# Generalized Rough Γ-Hyperideals in Γ-Semihypergroups

Authors: Naveed Yaqoob; Shamsul Haq
Journal: Journal of Applied Mathematics
 (2014)

---

## Abstract

Davvaz (2008) introduced the concept of set-valued homomorphism andT-rough sets in a group. In this paper, we consider the set-valued homomorphism T on Γ-semihypergroup H to interpret the lower and upper approximations. We study the roughness of (
m
,
n
) bi-Γ-hyperideals and (
m
,
n
) quasi-Γ-hyperideals in terms of set-valued homomorphisms, which are extended notions of (
m
,
n
) bi-Γ-hyperideals and (
m
,
n
) quasi-Γ-hyperideals of Γ-semihypergroups.


---

## Body

## 1. Introduction

Hyperstructure, in particular hypergroups, was introduced in 1934, by Marty [1]. Nowadays, hyperstructures have a lot of applications to several domains of mathematics and computer science and they are studied in many countries in the world. Recently, Anvariyeh et al. [2] introduced the notion of Γ-semihypergroup as a generalization of a semigroup, a generalization of a semihypergroup, and a generalization of a Γ-semigroup. Heidari et al. [3] studied the structure further and added some useful results to the theory of Γ-semihypergroups. Abdullah et al. [4–6] studied some properties of M-hypersystems and bi-Γ-hyperideals in Γ-semihypergroups; also Hila et al. [7] studied the structures of Γ-semihypergroups.In 1982, Pawlak [8] introduced the notion of rough sets as a tool to model uncertainty and vague and incomplete information system. The theory of rough sets is an extension of set theory, in which a subset of a universe is described by a pair of ordinary sets called the lower and upper approximations. At present, this concept has been applied to many directions, such as groups, probability theory, graph theory, automata theory, topology, cognitive sciences, machine learning, knowledge acquisition, and pattern recognition. The algebraic approach to rough sets has been studied by some authors; for instance, Biswas and Nanda [9] studied the classical group theory in terms of rough sets and introduced the notion of rough subgroups. Xiao and Zhang [10] introduced the notion of rough prime ideals and rough fuzzy prime ideals in a semigroup. Kuroki [11] introduced the notion of a rough ideal in a semigroup. In [12] the authors investigated Pawlak’s approximations in Γ-semihypergroups. Aslam et al. [13] introduced the concept of rough M-hypersystems and fuzzy M-hypersystems in Γ-semihypergroups. Yaqoob et al. [14–16] applied rough set theory to (
m
,
n
) bi-Γ-hyperideals and (
m
,
n
) quasi-Γ-hyperideals in Γ-semihypergroups. Fotea [17] and Leoreanu-Fotea and Davvaz  [18]  discussed the lower and upper approximations of hypergroups and n-ary hypergroups. Jun [19] studied the roughness of Γ-subsemigroups/ideals in Γ-semigroups.In 2008, Davvaz [20] introduced the concept of set-valued homomorphism and T-rough sets in a group. This idea attracted several mathematicians. Xiao [21] studied the properties of T-roughness in semigroups. Yamak et al. [22] introduced generalized lower and upper approximations in a ring. Yaqoob and Aslam [23] applied generalized rough set theory (in terms of set-valued homomorphisms) to the theory of Γ-semihypergroups. Hosseini et al. [24] applied T-rough set theory to semigroups.In this paper, we study the roughness of(
m
,
n
) bi-Γ-hyperideals and (
m
,
n
) quasi-Γ-hyperideals in terms of set-valued homomorphisms.


## 2. Preliminaries and Basic Definitions

In this section, we will recall some concepts related toΓ-semihypergroups and generalized rough sets. Throughout the paper, S denotes a Γ-semihypergroup unless otherwise specified.Definition 1.
A map∘
:
S
×
S
→
P
*
(
S
) is called hyperoperation or join operation on the set S, where S is a nonempty set and P
*
(
S
) denotes the set of all nonempty subsets of S. A hypergroupoid is a set S together with a (binary) hyperoperation. A hypergroupoid (
S
,
∘
), which is associative, that is, x
∘
(
y
∘
z
)
=
(
x
∘
y
)
∘
z, for all x
,
y
,
z
∈
S, is called a semihypergroup.Definition 2 (see [2]).
LetS and Γ be two nonempty sets. S is called a Γ-semihypergroup if every γ
∈
Γ is a hyperoperation on S, that is, x
γ
y
⊆
S, for every x
,
y
∈
S, and, for every γ
,
β
∈
Γ and x
,
y
,
z
∈
S, we have x
γ
(
y
β
z
)
=
(
x
γ
y
)
β
z.LetA and B be two nonempty subsets of S. Then, we define
(1)
A
Γ
B
=
⋃
γ
∈
Γ
‍
A
γ
B
=
⋃
‍
{
a
γ
b
∣
a
∈
A
,
b
∈
B
and
γ
∈
Γ
}
.
Let (
S
,
∘
) be a semihypergroup and let Γ
=
{
∘
}. Then, S is a Γ-semihypergroup. So, every semihypergroup is Γ-semihypergroup.Definition 3 (see [2]).
LetS be a Γ-semihypergroup and γ
∈
Γ. A nonempty subset A of S is called a sub-Γ-semihypergroup of S if x
γ
y
⊆
A for every x
,
y
∈
A.Definition 4 (see [2]).
A subsetA of a Γ-semihypergroup S is called an interior Γ-hyperideal of S if S
Γ
A
Γ
S
⊆
A.LetA be a nonempty subset of S and n
∈
N. A set A
n is defined to be the set
(2)
A
n
=
A
Γ
A
Γ
⋯
Γ
A
︸
n
-times
.
For example, A
2
=
A
Γ
A and A
3
=
A
Γ
A
Γ
A.Definition 5 (see [14]).
A subsetA of a Γ-semihypergroup S is called an (
m
,
0
)  Γ-hyperideal (resp., (
0
,
n
)  Γ-hyperideal) of S if A
m
Γ
S
⊆
A (resp., S
Γ
A
n
⊆
A).Definition 6 (see [14]).
A sub-Γ-semihypergroup A of a Γ-semihypergroup S is called an (
m
,
n
) bi-Γ-hyperideal of S, if A
m
Γ
S
Γ
A
n
⊆
A, where m, n are nonnegative integers (
A
m is suppressed if m
=
0
).Definition 7 (see [15]).
An(
m
,
n
) bi-Γ-hyperideal B of a Γ-semihypergroup S is called prime if for x
,
y
∈
S,   x
m
α
S
β
y
n
⊆
B  (or x
m
α
z
β
y
n
⊆
B, for all z
∈
S
) implies x
∈
B or y
∈
B, for all α
,
β
∈
Γ.Definition 8 (see [15]).
An(
m
,
n
) bi-Γ-hyperideal B of a Γ-semihypergroup S is called semiprime if for  x
∈
S,  x
m
α
S
β
x
n
⊆
B  (or x
m
α
z
β
x
n
⊆
B, for all z
∈
S
) implies x
∈
B, for all α
,
β
∈
Γ.Definition 9 (see [6]).
LetS be a Γ-semihypergroup and L a sub-Γ-semihypergroup of S. Then L is called an m-left Γ-hyperideal of S if S
m
Γ
L
⊆
L where m is any positive integer. Dually, R
Γ
S
n
⊆
R, and then R is called an n-right Γ-hyperideal of S, where n is any positive integer.Definition 10 (see [6]).
LetS be a Γ-semihypergroup and Q a nonempty subset of S. Then Q is called an (
m
.
n
) quasi-Γ-hyperideal of S if S
m
Γ
Q
∩
Q
Γ
S
n
⊆
Q.Now, we will recall some notions in generalized rough sets.Definition 11.
LetX and Y be two nonempty universes. Let T be a set-valued mapping given by T
:
X
→
P
(
Y
). Then, the triple (
X
,
Y
,
T
) is referred to as a generalized approximation space or generalized rough set. Any set-valued function from X to P
(
Y
) defines a binary relation from X to Y by setting R
T
=
{
(
x
,
y
)
∣
y
∈
T
(
x
)
}. Obviously, if R is an arbitrary relation from X to Y, then it can be defined as a set-valued mapping T
R
:
X
→
P
(
Y
) by T
R
(
x
)
=
{
y
∈
Y
∣
(
x
,
y
)
∈
R
} where x
∈
X. For any set A
⊆
Y, the lower and upper approximations T
_
(
A
) and T
¯
(
A
) are defined by
(3)
T
_
(
A
)
=
{
x
∈
X
∣
T
(
x
)
⊆
A
}
,
T
¯
(
A
)
=
{
x
∈
X
∣
T
(
x
)
∩
A
≠
∅
}
.
The pair (
T
_
(
A
)
,
T
¯
(
A
)
) is referred to as a generalized rough set, and T
_, T
¯ are referred to as lower and upper generalized approximation operators, respectively.If a subsetA
⊆
Y satisfies that T
_
(
A
)
=
T
¯
(
A
), then A is called a definable set of (
X
,
Y
,
T
). We denote all the definable sets of (
X
,
Y
,
T
) by Def (
T
).Theorem 12 (see [22]).
Let(
X
,
Y
,
T
) be a generalized approximation space; its lower and upper approximation operators satisfy the following properties: for all A
,
B
∈
P
(
Y
),
(4)
(
L
1
)
T
_
(
A
)
=
(
T
¯
(
A
c
)
)
c
,
(
L
2
)
T
_
(
Y
)
=
X
,
(
L
3
)
T
_
(
A
∩
B
)
=
T
_
(
A
)
∩
T
_
(
B
)
,
(
L
4
)
A
⊆
B
⟹
T
_
(
A
)
⊆
T
_
(
B
)
,
(
L
5
)
T
_
(
A
∪
B
)
⊇
T
_
(
A
)
∪
T
_
(
B
)
,
k
j
j
l
j
j
j
j
j
(
U
1
)
T
¯
(
A
)
=
(
T
_
(
A
c
)
)
c
,
(
U
2
)
T
¯
(
∅
)
=
∅
,
(
U
3
)
T
¯
(
A
∪
B
)
=
T
¯
(
A
)
∪
T
¯
(
B
)
,
(
U
4
)
A
⊆
B
⟹
T
¯
(
A
)
⊆
T
¯
(
B
)
,
(
U
5
)
T
¯
(
A
∩
B
)
⊆
T
¯
(
A
)
∩
T
¯
(
B
)
,
where A
c is the complement of the set A.Theorem 13 (see [22]).
Let(
X
,
X
,
T
) be a generalized approximation space; its lower and upper generalized approximation operators satisfy the following properties: for all A
∈
P
(
X
),
(5)
(
1
)
R
T
i
s
s
e
r
i
a
l
⟺
(
L
0
)
T
_
(
∅
)
=
∅
⟺
(
U
0
)
T
¯
(
X
)
=
X
⟺
(
L
U
0
)
T
_
(
A
)
⊆
T
¯
(
A
)
(
2
)
R
T
i
s
r
e
f
l
e
x
i
v
e
⟺
(
L
6
)
T
_
(
A
)
⊆
A
⟺
(
U
6
)
A
⊆
T
¯
(
A
)
(
3
)
R
T
i
s
s
y
m
m
e
t
r
i
c
⟺
(
L
7
)
T
¯
(
T
_
(
A
)
)
⊆
A
⟺
(
U
7
)
A
⊆
T
_
(
T
¯
(
A
)
)
(
4
)
R
T
i
s
t
r
a
n
s
i
t
i
v
e
⟺
(
L
8
)
T
_
(
A
)
⊆
T
_
(
T
_
(
A
)
)
(6)
⟺
(
U
8
)
T
¯
(
T
¯
(
A
)
)
⊆
T
¯
(
A
)IfR is an equivalence relation on X, then the pair (
X
,
R
) is the Pawlak approximation space. Therefore, a generalized rough set is an extended notion of Pawlak’s rough sets.Definition 14.
LetS be a Γ-semihypergroup. An equivalence relation ρ on S is called regular on S if
(7)
(
a
,
b
)
∈
ρ
implies
(
a
γ
x
,
b
γ
x
)
∈
ρ
,
(
x
γ
a
,
x
γ
b
)
∈
ρ
,
for all x
∈
S and γ
∈
Γ.Ifρ is a regular relation on S, then, for every x
∈
S, [
x
]
ρ stands for the class of x with the representation ρ. A regular relation ρ on S is called complete if [
a
]
ρ
γ
[
b
]
ρ
=
[
a
γ
b
]
ρ for all a
,
b
∈
S and γ
∈
Γ. In addition, ρ on S is called congruence if, for every a
,
b
∈
S and γ
∈
Γ, we have c
∈
[
a
]
ρ
γ
[
b
]
ρ
⇒
[
c
]
ρ
⊆
[
a
]
ρ
γ
[
b
]
ρ. It is obvious that, for a regular relation ρ on S,  [
a
]
ρ
γ
[
b
]
ρ
⊆
[
a
γ
b
]
ρ for all a
,
b
∈
S and γ
∈
Γ.


## 3. Generalized Rough Sets inΓ-Semihypergroups

In this section, we will present some results on generalized rough sets inΓ-semihypergroups.Definition 15 (see [23]).
A set-valued homomorphismT from a Γ-semihypergroup S to a Γ
˙-semihypergroup S
˙ is a mapping from S to ℘
*
(
S
˙
) which preserves the operation; that is, T
(
a
)
β
˙
T
(
b
)
⊆
T
(
a
β
b
) for all a
,
b
∈
S,   β
∈
Γ, and β
˙
∈
Γ
˙.  T is called a strong set-valued homomorphism, if T
(
a
)
β
˙
T
(
b
)
=
T
(
a
β
b
) for all a
,
b
∈
S,  β
∈
Γ, and β
˙
∈
Γ
˙.Theorem 16 (see [23]).
LetS be a Γ-semihypergroup, let S
˙ be a Γ
˙-semihypergroup, and let T
:
S
→
℘
*
(
S
˙
) be a set-valued homomorphism. If A, B are two nonempty subsets of S
˙, then
(1)T
¯
(
A
)
Γ
T
¯
(
B
)
⊆
T
¯
(
A
Γ
˙
B
);
(2) ifT is strong, then T
_
(
A
)
Γ
T
_
(
B
)
⊆
T
_
(
A
Γ
˙
B
).Theorem 17 (see [23]).
LetS be a Γ-semihypergroup, let S
˙ be a Γ
˙-semihypergroup, and let T
:
S
→
℘
*
(
S
˙
) be a set-valued homomorphism.
(1) IfA is a sub-Γ
˙-semihypergroup of S
˙, then T
¯
(
A
) is, if it is nonempty, a sub-Γ-semihypergroup of S.
(2) IfA is a left (resp., right) Γ
˙-hyperideal of S
˙, then T
¯
(
A
) is, if it is nonempty, a left (resp., right) Γ-hyperideal of S.Theorem 18 (see [23]).
LetS be a Γ-semihypergroup, let S
˙ be a Γ
˙-semihypergroup, and let T
:
S
→
℘
*
(
S
˙
) be a strong set-valued homomorphism.
(1) IfA is a sub-Γ
˙-semihypergroup of S
˙, then T
_
(
A
) is, if it is nonempty, a sub-Γ-semihypergroup of S.
(2) If A is a left (resp., right)Γ
˙-hyperideal of S
˙, then T
_
(
A
) is, if it is nonempty, a left (resp., right) Γ-hyperideal of S.Theorem 19.
LetS be a Γ-semihypergroup, let S
˙ be a Γ
˙-semihypergroup, and let T
:
S
→
℘
*
(
S
˙
) be a set-valued homomorphism. If A is an interior Γ
˙-hyperideal of S
˙, then
(1)T
¯
(
A
) is, if it is nonempty, an interior Γ-hyperideal of S;
(2) ifT is strong, then T
_
(
A
) is, if it is nonempty, an interior Γ-hyperideal of S.Proof.
The proof is straightforward.Lemma 20.
LetS be a Γ-semihypergroup, let S
˙ be a Γ
˙-semihypergroup, and let T
:
S
→
℘
*
(
S
˙
) be a set-valued homomorphism. Then, for a nonempty subset A of S
˙,
(1)(
T
¯
(
A
)
)
n
⊆
T
¯
(
A
n
) for all n
∈
N;
(2) ifT is strong, then (
T
_
(
A
)
)
n
⊆
T
_
(
A
n
) for all n
∈
N.Proof.
The proof is straightforward.


## 4. Generalized Rough(
m
,
n
) (Bi-)Quasi-Γ-Hyperideals

We will study here some properties of generalized lower and upper approximations of(
m
,
n
) bi-Γ-hyperideals in Γ-semihypergroup.A subsetA of a Γ
˙-semihypergroup S
˙ is called a generalized upper (resp., generalized lower) rough (
m
,
n
) bi-Γ-hyperideal of S if T
¯
(
A
) (resp., T
_
(
A
)) is an (
m
,
n
) bi-Γ-hyperideal of S.Theorem 21.
LetS be a Γ-semihypergroup, let S
˙ be a Γ
˙-semihypergroup, and let T
:
S
→
℘
*
(
S
˙
) be a set-valued homomorphism. If A is an (
m
,
n
) bi-Γ
˙-hyperideal of S
˙, then
(1)T
¯
(
A
) is, if it is nonempty, an (
m
,
n
) bi-Γ-hyperideal of S;
(2) ifT is strong, then T
_
(
A
) is, if it is nonempty, an (
m
,
n
) bi-Γ-hyperideal of S.Proof.
(1) LetA be an (
m
,
n
) bi-Γ
˙-hyperideal of S
˙. Then, by Theorem 16
(
1
) and Lemma 20
(
1
), we have
(8)
(
T
¯
(
A
)
)
m
Γ
S
Γ
(
T
¯
(
A
)
)
n
=
(
T
¯
(
A
)
)
m
Γ
T
¯
(
S
˙
)
Γ
(
T
¯
(
A
)
)
n
⊆
T
¯
(
A
m
)
Γ
T
¯
(
S
˙
)
Γ
T
¯
(
A
n
)
⊆
T
¯
(
A
m
Γ
˙
S
˙
)
Γ
T
¯
(
A
n
)
⊆
T
¯
(
A
m
Γ
˙
S
˙
Γ
˙
A
n
)
⊆
T
¯
(
A
)
.
From this and Theorem 17
(
1
), we obtain that T
¯
(
A
) is an (
m
,
n
) bi-Γ-hyperideal of S.
(2) LetA be an (
m
,
n
) bi-Γ
˙-hyperideal of S
˙. Then, by Theorem 16
(
2
) and Lemma 20
(
2
), we have
(9)
(
T
_
(
A
)
)
m
Γ
S
Γ
(
T
_
(
A
)
)
n
=
(
T
_
(
A
)
)
m
Γ
T
_
(
S
˙
)
Γ
(
T
_
(
A
)
)
n
⊆
T
_
(
A
m
)
Γ
T
_
(
S
˙
)
Γ
T
_
(
A
n
)
⊆
T
_
(
A
m
Γ
˙
S
˙
)
Γ
T
_
(
A
n
)
⊆
T
_
(
A
m
Γ
˙
S
˙
Γ
˙
A
n
)
⊆
T
_
(
A
)
.
From this and Theorem 18
(
1
), we obtain that T
_
(
A
) is, if it is nonempty, an (
m
,
n
) bi-Γ-hyperideal of S. This completes the proof.Corollary 22.
LetS be a Γ-semihypergroup, let S
˙ be a Γ
˙-semihypergroup, and let T
:
S
→
℘
*
(
S
˙
) be a set-valued homomorphism. If A is an (
m
,
0
)   Γ
˙-hyperideal (resp., (
0
,
n
)  Γ
˙-hyperideal, m-left Γ
˙-hyperideal, and n-right Γ
˙-hyperideal) of S
˙, then
(1)T
¯
(
A
) is, if it is nonempty, an (
m
,
0
)   Γ-hyperideal (resp., (
0
,
n
)  Γ-hyperideal, m-left Γ-hyperideal, and n-right Γ-hyperideal) of S;
(2) ifT is strong, then T
_
(
A
) is, if it is nonempty, an (
m
,
0
)  Γ-hyperideal (resp., (
0
,
n
)  Γ-hyperideal, m-left Γ-hyperideal, and n-right Γ-hyperideal) of S.Proof.
The proof is straightforward.A subsetA of a Γ
˙-semihypergroup S
˙ is called a generalized upper (resp., generalized lower) rough prime (
m
,
n
) bi-Γ-hyperideal of S if T
¯
(
A
) (resp., T
_
(
A
)) is a prime (
m
,
n
) bi-Γ-hyperideal of S.Theorem 23.
LetS be a Γ-semihypergroup, let S
˙ be a Γ
˙-semihypergroup, and let T
:
S
→
℘
*
(
S
˙
) be a strong set-valued homomorphism. If A is a prime (
m
,
n
) bi-Γ
˙-hyperideal of S
˙, then
(1)T
¯
(
A
) is, if it is nonempty, a prime (
m
,
n
) bi-Γ-hyperideal of S;
(2)T
_
(
A
) is, if it is nonempty, a prime (
m
,
n
) bi-Γ-hyperideal of S.Proof.
SinceA is an (
m
,
n
) bi-Γ
˙-hyperideal of S
˙. by Theorem 21, we know that T
¯
(
A
) and T
_
(
A
) are (
m
,
n
) bi-Γ-hyperideals of S.
(1) Letw be any element of S. Let x
,
y
∈
S and β
,
γ
∈
Γ such that x
m
β
w
γ
y
n
⊆
T
¯
(
A
). Thus,
(10)
T
(
x
m
β
w
γ
y
n
)
∩
A
=
(
T
(
x
m
)
β
˙
T
(
w
)
γ
˙
T
(
y
n
)
)
∩
A
≠
∅
,
where β
˙
,
γ
˙
∈
Γ
˙. Thus, there exist a
m
⊆
T
(
x
m
)
=
T
(
x
)
m,  w
′
∈
T
(
w
), and b
n
⊆
T
(
y
n
)
=
T
(
y
)
n such that a
m
β
˙
w
′
γ
˙
b
n
⊆
A. Since A is a prime (
m
,
n
) bi-Γ
˙-hyperideal, we have a
∈
A or b
∈
A. Now,
(11)
a
m
⊆
T
(
x
)
m
⟹
a
∈
T
(
x
)
also
b
n
⊆
T
(
y
)
n
⟹
b
∈
T
(
y
)
.
Thus, a
∈
T
(
x
)
∩
A or b
∈
T
(
y
)
∩
A. So x
∈
T
¯
(
A
) or y
∈
T
¯
(
A
). Therefore, T
¯
(
A
) is a prime (
m
,
n
) bi-Γ-hyperideal of S.
(2) We suppose thatT
_
(
A
) is not a prime (
m
,
n
) bi-Γ-hyperideal; then for β
,
γ
∈
Γ there exist x
,
y
∈
S and any element w
∈
S, such that x
m
β
w
γ
y
n
⊆
T
_
(
A
), but x
∉
T
_
(
A
) and y
∉
T
_
(
A
). Thus T
(
x
)
⊈
A and T
(
y
)
⊈
A. Then, there exist
(12)
a
∈
T
(
x
)
but
a
∉
A
,
b
∈
T
(
y
)
but
b
∉
A
.
Now, for w
′
∈
S,  β
,
γ
∈
Γ, and β
˙
,
γ
˙
∈
Γ
˙, we have
(13)
a
m
β
˙
w
′
γ
˙
b
n
⊆
T
(
x
)
m
β
˙
T
(
w
)
γ
˙
T
(
y
)
n
=
T
(
x
m
)
β
˙
T
(
w
)
γ
˙
T
(
y
n
)
=
T
(
x
m
β
w
γ
y
n
)
⊆
A
.
This implies that a
m
β
˙
w
′
γ
˙
b
n
⊆
A. Since A is a prime (
m
,
n
) bi-Γ
˙-hyperideal of S
˙, we have a
∈
A or b
∈
A. It contradicts the supposition. This means that T
_
(
A
) is, if it is nonempty, a prime (
m
,
n
) bi-Γ-hyperideal of S.The following example shows that the converse of Theorem23 does not hold.Example 24.
LetS
=
{
x
,
y
,
z
} and Γ
=
{
β
,
γ
} be the sets of binary hyperoperations defined as follows:
(14)
β
x
y
z
x
x
x
x
y
x
y
{
y
,
z
}
z
x
{
y
,
z
}
{
y
,
z
}
γ
x
y
z
x
x
x
x
y
x
y
{
y
,
z
}
z
x
{
y
,
z
}
z
.
Clearly S is a Γ-semihypergroup. Let S
˙
=
{
a
,
b
,
c
,
d
,
e
} and Γ
˙
=
{
β
˙
,
γ
˙
} be the sets of binary hyperoperations defined as follows:
(15)
β
˙
a
b
c
d
e
a
{
a
,
b
}
{
b
,
c
}
c
{
d
,
e
}
e
b
{
b
,
c
}
c
c
{
d
,
e
}
e
c
c
c
c
{
d
,
e
}
e
d
{
d
,
e
}
{
d
,
e
}
{
d
,
e
}
d
e
e
e
e
e
e
e
γ
˙
a
b
c
d
e
a
{
b
,
c
}
c
c
{
d
,
e
}
e
b
c
c
c
{
d
,
e
}
e
c
c
c
c
{
d
,
e
}
e
d
{
d
,
e
}
{
d
,
e
}
{
d
,
e
}
d
e
e
e
e
e
e
e
.
Clearly S
˙ is a Γ
˙-semihypergroup. Assume that T
(
x
)
=
{
d
,
e
},  T
(
y
)
=
{
a
,
b
,
c
}, and T
(
z
)
=
{
a
,
b
,
c
,
d
,
e
}. Here, T is a strong set-valued homomorphism from S to S
˙. Now for A
=
{
b
,
d
,
e
}
⊆
S
˙,  T
¯
(
A
)
=
{
x
,
y
,
z
} and T
_
(
A
)
=
{
x
}. It is clear that T
¯
(
A
) and T
_
(
A
) are prime (
m
,
n
) bi-Γ-hyperideals of S. But A is not a sub-Γ
˙-semihypergroup of S
˙; hence, A is not a prime (
m
,
n
) bi-Γ
˙-hyperideal of S
˙.A subsetQ of a Γ
˙-semihypergroup S
˙ is called a generalized upper (resp., generalized lower) rough (
m
,
n
) quasi-Γ-hyperideal of S if T
¯
(
A
) (resp., T
_
(
A
)) is an (
m
,
n
) quasi-Γ-hyperideal of S.Theorem 25.
LetS be a Γ-semihypergroup, let S
˙ be a Γ
˙-semihypergroup, and let T
:
S
→
℘
*
(
S
˙
) be a strong set-valued homomorphism. If Q is an (
m
,
n
) quasi-Γ
˙-hyperideal of S
˙, then T
_
(
A
) is, if it is nonempty, an (
m
,
n
) quasi-Γ-hyperideal of S.Proof.
LetQ be an (
m
,
n
) quasi-Γ
˙-hyperideal of S
˙; that is, S
˙
m
Γ
˙
Q
∩
Q
Γ
˙
S
˙
n
⊆
Q. Note that T
_
(
S
˙
)
=
S. Then, by Theorem 12(L3), Theorem 16(2), and Lemma 20(2), we have
(16)
S
m
Γ
T
_
(
Q
)
∩
T
_
(
Q
)
Γ
S
n
=
(
T
_
(
S
˙
)
)
m
Γ
T
_
(
Q
)
∩
T
_
(
Q
)
Γ
(
T
_
(
S
˙
)
)
n
⊆
T
_
(
S
˙
m
)
Γ
T
_
(
Q
)
∩
T
_
(
Q
)
Γ
T
_
(
S
˙
n
)
⊆
T
_
(
S
˙
m
Γ
˙
Q
)
∩
T
_
(
Q
Γ
˙
S
˙
n
)
=
T
_
(
S
˙
m
Γ
˙
Q
∩
Q
Γ
˙
S
˙
n
)
⊆
T
_
(
Q
)
.
This shows that T
_
(
Q
) is an (
m
,
n
) quasi-Γ-hyperideal of S.The next theorem shows that the intersection of a generalized lower roughm-left Γ-hyperideal and a generalized lower rough n-right Γ-hyperideal of a Γ-semihypergroup S is a generalized lower rough (
m
,
n
) quasi-Γ-hyperideal of S.Theorem 26.
LetS be a Γ-semihypergroup, let S
˙ be a Γ
˙-semihypergroup, and let T
:
S
→
℘
*
(
S
˙
) be a strong set-valued homomorphism. Let L and R be a generalized lower rough m-left Γ-hyperideal and a generalized lower rough n-right Γ-hyperideal of S, respectively. Then T
_
(
L
∩
R
) is, if it is nonempty, an (
m
,
n
) quasi-Γ-hyperideal of S.Proof.
LetL and R be a generalized lower rough m-left Γ-hyperideal and a generalized lower rough n-right Γ-hyperideal of S, respectively. Then,
(17)
S
m
Γ
T
_
(
L
)
⊆
T
_
(
L
)
,
T
_
(
R
)
Γ
S
n
⊆
T
_
(
R
)
.
Now, we have
(18)
S
m
Γ
T
_
(
L
∩
R
)
∩
T
_
(
L
∩
R
)
Γ
S
n
⊆
S
m
Γ
T
_
(
L
)
∩
T
_
(
R
)
Γ
S
n
⊆
T
_
(
L
)
∩
T
_
(
R
)
=
T
_
(
L
∩
R
)
.
Hence, this shows that T
_
(
L
∩
R
) is a generalized lower rough (
m
,
n
) quasi-Γ-hyperideal of S.



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*Source: 658252-2014-07-08.xml*