈ S∪AG(U). Then, for all a, b ∈ G,

fA(ab−1) ⊆ fA(a) ∪ fA(b−1)

= fA(a) ∪ fA(b)

⇒ fA(ab−1) ⊆ fA(a) ∪ fA(b).

Conversely, suppose that for all a, b ∈ G,

fA(ab−1) ⊆ fA(a) ∪ fA(b).

Then by choosing a = e we get

fA(b−1) ⊆ fA(b). (by Lemma 2)

Thus,
fA(b) = fA((b−1)−1) ⊆ fA(b−1).

Consequently, fA(b) = fA(b−1) ∀ b ∈ G. Now,

fA(ab) = fA(a(b−1)−1)

⊆ fA(a) ∪ fA(b−1)

= fA(a) ∪ fA(b).

Hence, A ∈ S∪AG(U).



A. Ullah, F. Karaaslan, I. Ahmad / Eur. J. Pure Appl. Math, 11 (2) (2018), 517-536 522

Lemma 4. Let A ∈ S∪AG(U). Then, for all a, b ∈ G, fA(ab) = fA(b) if and only if
fA(a) = fA(e).

Proof. Let A ∈ S∪AG(U) and fA(ab) = fA(b) for all a, b ∈ G. By choosing b = e we
get

fA(ae) = fA(e)

⇒ fA(ea) = fA(e) (by Lemma 3)

⇒ fA(a) = fA(e).

Conversely, suppose that fA(a) = fA(e) ∀ a ∈ G. Then,

fA(ab) ⊆ fA(a) ∪ fA(b)

= fA(e) ∪ fA(b)

= fA(b) (by Lemma 2)

This implies that
fA(ab) ⊆ fA(b). (1)

Also,

fA(b) = fA(eb) = fA((a−1a)b)

= fA
(
(ba)a−1

)
(by the left invertive law)

⊆ fA(ba) ∪ fA(a−1)

= fA(ab) ∪ fA(a) (by Lemma 3)

= fA(ab) ∪ fA(e)

= fA(ab). (by Lemma 2)

This implies that
fA(b) ⊆ fA(ab). (2)

Consequently, from Equation (1) and (2) we get, fA(ab) = fA(b).

Lemma 5. Let A ∈ S∪AG(U). Then fA(a) = fA(b), if fA(ab−1) = fA(e) for all a, b ∈ G.

Proof. Let A ∈ S∪AG(U) such that fA(ab−1) = fA(e). Then, for all a, b ∈ G

fA(a) = fA(e · a) = fA((bb−1)a)

= fA
(
(ab−1)b

)
(by the left invertive law)

⊆ fA(ab−1) ∪ fA(b)

= fA(e) ∪ fA(b)

= fA(b). (by assumption and Lemma 2)

Thus
fA(a) ⊆ fA(b). (3)



A. Ullah, F. Karaaslan, I. Ahmad / Eur. J. Pure Appl. Math, 11 (2) (2018), 517-536 523

And

fA(b) = fA(b−1) = fA(e · b−1) = fA((a−1a)b−1)

= fA
(
(b−1a)a−1

)
(by the left invertive law)

⊆ fA(b−1a) ∪ fA(a−1)

= fA(ab−1) ∪ fA(a) (by Lemma 3)

= fA(a). (by Lemma 2)

Thus
fA(b) ⊆ fA(a). (4)

Hence, fA(a) = fA(b) for all a, b ∈ G using Equations (3) and (4).

Theorem 2. Let A,B ∈ S∪AG(U). Then, A ∨B ∈ S∪AG(U).

Proof. Let (x1, y1), (x2, y2) ∈ G1 ×G2. Then, by Definition 3 and Theorem 1,

(fA ∨ fB)
(
(x1, y1) · (x2, y2)−1

)
= (fA ∨ fB)

(
(x1, y1) · (x−12 , y−12 )

)
= (fA ∨ fB)

(
x1x

−1
2 , y1y

−1
2

)
= fA(x1x

−1
2 ) ∪ fB(y1y

−1
2 )

⊆
(
fA(x1) ∪ fA(x−12 )

)
∪
(
fB(y1) ∪ fB(y−12 )

)
= (fA(x1) ∪ fA(x2)) ∪ (fB(y1) ∪ fB(y2))

= (fA(x1) ∪ fB(y1)) ∪ (fA(x2) ∪ fB(y2))

= (fA ∨ fB) (x1, y1) ∪ (fA ∨ fB) (x2, y2).

Therefore, A ∨B ∈ S∪AG(U).

The following counter example shows that A ∧ B of any two soft sets A and B may not
be a soft uni-AG-group.

Example 3. Consider a non-associative AG-group G = {0, 1, 2, 3} of order 4 with left
identity 0 defined by:

. 0 1 2 3

0 0 1 2 3
1 1 0 3 2
2 3 2 1 0
3 2 3 0 1

Let A and B be any two soft sets over U = Z10 as follow:

A = {(0, fA(0)) , (1, fA(1)) , (2, fA(3)) , (3, fA(4))}
= {(0, {0, 1, 2}) , (1, {0, 1, 2, 3, 4, 5}) , (2, {0, 1, 2, 3, 4, 5, 6, 7}) , (3, {0, 1, 2, 3, 4, 5, 6, 7})} .



A. Ullah, F. Karaaslan, I. Ahmad / Eur. J. Pure Appl. Math, 11 (2) (2018), 517-536 524

B = {(0, fB(0)) , (1, fB(1)) , (2, fB(3)) , (3, fB(4))}
= {(0, {5, 6}) , (1, {5, 6, 7, 8}) , (2, {5, 6, 7, 8, 9, 10}) , (3, {5, 6, 7, 8, 9, 10})} .

It is clear that both A,B ∈ S∪AG(U). Now, take

(fA ∧ fB)
(
(1, 1) · (0, 2)−1

)
= (fA ∧ fB) ((1, 1) · (0, 3))

(fA ∧ fB) (1 · 0, 1 · 3)

= (fA ∧ fB) (1, 2)

= (fA) (1) ∧ (fB) (2) = {5},

and
(fA ∧ fB) (1, 1) ∪ (fA ∧ fB) (0, 2) = φ ∪ φ = φ,

this implies that

(fA ∧ fB)
(
(1, 1) · (0, 2)−1

)
 (fA ∧ fB) (1, 1) ∩ (fA ∧ fB) (0, 2) .

Hence, A ∧B /∈ S∪AG(U).

Definition 5. Let A,B ∈ S∪AG(U) on AG-groups G1 and G2 respectively. Then, the
product of A and B is denoted by A×B and is defined by

A×B = {((a, b), (fA×B) (a, b)) ∀ (a, b) ∈ G1 ×G2}
= {((a, b), (fA(a)× fB(b))) ∀ (a, b) ∈ G1 ×G2} .

Example 4. Let U = Z10 be a universal set, and G1 = {a, b, c, d} and G2 = {x, y, z} are
AG-groups of order 4 and 3 defined in the following tables (i) and (ii) respectively:

. a b c d . x y z
a d a b c x x y z
b c d a b y z x y
c b c d a z y z x
d a b c d

(i) (ii)

Let A,B ∈ S∪AG(U) on AG-groups G1 and G2 respectively defined by:

fA(a) = {0, 1, 2} = fA(c), fA(b) = {0, 1}, fA(d) = {0},

and
fB(x) = {0} , fB(y) = {0, 1} = fB(z).



A. Ullah, F. Karaaslan, I. Ahmad / Eur. J. Pure Appl. Math, 11 (2) (2018), 517-536 525

Then

A×B = {(a, b), (fA(a)× fB(b)) ∀ (a, b) ∈ G1 ×G2} ,
= {{(a, x) , ((0, 0) , (1, 0) , (2, 0))} , {(a, y) , ((0, 0) , (0, 1) , (1, 0) , (1, 1), (2, 0), (2, 1))} ,
{(a, z) , ((0, 0) , (0, 1) , (1, 0) , (1, 1), (2, 0), (2, 1))} , {(b, x) , ((0, 0) , (1, 0))} ,
{(b, y) , ((0, 0) , (0, 1) , (1, 0), (1, 1))} , {(b, z) , ((0, 0) , (0, 1) , (1, 0), (1, 1))} ,
{{(c, x) , ((0, 0) , (1, 0) , (2, 0))} , {(c, y) , ((0, 0) , (0, 1) , (1, 0) , (1, 1), (2, 0), (2, 1))} ,
{(c, z) , ((0, 0) , (0, 1) , (1, 0) , (1, 1), (2, 0), (2, 1))} , {(d, x), ((0, 0))} ,
{(d, y) , ((0, 0) , (0, 1))} , {(d, z) , ((0, 0) , (0, 1))}}.

Theorem 3. Let A,B ∈ S∪AG(U) with respect to AG-groups G1 and G2. Then A×B ∈
S∪AG(U × U).

Proof. For any (x1, y1), (x2, y2) ∈ G1 ×G2,

(fA×B) ((x1, y1), (x2, y2)
−1) = (fA×B) ((x1, y1), (x

−1
2 , y−12 ))

= (fA×B) ((x1x
−1
2 , y1y

−1
2 )

= fA(x1x
−1
2 )× fB(y1y

−1
2 ) (by Definition 5)

⊆ (fA(x1) ∪ fA(x2))× (fB(y1) ∪ fB(y2))

= (fA(x1)× fB(y1)) ∪ (fA(x2)× fB(y2))

= (fA×B) (x1, y1) ∪ (fA×B) (x2, y2).

Hence, A×B ∈ S∪AG(U × U).

Theorem 4. Let A,B ∈ S∪AG(U), then A∪̃B ∈ S∪AG(U).

Proof. Since A,B ∈ S∪AG(U). Therefore, A∪̃B 6= ∅. For any a, b ∈ A∪̃B, we have

(fA∪̃fB) (ab−1) = fA(ab−1) ∪ fB(ab−1) (by Definition 2-(v))

⊆ (fA(a) ∪ fA(b)) ∪ (fB(a) ∪ fB(b))

= (fA(a) ∪ fB(a)) ∪ (fA(b) ∪ fB(b))

= (fA∪̃fB) (a) ∪ (fA∪̃fB) (b).

Hence, A∪̃B ∈ S∪AG(U).

The following counter example, depicts that A∩̃B /∈ S∪AG(U) for any A,B ∈ S∪AG(U).

Example 5. Let G = {0, 1, 2, 3, 4, 5, 6, 7, 8} be an AG-group of order 9 defined in the
following table:



A. Ullah, F. Karaaslan, I. Ahmad / Eur. J. Pure Appl. Math, 11 (2) (2018), 517-536 526

· 0 1 2 3 4 5 6 7 8

0 0 1 2 3 4 5 6 7 8
1 2 0 1 4 5 3 7 8 6
2 1 2 0 5 3 4 8 6 7
3 7 6 8 0 2 1 5 3 4
4 6 8 7 1 0 2 4 5 3
5 8 7 6 2 1 0 3 4 5
6 4 3 5 8 6 7 0 2 1
7 3 5 4 7 8 6 1 0 2
8 5 4 3 6 7 8 2 1 0

Let A,B ∈ S∪AG(Z10), defined by

fA(0) = ∅, fA(1) = {0, 1} = fA(2),

fA(3) = {0, 1, 2, 3, 4, 5, 6} = fA(4) = fA(5) = fA(6) = fA(7) = fA(8),

and

fB(0) = ∅, fB(3) = {0, 1, 2, 3} = fB(7),

fB(1) = {0, 1, 2, 3, 4, 5, 6, 7, 8} = fB(2) = fB(4) = fB(5) = fB(6) = fB(7) = fB(8),

It is clear that

(fA∩̃fB)
(
2 · 3−1

)
= (fA∩̃fB) (5) = fA(5) ∩ fB(5) = {0, 1, 2, 3, 4, 5, 6} , (5)

and

((fA∩̃fB) (2)) ∪ ((fA∩̃fB) (3)) = (fA(2) ∩ fB(2)) ∪ (fA(3) ∩ fB(3))

= {0, 1, 2, 3},

this implies that
((fA∩̃fB) (2)) ∪ ((fA∩̃fB) (3)) = {0, 1, 2, 3}. (6)

From Equations (5) and (6) it is clear that

(fA∩̃fB)
(
2 · 3−1

)
* ((fA∩̃fB) (2)) ∪ ((fA∩̃fB) (3)) .

Hence, A∩̃B /∈ S∪AG(U).

Definition 6. Let H be an AG-subgroup of an AG-group G. Then a soft subset B over H
is called a soft uni-AG-subgroup of a soft subset A over G if B is a nonempty soft subset
of A. We denote this by B≤̃A.



A. Ullah, F. Karaaslan, I. Ahmad / Eur. J. Pure Appl. Math, 11 (2) (2018), 517-536 527

Example 6. Let U = Z10 be the universal set and G be any AG-group of order 9 defined
as in Example 5. Define a soft uni-AG-group A as follows:

fA(0) = {0, 1, 2}, fA(3) = {0, 1, 2, 3, 4} = fA(7),

fA(1) = {0, 1, 2, 3, 4, 5, 6} = fA(2) = fA(4) = fA(5) = fA(6) = fA(8).

Let H1 = {0, 3, 7} and H2 = {0, 1, 2} be two AG-subgroups of G. Define soft uni-AG-
groups B and C over U , w. r. t. H1 and H2 respectively as follow:

B = {(0, {0, 1}) , (3, {0, 1, 2}) , (7, {0, 1, 2})} ,

and
C = {(0, {0, 2}) , (1, {0, 2, 4}) , (2, {0, 2, 4})} .

As B⊆̃A and C⊆̃A. Therefore, B≤̃A and C≤̃A.

Theorem 5. Let B≤̃A and C≤̃A. Then, B∪̃C≤̃A.

Proof. Since, B≤̃A and C≤̃A. Therefore, B∪̃C 6= ∅. Let x, y ∈ B∪̃C. Then by
Theorem 1

(fB∪̃fC) (xy−1) =
(
(fB∪̃C) (xy−1)

)
= fB(xy−1) ∪ fC(xy−1)

⊆ (fB(x) ∪ fB(y)) ∪ (fC(x) ∪ fC(y))

= (fB(x) ∪ fC(x)) ∪ (fB(y) ∪ fC(y))

= fB∪̃C(x) ∪ fB∪̃C(y)

= (fB∪̃fC)(x) ∪ (fB∪̃fC)(y).

Hence, B∪̃C≤̃A.

Theorem 6. Let {Bi : i ∈ I} ≤̃A for all i ∈ I. Then ∪̃
i∈I
Bi≤̃A.

Proof. Since, {Bi : i ∈ I} ≤̃A for all i ∈ I. Therefore, ∪
i∈I
Bi 6= ∅. Let x, y ∈ ∪

i∈I
Bi.

Then by Theorem 1 we get(
∪̃
i∈I
fBi

)
(xy−1) =

((
f ∪̃
i∈I

Bi

)
(xy−1)

)
= ∪

i∈I

(
fBi(xy

−1) : i ∈ I
)

⊆ ∪
i∈I

((fBi(x) ∪ fBi(y)) : i ∈ I)

=

(
∪
i∈I

(fBi(x) : i ∈ I)

)
∪
(
∪
i∈I

(fBi(y) : i ∈ I)

)



A. Ullah, F. Karaaslan, I. Ahmad / Eur. J. Pure Appl. Math, 11 (2) (2018), 517-536 528

=

((
f ∪̃
i∈I

Bi

)
(x)

)
∪
((

f ∪̃
i∈I

Bi

)
(y)

)
.

Hence, ∪̃
i∈I
Bi≤̃A.

The following counter example clearly shows that B∩̃C�̃A.

Example 7. From, Example 5, we have

(fB∩̃fC) (4 · 4−1) = (fB∩̃C) (4 · 4) = (fB∩̃C) (0) = fB(0) ∩ fC(0) = {0}, (7)

and

((fB∩̃fC) (4)) ∪ ((fB∩̃fC) (4)) = (fB(4) ∩ fC(4)) ∪ (fB(4) ∩ fC(4))

= ∅,

this implies that
((fB∩̃fC) (4)) ∪ ((fB∩̃fC) (4)) = ∅. (8)

By Equations (7) and (8), we get

(fB∩̃fC) (4 · 4−1)  ((fB∩̃fC) (4)) ∪ ((fB∩̃fC) (4)) .

Hence, B∩̃C�̃A.

3. Conjugate Soft Uni-AG-groups

Definition 7. Let A ∈ S∪AG(U) and x ∈ G. Then Ax is called conjugate soft uni-
AG-group of A (with respect to x) denoted by Ax

c∼ A, and is given by

fAx(g) = fA
(
(xg)x−1

)
, for all g ∈ G.

Remark 1. It is noted that a conjugate soft uni-AG-group may or may not be a soft-uni-
AG-group.

Example 8. Consider an AG-group G of order 6 defined by

· 0 1 2 3 4 5

0 0 1 2 3 4 5
1 5 0 1 2 3 4
2 4 5 0 1 2 3
3 3 4 5 0 1 2
4 2 3 4 5 0 1
5 1 2 3 4 5 0



A. Ullah, F. Karaaslan, I. Ahmad / Eur. J. Pure Appl. Math, 11 (2) (2018), 517-536 529

Let A ∈ S∪AG(Z), defined as follows:

fA(0) = {−1, 0, 1} ,
fA(2) = {−2,−1, 0, 1, 2} = fA(4),

fA(1) = {−3,−2,−1, 0, 1, 2, 3} = fA(3) = fA(5).

The conjugates soft uni-AG-groups of A is given by:

fA0(0) = fA3(0) = fA(0) = {−1, 0, 1} ,
fA0(1) = fA3(1) = fA(5) = {−3,−2,−1, 0, 1, 2, 3} ,
fA0(2) = fA3(2) = fA(4) = {−2,−1, 0, 1, 2} ,
fA0(3) = fA3(3) = fA(3) = {−3,−2,−1, 0, 1, 2, 3} ,
fA0(4) = fA3(4) = fA(2) = {−2,−1, 0, 1, 2} ,
fA0(5) = fA3(5) = fA(1) = {−3,−2,−1, 0, 1, 2, 3} .

fA1(0) = fA4(0) = fA(2) = {−2,−1, 0, 1, 2} ,
fA1(1) = fA4(1) = fA(1) = {−3,−2,−1, 0, 1, 2, 3} ,
fA1(2) = fA4(2) = fA(0) = {−1, 0, 1} ,
fA1(3) = fA4(3) = fA(5) = {−3,−2,−1, 0, 1, 2, 3} ,
fA1(4) = fA4(4) = fA(4) = {−2,−1, 0, 1, 2} ,
fA1(5) = fA4(5) = fA(3) = {−3,−2,−1, 0, 1, 2, 3} .

fA2(0) = fA5(0) = fA(4) = {−2,−1, 0, 1, 2} ,
fA2(1) = fA5(1) = fA(3) = {−3,−2,−1, 0, 1, 2, 3} ,
fA2(2) = fA5(2) = fA(2) = {−2,−1, 0, 1, 2} ,
fA2(3) = fA5(3) = fA(1) = {−3,−2,−1, 0, 1, 2, 3} ,
fA2(4) = fA5(4) = fA(0) = {−1, 0, 1} ,
fA2(5) = fA5(5) = fA(5) = {−3,−2,−1, 0, 1, 2, 3} .

A1 and A2 are conjugate soft uni-AG-groups but are not soft uni-AG-groups over Z, as

fA1(2 · 2) = fA1(0) = {−2,−1, 0, 1, 2}  fA1(2) ∪ fA1(2) = {−1, 0, 1} ,

and
fA2(4 · 4) = fA2(0) = {−2,−1, 0, 1, 2}  fA2(4) ∪ fA2(4) = {−1, 0, 1} .

Definition 8. Let A ∈ S∪AG(U). Then A is called a normal soft uni-AG-subgroup
over U if

fAx(y) = fA
(
(xy)x−1

)
= fA(y) ∀ x, y ∈ G.

In other words A is a normal soft uni-AG-subgroup over U , if A is self-conjugate soft
uni-AG-group.



A. Ullah, F. Karaaslan, I. Ahmad / Eur. J. Pure Appl. Math, 11 (2) (2018), 517-536 530

The set of all normal soft uni-AG-subgroups over U is represented by NS∪AG(U).

Example 9. Let G be an AG-group of order 6 defined as in Example 8. Let A ∈ S∪AG(Z),
defined by

fA(0) = {−1, 0, 1} = fA(2) = fA(4),

fA(1) = {−2,−1, 0, 1, 2} = fA(3) = fA(5).

The conjugates soft uni-AG-groups of A are given by:

fA0(0) = fA3(0) = fA(0) = {−1, 0, 1} ,
fA0(1) = fA3(1) = fA(5) = {−2,−1, 0, 1, 2} ,
fA0(2) = fA3(2) = fA(4) = {−1, 0, 1} ,
fA0(3) = fA3(3) = fA(3) = {−2,−1, 0, 1, 2} ,
fA0(4) = fA3(4) = fA(2) = {−1, 0, 1} ,
fA0(5) = fA3(5) = fA(1) = {−2,−1, 0, 1, 2} .

fA1(0) = fA4(0) = fA(2) = {−1, 0, 1} ,
fA1(1) = fA4(1) = fA(1) = {−2,−1, 0, 1, 2} ,
fA1(2) = fA4(2) = fA(0) = {−1, 0, 1} ,
fA1(3) = fA4(3) = fA(5) = {−2,−1, 0, 1, 2} ,
fA1(4) = fA4(4) = fA(4) = {−1, 0, 1} ,
fA1(5) = fA4(5) = fA(3) = {−2,−1, 0, 1, 2} .

fA2(0) = fA5(0) = fA(4) = {−1, 0, 1} ,
fA2(1) = fA5(1) = fA(3) = {−2,−1, 0, 1, 2} ,
fA2(2) = fA5(2) = fA(2) = {−1, 0, 1} ,
fA2(3) = fA5(3) = fA(1) = {−2,−1, 0, 1, 2} ,
fA2(4) = fA5(4) = fA(0) = {−1, 0, 1} ,
fA2(5) = fA5(5) = fA(5) = {−2,−1, 0, 1, 2} .

Hence, A ∈ NS∪AG(Z), as A is self conjugate soft uni-AG-subgroup.

Lemma 6. Let A ∈ NS∪AG(U). Then for all x, y ∈ G, the following assertions are
equivalent:

1. fA
(
(xy)x−1

)
= fA(y),

2. fA
(
(xy)x−1

)
⊇ fA(y),

3. fA
(
(xy)x−1

)
⊆ fA(y).



A. Ullah, F. Karaaslan, I. Ahmad / Eur. J. Pure Appl. Math, 11 (2) (2018), 517-536 531

Proof. (i)⇒ (ii): Obvious.
(ii)⇒ (iii): Assume that (ii) holds. Consider

fA
(
(xy)x−1

)
⊆ fA

((
x−1

(
(xy)x−1

)) (
(x−1)−1

))
= fA

((
x−1

(
(xy)x−1

))
x
)

= fA
((
x
(
(xy)x−1

))
x−1

)
(by the left invertive law)

= fA
((

(xy)
(
xx−1

))
x−1

)
(by Lemma 1-(ii))

= fA
(
((xy) e)x−1

)
= fA

(
((ey)x)x−1

)
(by the left invertive law)

= fA
(
(yx)x−1

)
= fA

((
x−1x

)
y
)

(by the left invertive law)

= fA (ey) = fA(y)

⇒ fA
(
(xy)x−1

)
⊆ fA(y) ∀ x, y ∈ G.

(iii)⇒ (i): Assume that (iii) holds. Consider,

fA
(
(xy)x−1

)
⊇ fA

((
x−1

(
(xy)x−1

)) ((
x−1

)−1))
= fA(y), as in the proof (ii)⇒ (iii)

⇒ fA
(
(xy)x−1

)
⊇ fA(y) ∀ x, y ∈ G.

Consequently, fA
(
(xy)x−1

)
⊆ fA(y) ⊆ fA

(
(xy)x−1

)
. Hence, fA

(
(xy)x−1

)
= fA(y).

Theorem 7. Let A ∈ S∪AG(U). Then A ∈ NS∪AG(U) if and only if fA([x, y]) ⊆
fA(x) ∀ x, y ∈ G, where [x, y] = xy · y−1x−1 is a commutator of x and y in AG-group G.

Proof. Let A ∈ NS∪AG(U). Then,

fA([x, y]) = fA
(
(xy)

(
y−1x−1

))
(by Definition of commutator in G)

= fA
((
y−1x−1

)
(xy)

)
(by Lemma 3 )

= fA
(
(yx)

(
x−1y−1

))
(by Lemma 1-(iv))

= fA
(
x−1

(
(yx) y−1

))
(by Lemma 1-(ii))

⊆ fA
(
x−1

)
∪ fA

(
(yx) y−1

)
= fA(x) ∪ fA(x) (as A ∈ NS∪AG(U))

= fA(x).

Hence, fA([x, y]) ⊆ fA(x) ∀ x, y ∈ G.
Conversely, assume that fA([x, y]) ⊆ fA(x) ∀ x, y ∈ G. Then, for any z ∈ G,

fA
(
(xz)x−1

)
= fA

(
e
(
(xz)x−1

))
= fA

((
zz−1

) (
(xz)x−1

))
= fA

(((
(xz)x−1

)
z−1
)
z
)

(by the left invertive law)



A. Ullah, F. Karaaslan, I. Ahmad / Eur. J. Pure Appl. Math, 11 (2) (2018), 517-536 532

= fA
(((

z−1x−1
)

(xz)
)
z
)

(by the left invertive law)

= fA
((

(zx)
(
x−1z−1

))
z
)

(by Lemma 1-(iv))

= fA ([z, x]z)

⊆ fA([z, x]) ∪ fA(z)

⊆ fA(z) ∪ fA(z) = fA(z).

This implies that fA
(
(xz)x−1

)
⊆ fA(z) ∀ x ∈ G. Now by Theorem 6, we have fA

(
(xz)x−1

)
=

fA(z) ∀ x ∈ G. Hence, A ∈ NS∪AG(U).

Proposition 1. Let A ∈ S∪AG(U). Then fA([x, y]) = fA(e) ∀ x, y ∈ G if and only if
A ∈ NS∪AG(U).

Proof. A ∈ NS∪AG(U), if and only if

fA
(
(yx) y−1

)
= fA(x) ∀ x, y ∈ G

⇔ fA
(
e
(
(yx) y−1

))
= fA(x)

⇔ fA
((
xx−1

) (
(yx) y−1

))
= fA(x)

⇔ fA
(((

(yx) y−1
)
x−1

)
x
)

= fA(x) (by the left invertive law)

⇔ fA
(((

x−1y−1
)

(yx)
)
x
)

= fA(x) (by the left invertive law)

⇔ fA
((

(xy)
(
y−1x−1

))
x
)

= fA(x) (by Lemma 1-(iv))

⇔ fA (([x, y])x) = fA(x)

⇔ fA([x, y]) = fA(e). (by Lemma 4)

Hence, A ∈ NS∪AG(G) if and only if fA([x, y]) = fA(e) ∀ x, y ∈ G.

4. α-inclusion of Soft Uni-AG-groups

Definition 9. Let A ∈ S∪AG(U). Then, e-set of A is denoted by Aẽ and defined as

Aẽ = {x ∈ G : fA(x) = fA(e)} .

Example 10. In Example 2, Aẽ = {0}.

Theorem 8. Let A ∈ S∪AG(U). Then, Aẽ is an AG-subgroup of G.

Proof. By definition of Aẽ, it is obvious that Aẽ 6= ∅. Let x, y ∈ Aẽ. Then, fA(x) =
fA(e) = fA(y). Consider,

fA(xy−1) ⊆ fA(x) ∪ fA(y)

= fA(e) ∪ fA(e)

= fA(e),

also by Theorem 2, fA(e) ⊆ fA(xy−1) ∀ x, y ∈ G. Consequently, fA(xy−1) = fA(e). This
implies that xy−1 ∈ Aẽ. Hence Aẽ is an AG-subgroup of G.



A. Ullah, F. Karaaslan, I. Ahmad / Eur. J. Pure Appl. Math, 11 (2) (2018), 517-536 533

Definition 10. Let A ∈ S∪AG(U) and α ∈ P (U). Then α-inclusion of A, is denoted by
Aα̃, and defined by

Aα̃ = {x ∈ G : fA(x) ⊆ α} ,

while the set
Aα̃+ = {x ∈ G : fA(x) ⊂ α} ,

is called the strong α-inclusion of A.

Note that if α = U . Then Aα̃ = {x ∈ G : fA(x) 6= U }, and is called support of A, and is
denoted by supp(A).

Example 11. Let U = {u1, u2, u3, u4, u5, u6, u7} be the universal set and G = {0, 1, 2, 3, 4, 5}
be an AG-group of order 6 defined as in Example 8. If we define soft uni-AG-group A over
U by

fA(0) = {u1, u2, u3} ,
fA(2) = {u1, u2, u3, u4, u5} = fA(4),
fA(1) = {u1, u2, u3, u4, u5, u6} = fA(3) = fA(5)

Let α = {u1, u2, u3, u4, u5}, then Aα̃ = {0, 2, 4} and Aα̃+ = {0}.

Corollary 1. Let B≤̃A and C≤̃A. Then, the following assertions hold;

1. If B⊆̃C, α ∈ P (U). Then Cα̃ ⊆ Bα̃,

2. Let α2 ⊆ α1, with α1, α2 ∈ P (U). Then Bα̃2 ⊆ Bα̃1,

3. B=̃C ⇔ Bα̃ = Cα̃, for all α ∈ P (U).

Proof. Let B≤̃A and C≤̃A.

1. Let x ∈ Cα̃, then, fC(x) ⊆ α. Since B⊆̃C, α ∈ P (U). This implies that fB(x) ⊆
fC(x) ⊆ α⇒ fB(x) ⊆ α⇒ x ∈ Bα̃. Hence Cα̃ ⊆ Bα̃.

2. Let α2 ⊆ α1, α1, α2 ∈ P (U), and x ∈ Bα̃2 . Then fB(x) ⊆ α2. Since, α2 ⊆ α1 implies
that fB(x) ⊆ α1 ⇒ x ∈ Bα̃1 . Therefore, Bα̃2 ⊆ Bα̃1 .

Proof. The proof is straight forward.

Theorem 9. Let B,C are any two soft sets of G over U and α ∈ P (U). Then,

1. Bα̃ ∪ Cα̃ ⊆ (B∪̃C)α̃,

2. Bα̃ ∩ Cα̃ = (B∩̃C)α̃.

Theorem 10. Let {Bi : i ∈ I} be the family of soft sets of G over U . Then, for any
α ∈ P (U)



REFERENCES 534

1.
⋃
i∈I

(Biα̃) ⊆
(
∪̃
i∈I
Bi

)
α̃

,

2. ∩
i∈I

(Biα̃) =

(
∩̃
i∈I
Bi

)
α̃

.

Theorem 11. Let G be an AG-group and α ∈ P (U). Then A ∈ S∪AG(U) if and only if
Aα̃ is a subgroup of G, where Aα̃ 6= ∅.

Proof. Let A ∈ S∪AG(U) and Aα̃ 6= ∅. Suppose that x, y ∈ Aα̃, then fA(x) ⊆ α and
fA(y) ⊆ α. Therefore,

fA(xy−1) ⊆ fA(x) ∪ fA(y) ⊆ α.

This implies that, xy−1 ∈ Aα̃. Hence, Aα̃ is a subgroup of G.
Conversely, suppose that Aα̃ is a subgroup of G for any Aα̃ 6= ∅. Let x, y ∈ G such that

fA(x) = β and fA(y) = γ and let δ = β ∪ γ. Then x, y ∈ Aδ̃ and Aδ̃ ≤ G by hypothesis.
So xy−1 ∈ Aδ̃. Therefore, fA(xy−1) ⊆ δ = β ∪ γ = fA(x) ∪ fA(y). Hence, A ∈ S∪AG(U).

Theorem 12. Let A ∈ NS∪AG(U). Then, Aẽ is a normal AG-subgroup of G.

Proof. By Theorem 8, Aẽ ≤ G. Let x ∈ Aẽ and g ∈ G. Then, by Definition 8, we get

fA(gx · g−1) = fA(x) = fA(e) this implies that gx · g−1 ∈ Aẽ.

Hence, Aẽ is a normal AG-subgroup of G.

5. Conclusion

In this paper, the concepts of “soft uni-groups” are extended to soft uni-AG-groups.
The notion of conjugates soft uni-AG-groups, normal soft uni-AG-groups, e-set and α-
inclusion of soft uni-AG-groups are presented and investigated. In future, these concepts
can further be generalized to bipolar soft uni-AG-groups, soft uni-LA-rings and soft uni-
LA-near-rings. Moreover, the study of isomorphism theorems may also be a nice work in
this area.

Acknowledgements

This research is financially supported by HEC through NRPU Project-3509.

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