# Some Generalized Forms of Fuzzy Interval Valued Hyperideals in a Hyperring

Authors: Hongjie Li; Zeyuan Li; Yunqiang Yin
Journal: Journal of Applied Mathematics
 (2014)

---

## Abstract

Some generalized forms of the hyperideals of a hyperring in the paper of Zhan et al.  (2008)  will be given. As a generalization of the interval valued(
α
,
β
)-fuzzy hyperideals of a hyperring with α
,
β
∈
{
∈
,
q
,
∈
∧
q
,
∈
∨
q
} and α
≠
∈
∧
q, the notion of generalized interval valued  (
α
,
β
)-fuzzy  hyperideals of a hyperring is also introduced. Special attention is concentrated on the interval valued (
∈
γ
~
,
∈
γ
~
∨
q
δ
~
)-fuzzy hyperideals. As a consequence, some characterizations theorems of interval valued (
∈
γ
~
,
∈
γ
~
∨
q
δ
~
)-fuzzy hyperideals will be provided.


---

## Body

## 1. Introduction

The concept of hyperstructure was first introduced by Marty [1] in 1934. People later observed that hyperstructures have many applications in pure and applied sciences. A comprehensive review of hyperstructures can be found in [2, 3]. In a recent monograph of Corsini and Leoreanu [4], the authors have collected numerous applications of algebraic hyperstructures, especially those from the last decade to the following subjects: geometry, hypergraphs, binary relations, lattices, fuzzy sets and rough sets, automata, cryptography, codes, median algebras, relation algebras, artificial intelligence, and probabilities.After introducing the concept of fuzzy sets of Zadeh in 1965 (see [5]), there are many papers devoted to fuzzify the classical mathematics into fuzzy mathematics. The relationships between the fuzzy sets and algebraic hyperstructures (structures) have been considered by Corsini, Davvaz, Kehagias, Leoreanu, Yin, Zahedi, Zhan, and others. The reader is referred to the papers [6–24]. By using the notion “belongingness (∈)” and “quasicoincidence (
q
)” of a fuzzy point with a fuzzy set introduced by Pu and Liu [25], the concept of (
α
,
β
)-fuzzy subgroups, where α, β are any two in {
∈
,
q
,
∈
∨
q
,
∈
∧
q
} with α
≠
∈
∧
q, was introduced by Bhakat and Das [26] in 1992, in which they first generalized Rosenfeld’s fuzzy subgroup [27]. The detailed study of (
∈
,
∈
∨
q
)-fuzzy subgroups has been considered by Bhakat and Das in [28] and Bhakat in [29, 30]. The concept of the (
∈
,
∈
∨
q
)-fuzzy subhyperquasigroups of hyperquasigroups was introduced by Davvaz and Corsini [12]. By using the idea of “quasicoincidence” of a fuzzy interval value with an interval valued fuzzy set, Zhan et al. [21] introduced and investigated the notion of interval valued (
α
,
β
)-fuzzy hyperideals of a hyperring.As a generalization of the concepts of “belongingness (∈)” and “quasicoincidence (
q
)” of a fuzzy point with a fuzzy set introduced by Pu and Liu [25], Yin and Zhan [31] introduced the concept of “γ-belongingness (∈
γ)” and “δ-quasicoincidence (q
δ)” of a fuzzy point with a fuzzy set. Using these new concepts, Yin and Zhan [31] introduced the concepts of (
α
,
β
)-fuzzy (implicative, positive implicative, and fantastic) filters and (
β
¯
,
α
¯
)-fuzzy (implicative, positive implicative, and fantastic) filters of BL-algebras, where α
,
β
∈
{
∈
γ
,
q
δ
,
∈
γ
∧
q
δ
,
∈
γ
~
∨
q
δ
~
}, α
¯
,
β
¯
∈
{
∈
γ
¯
,
q
δ
¯
,
∈
γ
¯
∧
q
δ
¯
,
∈
γ
¯
∨
q
δ
¯
}, α
≠
∈
γ
∧
q
δ, and β
¯
≠
∈
γ
¯
∧
q
δ
¯, and some related properties were investigated. This idea is continued and studied by Ma et al. [32], Ma and Zhan [33], Yin and Zhan [34], and so on. Following this idea, in this paper, we will give some of the generalized forms of the interval fuzzy hyperideals given by Zhan et al. in [21]. The notion of generalized interval valued (
α
,
β
)-fuzzy hyperideals is introduced which is the generalization of the notion of interval valued (
α
,
β
)-fuzzy hyperideals. Special attention will be paid to the interval valued (
∈
γ
~
,
∈
γ
~
∨
q
δ
~
)-fuzzy hyperideals. It is noteworthy that the notion of an interval valued (
α
,
β
)-fuzzy hyperideal is a special generalized interval valued (
α
,
β
)-fuzzy hyperideal and hence many results in [21] will become easy corollaries of our results given in this paper.


## 2. Hyperrings

We first recall that a hyperstructure is a nonempty setH together with a mapping “∘”: H
×
H
→
P
*
(
H
), where P
*
(
H
) is the set of all nonempty subsets of H, written as (
x
,
y
)
↦
x
∘
y. We note that if x
∈
H and A
,
B are nonempty subsets of H, then by A
∘
B
A
∘
x and x
∘
B, we mean that A
∘
B
=
⋃
a
∈
A
,
b
∈
B
a
∘
b
,
A
∘
x
=
A
∘
{
x
} and x
∘
B
=
{
x
}
∘
B, respectively.Now, we call a hyperstructure(
H
,
∘
)
a canonical hypergroup [35] if the following axioms are satisfied:(1)
for everyx
,
y
,
z
∈
H, (
x
∘
y
)
∘
z
=
x
∘
(
y
∘
z
);
(2)
for everyx
,
y
∈
H, x
∘
y
=
y
∘
x;
(3)
there is a unique0
∈
H such that 0
∘
x
=
x, for all x
∈
H;
(4)
for everyx
∈
H, there exists a unique element x
′
∈
H such that 0
∈
x
∘
x
′ (we call the element x
′ the opposite of x).Definition 1 (see [36]).
A hyperring is an algebraic structure(
R
,
+
,
·
) which satisfies the following axioms.(1)
(
R
,
+
) is a canonical hypergroup (we will write -
x for x
′).
(2)
(
R
,
·
) is a semigroup having zero as a bilaterally absorbing element.
(3)
The multiplication is distributive with respect to the hyperoperation “+.”Let(
R
,
+
,
·
) be a hyperring and let A be a nonempty subset of R. Then A is called asubhyperring of R if (
A
,
+
,
·
) itself is a hyperring. In what follows, let R denote a hyperring unless otherwise stated. A subhyperring A of R is aleft (right) hyperideal of R if r
a
∈
A
(
a
r
∈
A
), for all r
∈
R and a
∈
A. A subhyperring A is called ahyperideal if A is both left and right hyperideal.


## 3. Interval Valued Fuzzy Sets

By aninterval number
a
~ [37], we mean an interval [
a
-
,
a
+
], where 0
≤
a
-
≤
a
+
≤
1. The set of all interval numbers is denoted by D
[
0,1
]. The interval [
a
,
a
] can be simply identified by the number a. For the interval numbers a
~
i
=
[
a
i
-
,
a
i
+
], b
~
i
=
[
b
i
-
,
b
i
+
]
∈
D
[
0,1
]
,
i
∈
I, where I is an index set, we define
(1)
rmax
{
a
~
i
,
b
~
i
}
=
[
max
⁡
(
a
i
-
,
b
i
-
)
,
max
⁡
(
a
i
+
,
b
i
+
)
]
,
rmin
{
a
~
i
,
b
~
i
}
=
[
min
⁡
(
a
i
-
,
b
i
-
)
,
min
⁡
(
a
i
+
,
b
i
+
)
]
,
⋁
i
∈
I
a
~
i
=
[
⋁
i
∈
I
a
i
-
,
⋁
i
∈
I
a
i
+
]
,
⋀
i
∈
I
a
~
i
=
[
⋀
i
∈
I
a
i
-
,
⋀
i
∈
I
a
i
+
]
.For any intervalsa
~
=
[
a
-
,
a
+
] and b
~
=
[
b
-
,
b
+
], where 0
≤
a
-
≤
a
+ and 0
≤
b
-
≤
b
+, we define
(2)
a
~
+
b
~
=
[
a
-
+
b
-
,
a
+
+
b
+
]
,
a
~
-
b
~
=
[
a
-
-
b
-
,
a
+
-
b
+
]
,
whenever
a
-
≤
b
-
,
a
+
≤
b
+
,
k
a
~
=
[
k
a
-
,
k
a
+
]
,
whenever
k
≥
0
,
and we put(i)
a
~
≤
b
~
⇔
a
-
≤
b
- and a
+
≤
b
+,
(ii)
a
~
=
b
~
⇔
a
-
=
b
- and a
+
=
b
+,
(iii)
a
~
<
b
~
⇔
a
~
≤
b
~ and a
~
≠
b
~.Denote by D
′
[
0,1
] the set all of interval numbers such that any two elements of D
′
[
0,1
] are comparable. Then, it is clear that both (
D
[
0,1
]
,
≤
,
∧
,
∨
) and (
D
[
0,1
]
′
,
≤
,
∧
,
∨
) form a complete lattice with 0
=
[
0,0
] as their least element and 1
=
[
1,1
] as their greatest element.Recall that an interval valued fuzzy subsetF on X is the set
(3)
F
=
{
(
x
,
[
μ
F
-
(
x
)
,
μ
F
+
(
x
)
]
)
∣
x
∈
X
}
,
where μ
F
- and μ
F
+ are two fuzzy subsets of X such that μ
F
-
(
x
)
≤
μ
F
+
(
x
) for all x
∈
X. Putting μ
F
~
(
x
)
=
[
μ
F
-
(
x
)
,
μ
F
+
(
x
)
], we see that F
=
{
(
x
,
μ
F
~
(
x
)
)
∣
x
∈
X
}, where μ
F
~
:
X
→
D
[
0,1
]. Denote by I
F
(
X
) the set of all interval valued fuzzy subsets of X such that, for any F
∈
I
F
(
X
), ImF
=
{
[
μ
F
-
(
x
)
,
μ
F
+
(
x
)
]
∣
x
∈
X
}
⊆
D
′
[
0,1
].IfF and G are two interval valued fuzzy subsets of X, then we defineF
⊆
G if and only if, for all x
∈
X, μ
F
-
(
x
)
≤
μ
G
-
(
x
) and μ
F
+
(
x
)
≤
μ
G
+
(
x
),F
=
G if and only if, for all x
∈
X, μ
F
-
(
x
)
=
μ
G
-
(
x
) and μ
F
+
(
x
)
=
μ
G
+
(
x
).Also, the union and intersection are defined as follows:(4)
F
∪
G
=
{
(
x
,
[
max
⁡
{
μ
F
-
(
x
)
,
μ
G
-
(
x
)
}
,
m
m
m
max
⁡
{
μ
F
+
(
x
)
,
μ
G
+
(
x
)
}
]
)
∣
x
∈
X
}
,
F
∩
G
=
{
(
x
,
[
min
⁡
{
μ
F
-
(
x
)
,
μ
G
-
(
x
)
}
,
m
m
m
min
⁡
{
μ
F
+
(
x
)
,
μ
G
+
(
x
)
}
]
)
∣
x
∈
X
}
.An interval valued fuzzy subsetF of X of the form
(5)
μ
F
~
(
y
)
=
{
r
~
(
≠
[
0,0
]
)
if
y
=
x
,
[
0,0
]
otherwise
,
is said to be afuzzy interval value with support
x
and interval value
r
~ and is denoted by U
(
x
;
r
~
).Letγ
~
,
δ
~
∈
D
[
0,1
] be such that γ
~
<
δ
~. For a fuzzy interval value U
(
x
;
t
~
) and an interval valued fuzzy set F, we say that(1)
U
(
x
;
r
~
)
∈
γ
~
F if μ
F
~
(
x
)
≥
r
~
>
γ
~;
(2)
U
(
x
;
r
~
)
q
δ
~
F if μ
F
~
(
x
)
+
r
~
>
2
δ
~;
(3)
U
(
x
;
r
~
)
∈
γ
~
∨
q
δ
~
F if U
(
x
;
r
~
)
∈
γ
~
F or U
(
x
;
r
~
)
q
δ
~
F;
(4)
U
(
x
;
r
~
)
∈
γ
~
∧
q
δ
~
F if U
(
x
;
r
~
)
∈
γ
~
F and U
(
x
;
r
~
)
q
δ
~
F;
(5)
U
(
x
;
r
~
)
α
¯
F if U
(
x
;
r
~
)
α
F does not hold for α
∈
{
∈
γ
~
,
q
δ
~
,
∈
γ
~
∨
q
δ
~
,
∈
γ
~
∧
q
δ
~
}.Next we define a new ordering relation “⊆
∨
q
(
γ
~
,
δ
~
)” on I
F
(
X
), which is called theinterval valued fuzzy inclusion or quasicoincidence relation, as follows.For anyF
,
G
∈
I
F
(
X
) and γ
~
,
δ
~
∈
[
0,1
] such that γ
~
<
δ
~, by F
⊆
∨
q
(
γ
~
,
δ
~
)
G we mean that U
(
x
;
r
~
)
∈
γ
~
F
⇒
U
(
x
;
r
~
)
∈
γ
~
∨
q
δ
~
G for all x
∈
X and γ
~
<
r
~
≤
[
1,1
].And we define a relation “≈” on I
F
(
X
) as follows.For anyF
,
G
∈
I
F
(
X
), by F
≈
G we mean that F
⊆
∨
q
(
γ
~
,
δ
~
)
G and G
⊆
∨
q
(
γ
~
,
δ
~
)
F.In the sequel, unless otherwise stated, we letR be a hyperring, γ
~
,
δ
~
∈
D
[
0,1
], such that γ
~
<
δ
~ and denote by ⊆
∨
q
(
γ
~
,
δ
~
)
¯ that ⊆
∨
q
(
γ
~
,
δ
~
) is not true. We also emphasize that any interval valued fuzzy subset F of X must satisfy the following conditions:μ
F
~
(
x
)
≤
γ
~ or γ
~
<
μ
F
~
(
x
) and μ
F
~
(
x
)
≤
δ
~ or δ
~
<
μ
F
~
(
x
) for all x
∈
X.Lemma 2.
LetX be a nonempty set and F
,
G
∈
I
F
(
X
). Then F
⊆
∨
q
(
γ
~
,
δ
~
)
G if and only if rmax
{
μ
G
~
(
x
)
,
γ
~
}
≥
rmin
{
μ
F
~
(
x
)
,
δ
~
} for all x
∈
X.Proof.
Assume thatF
⊆
∨
q
(
γ
~
,
δ
~
)
G. Let x
∈
X. If rmax
{
μ
G
~
(
x
)
,
γ
~
}
<
rmin
{
μ
F
~
(
x
)
,
δ
~
}, then there exists r
~ such that rmax
{
μ
G
~
(
x
)
,
γ
~
}
<
r
~
<
rmin
{
μ
F
~
(
x
)
,
δ
~
}; that is, U
(
x
;
r
~
)
∈
γ
~
F but U
(
x
;
r
~
)
∈
γ
~
∨
q
δ
~
¯
G, a contradiction. Hence rmax
{
μ
G
~
(
x
)
,
γ
~
}
≥
rmin
{
μ
F
~
(
x
)
,
δ
~
}.
Conversely, assume thatrmax
{
μ
G
~
(
x
)
,
γ
~
}
≥
rmin
{
μ
F
~
(
x
)
,
δ
~
} for all x
∈
X. If F
⊆
∨
q
(
γ
~
,
δ
~
)
¯
G, then there exist x
∈
X and r
~
>
γ
~ such that U
(
x
;
r
~
)
∈
γ
~
F but U
(
x
;
r
~
)
∈
γ
~
∨
q
δ
~
¯
G, and so μ
F
~
(
x
)
≥
r
~, μ
G
~
(
x
)
<
r
~, and μ
G
~
(
x
)
+
r
~
<
2
δ
~. This gives that rmax
{
μ
G
~
(
x
)
,
γ
~
}
<
rmin
{
μ
F
~
(
x
)
,
δ
~
}, a contradiction. Hence F
⊆
∨
q
(
γ
~
,
δ
~
)
G.Lemma 3.
LetX be a nonempty set and F
,
G
,
H
∈
I
F
(
X
) such that F
⊆
∨
q
(
γ
~
,
δ
~
)
G
⊆
∨
q
(
γ
~
,
δ
~
)
H. Then F
⊆
∨
q
(
γ
~
,
δ
~
)
H.Proof.
It is straightforward by Lemma2.Note that Lemma2 gives that F
≈
G if and only if rmax
{
rmin
{
μ
F
~
(
x
)
,
δ
~
}
,
γ
~
}
=
rmax
{
rmin
{
μ
G
~
(
x
)
,
δ
~
}
,
γ
~
} for all x
∈
X and F
,
G
∈
I
F
(
X
), and it follows from Lemmas 2 and 3 that “≈” is an equivalence relation on I
F
(
X
).We now define some operations of interval valued fuzzy subsets ofR.Definition 4.
LetF
,
G
∈
I
F
(
R
). Define the sum and product of F and G, denoted by F
⊕
G and F
⊗
G, by
(6)
μ
F
⊕
G
~
(
x
)
=
⋁
x
∈
y
+
z
rmin
{
μ
F
~
(
y
)
,
μ
G
~
(
z
)
}
,
μ
F
⊗
G
~
(
x
)
=
{
⋁
x
=
y
z
rmin
{
μ
F
~
(
y
)
,
μ
G
~
(
z
)
}
if
there
exist
y
,
z
∈
R
such
that
x
=
y
z
[
0,0
]
otherwise
,
for all x
∈
R.By Lemma2 and Definition 4, we can easily deduce the following results.Lemma 5.
LetF
1
,
F
2
,
G
1
,
G
2
∈
I
F
(
R
) be such that F
1
⊆
∨
q
(
γ
~
,
δ
~
)
F
2 and G
1
⊆
∨
q
(
γ
~
,
δ
~
)
G
2. Then(1)
F
1
⊕
G
1
⊆
∨
q
(
γ
~
,
δ
~
)
F
2
⊕
G
2, F
1
⊗
G
1
⊆
∨
q
(
γ
~
,
δ
~
)
F
2
⊗
G
2;
(2)
F
1
∩
G
1
⊆
∨
q
(
γ
~
,
δ
~
)
F
2
∩
G
2.Lemma 6.
LetF
,
G
,
H
∈
I
F
(
R
). Consider(1)
F
⊕
G
=
G
⊕
F;
(2)
F
⊕
(
G
⊕
H
)
=
(
F
⊕
G
)
⊕
H
,
F
⊗
(
G
⊗
H
)
=
(
F
⊗
G
)
⊗
H;
(3)
F
⊕
(
G
∪
H
)
=
F
⊕
G
∪
F
⊕
H
,
F
⊗
(
G
∪
H
)
=
F
⊗
G
∪
F
⊗
H;
(4)
F
⊕
(
G
∩
H
)
⊆
F
⊕
G
∩
F
⊕
H
,
F
⊗
(
G
∩
H
)
⊆
F
⊗
G
∩
F
⊗
H.Lemma5 indicates that the equivalence relation “≈” is a congruence relation on (
I
F
(
R
)
,
⊕
).


## 4. Generalized Interval Valued(
α
,
β
)-Fuzzy Hyperideals

In this section, we define and investigate generalized interval valued(
α
,
β
)-fuzzy hyperideals of a hyperring, where α
∈
{
∈
γ
~
,
q
δ
~
,
∈
γ
~
∨
q
δ
~
} and β
∈
{
∈
γ
~
,
q
δ
~
,
∈
γ
~
∧
q
δ
~
,
∈
γ
~
∨
q
δ
~
}.Definition 7.
An interval valued fuzzy subsetF of R is called aninterval valued
(
α
,
β
)-fuzzy hyperideal of R if, for all γ
~
<
r
~
≤
[
1,1
], γ
~
<
t
~
≤
[
1,1
], and x
,
y
∈
R:(F1a)
U
(
x
;
r
~
)
α
F and U
(
y
;
t
~
)
α
F
⇒
U
(
z
;
rmin
{
r
~
,
t
~
}
)
β
F for all z
∈
x
+
y;
(F2a)
U
(
x
;
r
~
)
α
F
⇒
U
(
-
x
;
r
~
)
β
F for all x
∈
R;
(F3a)
U
(
x
;
r
~
)
α
F and U
(
y
;
t
~
)
α
F
⇒
U
(
x
y
;
rmin
{
r
~
,
t
~
}
)
β
F for all x
,
y
∈
R.An interval valued(
α
,
β
)-fuzzy hyperideal with γ
~
=
[
0,0
] and δ
~
=
[
0.5,0.5
] is called an interval valued (
α
,
β
)-fuzzy hyperideal studied in [21].The caseα
=
∈
γ
∧
q
δ must be omitted since for an interval valued fuzzy subset F of R such that μ
F
~
(
x
)
≤
δ
~ for any x
∈
R in the case U
(
x
;
r
~
)
∈
∧
q
F we have μ
F
~
(
x
)
≥
r
~ and μ
F
~
(
x
)
+
r
~
>
2
δ
~. Thus μ
F
~
(
x
)
+
μ
F
~
(
x
)
>
μ
F
~
(
x
)
+
r
~
>
2
δ
~, which implies μ
F
~
(
x
)
>
δ
~. This means that {
U
(
x
;
r
~
)
:
U
(
x
;
r
~
)
∈
∧
q
F
}
=
∅.As it is not difficult to see, each interval valued(
α
,
β
)-fuzzy hyperideal of R is an interval valued (
α
,
∈
γ
~
∨
q
δ
~
)-fuzzy hyperideal of R. So, in the theory of interval valued (
α
,
β
)-fuzzy hyperideals the central role is played by interval valued (
α
,
∈
γ
~
∨
q
δ
~
)-fuzzy hyperideals and we only need to investigate the properties of interval valued (
α
,
∈
γ
~
∨
q
δ
~
)-fuzzy hyperideals. In what follows, let α
∈
{
∈
γ
~
,
q
δ
~
,
∈
γ
~
∨
q
δ
~
} unless otherwise specified.Example 8.
LetR
=
{
a
,
b
,
c
,
d
} be a set with a hyperoperation “+” and a binary operation “·” as follows:
(7)
+
a
b
c
d
a
{
a
}
{
b
}
{
c
}
{
d
}
b
{
b
}
{
a
,
b
}
{
d
}
{
c
}
c
{
c
}
{
d
}
{
a
,
c
}
{
b
}
d
{
d
}
{
c
}
{
b
}
{
a
,
d
}
,
(8)
·
a
b
c
d
a
{
a
}
{
a
}
{
a
}
{
a
}
b
{
a
}
{
b
}
{
b
}
{
b
}
c
{
a
}
{
c
}
{
c
}
{
c
}
d
{
a
}
{
d
}
{
d
}
{
d
}
.
Then (
R
,
+
,
·
) is a hyperring ([21]). Define an interval valued fuzzy subset F of R by
(9)
μ
F
~
(
a
)
=
[
0.6,0.7
]
,
μ
F
~
(
b
)
=
[
0.2,0.4
]
,
μ
F
~
(
c
)
=
[
0.2,0.4
]
,
μ
F
~
(
d
)
=
[
0.2,0.4
]
.
Then F is an interval valued (
α
,
∈
[
0.2,0.4
]
∨
q
[
0.6,0.7
]
)-fuzzy hyperideal of R.Proposition 9.
An interval valued(
∈
γ
~
∨
q
δ
~
,
∈
γ
~
∨
q
δ
~
)-fuzzy hyperideal F of R is an interval valued (
∈
γ
~
,
∈
γ
~
∨
q
δ
~
)-fuzzy hyperideal of R.Proof.
This is straightforward sinceU
(
x
;
r
~
)
∈
γ
~
F implies U
(
x
;
r
~
)
∈
γ
~
∨
q
δ
~
F for all x
∈
R and γ
~
<
r
~
≤
[
1,1
].Proposition 10.
Any interval valued(
∈
γ
~
,
∈
γ
~
)-fuzzy hyperideal F of R is an interval valued (
q
δ
~
,
∈
γ
~
∨
q
δ
~
)-fuzzy hyperideal of R.Proof.
This part is straightforward.The following example shows that the converse of Propositions9 and 10 may not be true in general.Example 11.
Consider Example8. Define an interval valued fuzzy subset F of R by
(10)
μ
F
~
(
a
)
=
[
0.6,0.7
]
,
μ
F
~
(
b
)
=
[
0.7,0.8
]
,
μ
F
~
(
c
)
=
[
0.3,0.5
]
,
μ
F
~
(
d
)
=
[
0.2,0.4
]
.
Then(1)
it is routine to verify thatF is an (
∈
[
0.3,0.5
]
;
∈
[
0.3,0.5
]
∨
q
[
0.6,0.7
]
)-fuzzy hyperideal of R;
(2)
F is not an (
∈
[
0.3,0.5
]
;
∈
[
0.3,0.5
]
)-fuzzy hyperideal of R since U
(
b
;
[
0.7,0.8
]
)
∈
[
0.3,0.5
]
F but a
∈
b
+
b and U
(
a
;
[
0.7,0.8
]
)
∈
[
0.7,0.8
]
¯
F;
(3)
F is not an (
∈
[
0.3,0.5
]
∨
q
[
0.6,0.7
]
;
∈
[
0.3,0.5
]
∨
q
[
0.6,0.7
]
)-fuzzy hyperideal of R since U
(
b
;
[
1,1
]
)
∈
[
0.3,0.5
]
∨
q
[
0.6,0.7
]
F and U
(
c
;
[
1,1
]
)
∈
[
0.3,0.5
]
∨
q
[
0.6,0.7
]
F but U
(
b
+
c
;
[
1,1
]
)
=
U
(
d
;
[
1,1
]
)
∈
[
0.3,0.5
]
∨
q
[
0.6,0.7
]
¯
F.Theorem 12.
Let2
δ
~
=
[
1,1
]
+
γ
~ and F be an interval valued (
α
,
∈
γ
~
∨
q
δ
~
)-fuzzy hyperideal of R. Then the set F
γ
~
^ is a hyperideal of R, where F
γ
~
^
=
{
x
∈
R
|
μ
F
~
(
x
)
>
γ
~
}.Proof.
Assume thatF is an interval valued (
α
,
∈
γ
~
∨
q
δ
~
)-fuzzy hyperideal of R. Let x
,
y
∈
F
γ
~
^. Then μ
F
~
(
x
)
>
γ
~, μ
F
~
(
y
)
>
γ
~. We consider the following two cases.
Case
1. If α
∈
{
∈
γ
~
,
∈
γ
~
∨
q
δ
~
}, then U
(
x
;
μ
F
~
(
x
)
)
α
F and U
(
y
;
μ
F
~
(
y
)
)
α
F. By (F1a), U
(
z
;
rmin
{
μ
F
~
(
x
)
,
μ
F
~
(
y
)
}
)
∈
γ
~
∨
q
δ
~
F for all z
∈
x
+
y, that is, U
(
z
;
rmin
{
μ
F
~
(
x
)
,
μ
F
~
(
y
)
}
)
∈
γ
~
F or U
(
z
;
rmin
{
μ
F
~
(
x
)
,
μ
F
~
(
y
)
}
)
q
δ
~
F. It follows that F
(
z
)
≥
rmin
{
μ
F
~
(
x
)
,
μ
F
~
(
y
)
}
>
γ
~ or μ
F
~
(
z
)
+
rmin
{
μ
F
~
(
x
)
,
μ
F
~
(
y
)
}
>
2
δ
~, and so μ
F
~
(
z
)
≥
rmin
{
μ
F
~
(
x
)
,
μ
F
~
(
y
)
}
>
γ
~ or μ
F
~
(
z
)
>
2
δ
~
-
rmin
{
μ
F
~
(
x
)
,
μ
F
~
(
y
)
}
≥
2
δ
~
-
[
1,1
]
=
γ
~. Hence μ
F
~
(
z
)
>
γ
~ and so z
∈
F
γ
~
^. Therefore, x
+
y
⊆
F
γ
~.
By (F2a) and (F3a),U
(
-
x
;
μ
F
~
(
x
)
)
∈
γ
~
∨
q
δ
~
F and U
(
x
y
;
rmin
{
μ
F
~
(
x
)
,
μ
F
~
(
y
)
}
)
∈
γ
~
∨
q
δ
~
F. Analogous to the above proof, we have μ
F
~
(
x
)
>
γ
~ and μ
F
~
(
x
y
)
>
γ
~; that is, -
x
∈
F
γ
~ and x
y
∈
F
γ
~. Therefore, F
γ
~
^ is a hyperideal of R.
Case
2. If α
=
q
δ
~, then U
(
x
;
[
1,1
]
)
α
F and U
(
y
;
[
1,1
]
)
α
F since 2
δ
~
=
[
1,1
]
+
γ
~. Analogous to the proof of Case 1, we may prove that F
γ
~
^ is a hyperideal of R.If we takeγ
~
=
[
0,0
] and δ
~
=
[
0.5,0.5
] in Theorem 12, then we have the following corollary.Corollary 13.
LetF be an interval valued (
α
,
∈
∨
q
)-fuzzy hyperideal of R. Then the set F
[
0,0
]
^ is a hyperideal of R, where α
∈
{
∈
,
q
,
∈
∨
q
}.Theorem 14.
Let2
δ
~
=
[
1,1
]
+
γ
~ and A be a nonempty subset of R. Then A is a hyperideal of R if and only if the interval valued fuzzy subset F of R such that μ
F
~
(
x
)
≥
δ
~ for all x
∈
A and μ
F
~
(
x
)
≤
γ
~ otherwise is an interval valued (
α
,
∈
γ
~
∨
q
δ
~
)-fuzzy hyperideal of R.Proof.
Assume thatA is a hyperideal of R. Let x
,
y
∈
R and γ
~
<
r
~
≤
[
1,1
]
,
γ
~
<
t
~
≤
[
1,1
] be such that U
(
x
;
r
~
)
α
F and U
(
y
;
t
~
)
α
F. Then we have the following four cases.
Case
1. If U
(
x
;
r
~
)
∈
γ
~
F and U
(
y
;
t
~
)
∈
γ
~
F, then μ
F
~
(
x
)
≥
r
~
>
γ
~ and μ
F
~
(
y
)
≥
t
~
>
γ
~. Thus, μ
F
~
(
x
)
≥
δ
~ and μ
F
~
(
y
)
≥
δ
~; that is, x
,
y
∈
A.
Case
2. If U
(
x
;
r
~
)
q
δ
~
F and U
(
y
;
t
~
)
q
δ
~
F, then μ
F
~
(
x
)
+
r
~
>
2
δ
~ and μ
F
~
(
y
)
+
t
~
>
2
δ
~, and so μ
F
~
(
x
)
>
2
δ
~
-
r
~
≥
2
δ
~
-
[
1,1
]
=
γ
~ and μ
F
~
(
y
)
>
2
δ
~
-
t
~
≥
2
δ
~
-
[
1,1
]
=
γ
~. It follows that μ
F
~
(
x
)
≥
δ
~ and μ
F
~
(
y
)
≥
δ
~; that is, x
,
y
∈
A.
Case
3. If U
(
x
;
r
~
)
∈
γ
~
F and U
(
y
;
t
~
)
q
δ
~
F, then μ
F
~
(
x
)
≥
r
~
>
γ
~ and μ
F
~
(
y
)
+
t
~
>
2
δ
~. Analogous to the proof of Cases 1 and 2, μ
F
~
(
x
)
≥
δ
~ and μ
F
~
(
y
)
≥
δ
~; that is, x
,
y
∈
A.
Case
4. If U
(
x
;
r
~
)
q
δ
~
F and U
(
y
;
t
~
)
∈
γ
~
F, then μ
F
~
(
x
)
+
r
~
>
2
δ
~ and μ
F
~
(
y
)
≥
t
~
>
γ
~. Analogous to the proof of Cases 1 and 2, μ
F
~
(
x
)
≥
δ
~ and μ
F
~
(
y
)
≥
δ
~; that is, x
,
y
∈
A.
Thus, in any case,x
,
y
∈
A. Hence z
∈
A for all z
∈
x
+
y, which implies that μ
F
~
(
z
)
≥
δ
~. If rmin
{
μ
F
~
(
x
)
,
μ
F
~
(
y
)
}
≤
δ
~, then μ
F
~
(
z
)
≥
δ
~
≥
rmin
{
μ
F
~
(
x
)
,
μ
F
~
(
y
)
}
>
γ
~; it follows that U
(
z
;
rmin
{
μ
F
~
(
x
)
,
μ
F
~
(
y
)
}
)
∈
γ
~
F. If rmin
{
μ
F
~
(
x
)
,
μ
F
~
(
y
)
}
>
δ
~, then μ
F
~
(
z
)
+
rmin
{
μ
F
~
(
x
)
,
μ
F
~
(
y
)
}
>
δ
~
+
δ
~
=
2
δ
~; it follows that U
(
z
;
rmin
{
μ
F
~
(
x
)
,
μ
F
~
(
y
)
}
)
q
δ
~
f. Therefore, U
(
z
;
rmin
{
μ
F
~
(
x
)
,
μ
F
~
(
y
)
}
)
∈
γ
~
∨
q
δ
~
f. In a similar way, we may prove that U
(
-
x
;
μ
F
~
(
x
)
)
∈
γ
~
∨
q
δ
~
F and U
(
x
y
;
rmin
{
μ
F
~
(
x
)
,
μ
F
~
(
y
)
}
)
∈
γ
~
∨
q
δ
~
F. Therefore, F is an interval valued (
α
,
∈
γ
~
∨
q
δ
~
)-fuzzy hyperideal of R.
Conversely, assume thatF is an interval valued (
α
,
∈
γ
~
∨
q
δ
~
)-fuzzy hyperideal of R. It is easy to see that A
=
F
γ
~
^. It follows from Theorem 12 that A is a hyperideal of R.If we takeγ
~
=
[
0,0
] and δ
~
=
[
0.5,0.5
] in Theorem 14, then we have the following corollary.Corollary 15.
LetA be a nonempty subset of R. Then A is a hyperideal of R if and only if the interval valued fuzzy subset F of R such that μ
F
~
(
x
)
≥
[
0.5,0.5
] for all x
∈
A and μ
F
~
(
x
)
=
[
0,0
] otherwise is an (
α
,
∈
∨
q
)-fuzzy hyperideal of R, where α
∈
{
∈
,
q
,
∈
∨
q
}.Theorem 16.
LetF be an (
q
δ
~
,
∈
γ
~
∨
q
δ
~
)-fuzzy hyperideal of R. Then the following conditions hold:(F1b)
rmax
{
⋀
z
∈
x
+
y
μ
F
~
(
z
)
,
γ
~
}
≥
rmin
{
μ
F
~
(
x
)
,
μ
F
~
(
y
)
,
δ
~
} for all x
,
y
∈
R;
(F2b)
rmax
{
μ
F
~
(
-
x
)
,
γ
~
}
≥
rmin
{
μ
F
~
(
x
)
,
δ
~
} for all x
∈
R;
(F3b)
rmax
{
μ
F
~
(
x
y
)
,
γ
~
}
≥
rmin
{
μ
F
~
(
x
)
,
μ
F
~
(
y
)
,
δ
~
} for all x
∈
R.Proof.
LetF be an (
q
δ
~
,
∈
γ
~
∨
q
δ
~
)-fuzzy hyperideal of R. We only show (F1b). The other properties can be similarly proved. Assume that there exist x
,
y
,
z
∈
R such that z
∈
x
+
y and rmax
{
μ
F
~
(
z
)
,
γ
~
}
<
rmin
{
μ
F
~
(
x
)
,
μ
F
~
(
y
)
,
δ
~
}. Then, for all r
~
∈
D
[
0,1
] such that 2
δ
~
-
rmax
{
μ
F
~
(
z
)
,
γ
~
}
>
r
~
>
2
δ
~
-
rmin
{
μ
F
~
(
x
)
,
μ
F
~
(
y
)
,
δ
~
}, we have
(11)
2
δ
~
-
μ
F
~
(
z
)
≥
2
δ
~
-
rmax
{
μ
F
~
(
z
)
,
γ
~
}
>
r
~
>
rmax
{
2
δ
~
-
μ
F
~
(
x
)
,
2
δ
~
-
μ
F
~
(
y
)
,
δ
~
}
,
and so μ
F
~
(
x
)
+
r
~
>
2
δ
~, μ
F
~
(
y
)
+
r
~
>
2
δ
~, μ
F
~
(
z
)
+
r
~
<
2
δ
~, and μ
F
~
(
z
)
<
δ
~
<
r
~. Hence U
(
x
;
r
~
)
q
δ
~
F, U
(
y
;
r
~
)
q
δ
~
F but U
(
z
;
r
~
)
∈
γ
~
∨
q
δ
~
¯
F, a contradiction. Hence (F1b) is valid.


## 5. Interval Valued(
∈
γ
~
,
∈
γ
~
∨
q
δ
~
)-Fuzzy Hyperideals

In this section, using a new ordering relation onI
F
(
R
), we investigate the interval valued (
∈
γ
~
,
∈
γ
~
∨
q
δ
~
)-fuzzy hyperideals of a hyperring. In what follows, we take α
=
∈
γ
~ and β
=
∈
γ
~
∨
q
δ
~in Definition 7. Before proceeding, we present some characterizations of interval valued (
∈
γ
~
,
∈
γ
~
∨
q
δ
~
)-fuzzy hyperideals.Lemma 17.
LetF
∈
I
F
(
H
). Then (F1a) holds if and only if one of the following conditions holds:(F1b)
rmax
{
⋀
z
∈
x
+
y
μ
F
~
(
z
)
,
γ
~
}
≥
rmin
{
μ
F
~
(
x
)
,
μ
F
~
(
y
)
,
δ
~
} for all x
,
y
∈
R;
(F1c)
F
⊕
F
⊆
∨
q
(
γ
~
,
δ
~
)
F.Proof.
(F1a)⇒(F1b) Let x
,
y
∈
R. Let, if possible, z
∈
R be such that z
∈
x
+
y and rmax
{
μ
F
~
(
z
)
,
γ
~
}
<
r
~
=
rmin
{
μ
F
~
(
x
)
,
μ
F
~
(
y
)
,
δ
~
}. Then U
(
x
;
r
~
)
∈
γ
~
F, U
(
y
;
r
~
)
∈
γ
~
F, U
(
z
;
r
~
)
∈
γ
~
¯
F, and μ
F
~
(
z
)
+
r
~
<
r
~
+
r
~
≤
2
δ
~; that is, U
(
z
;
r
~
)
∈
γ
~
∨
q
δ
~
¯
F, a contradiction. Hence (F1b) is valid.
(F1b)⇒(F1c) For U
(
x
;
r
~
)
∈
γ
~
F
⊕
F, suppose, if possible, U
(
x
;
r
~
)
∈
γ
~
∨
q
δ
~
¯
F. Then U
(
x
;
r
~
)
∈
γ
~
¯
F and U
(
x
;
r
~
)
q
δ
~
¯
F; that is, μ
F
~
(
x
)
<
r
~ and μ
F
~
(
x
)
+
r
~
≤
2
δ
~ which implies that μ
F
~
(
x
)
<
δ
~. If x
∈
y
+
z for some y
,
z
∈
R, by (F1b), we have δ
~
>
rmax
{
μ
F
~
(
x
)
,
γ
~
}
≥
rmin
{
μ
F
~
(
y
)
,
μ
F
~
(
z
)
,
δ
~
}, it follows that rmax
{
μ
F
~
(
x
)
,
γ
~
}
≥
rmin
{
μ
F
~
(
y
)
,
μ
F
~
(
z
)
}. Hence we have
(12)
r
~
≤
(
F
⊕
F
)
(
x
)
=
⋁
x
∈
a
+
b
rmin
{
μ
F
~
(
y
)
,
μ
F
~
(
z
)
}
≤
⋁
x
∈
a
+
b
rmax
{
μ
F
~
(
x
)
,
γ
~
}
=
rmax
{
μ
F
~
(
x
)
,
γ
~
}
;
which contradicts γ
~
<
r
~ and μ
F
~
(
x
)
<
r
~. Hence (F1c) is satisfied.
(F1c)⇒(F1a) Let x
,
y
∈
R and γ
~
<
r
~
≤
[
1,1
]
,
γ
~
<
t
~
≤
[
1,1
] be such that U
(
x
;
r
~
)
∈
γ
~
F and U
(
y
;
t
~
)
∈
γ
~
F. Then for any z
∈
x
+
y, we have
(13)
(
F
⊕
F
)
(
z
)
=
⋁
z
∈
a
+
b
rmin
{
μ
F
~
(
a
)
,
μ
F
~
(
b
)
}
≥
rmin
{
μ
F
~
(
x
)
,
μ
F
~
(
y
)
}
≥
rmin
{
r
~
,
t
~
}
>
γ
~
.
Hence U
(
z
;
rmin
{
μ
F
~
(
x
)
,
μ
F
~
(
y
)
}
)
∈
γ
~
F
⊕
F and so (F1a) holds.Lemma 18.
LetF
∈
I
F
(
R
). Then (F2a) holds if and only if one of the following conditions holds:(F2b)
rmax
{
μ
F
~
(
-
x
)
,
γ
~
}
≥
rmin
{
μ
F
~
(
x
)
,
δ
~
} for all x
∈
R;
(F2c)
F
⊆
∨
q
(
γ
~
,
δ
~
)
-
F, where -
F is defined by μ
-
F
~
(
x
)
=
μ
F
~
(
-
x
) for all x
∈
R.Proof.
The proof is analogous to the proof of Lemma17.Lemma 19.
LetF
∈
I
F
(
R
). Then (F3a) holds if and only if one of the following conditions holds:(F3b)
rmax
{
μ
F
~
(
x
y
)
,
γ
~
}
≥
rmin
{
μ
F
~
(
x
)
,
μ
F
~
(
y
)
,
δ
~
} for all x
∈
R;
(F3c)
F
⊗
F
⊆
∨
q
(
γ
~
,
δ
~
)
F.Proof.
The proof is analogous to Lemma17.From Lemmas17–19, we obtain the following results.Theorem 20.
LetF
∈
I
F
(
R
). Then F is an (
∈
γ
~
,
∈
γ
~
∨
q
δ
~
)-fuzzy hyperideal of R if and only if it satisfies conditions (F1b)–(F3b) or conditions (F1c)–(F3c).Remark 21.
For any interval valued(
∈
γ
~
,
∈
γ
~
∨
q
δ
~
)-fuzzy hyperideal F of R, we can conclude that(1)
F is an interval valued fuzzy hyperideal of R when γ
~
=
[
0,0
] and δ
~
=
[
1,1
] ([21], Definition 4.1);
(2)
F is an interval valued (
∈
,
∈
∨
q
)-fuzzy hyperideal of R when γ
~
=
[
0,0
] and δ
~
=
[
0.5,0.5
] ([21], Definition 4.3);
(3)
F is an interval valued fuzzy hyperideal of R with thresholds (
γ
~
,
δ
~
) ([21], Definition 4.9).Combining Theorems16 and 20, we have the following result.Proposition 22.
Any interval valued(
q
δ
~
,
∈
γ
~
∨
q
δ
~
)-fuzzy hyperideal of R is an interval valued (
∈
γ
~
,
∈
γ
~
∨
q
δ
~
)-fuzzy hyperideal of R.The following example shows that the converse of Proposition22 is not true in general.Example 23.
LetR and F be as in Example 11. Then F is an (
∈
[
0.3,0.5
]
;
∈
[
0.3,0.5
]
∨
q
[
0.6,0.7
]
)-fuzzy hyperideal of R, but it is not an (
q
[
0.6,0.7
]
;
∈
[
0.3,0.5
]
∨
q
[
0.6,0.7
]
)-fuzzy hyperideal of R since U
(
a
;
[
0.7,0.8
]
)
q
[
0.6,0.7
]
F and U
(
c
;
[
1,1
]
)
∈
[
0.3,0.5
]
∨
q
[
0.6,0.7
]
F but U
(
a
+
c
;
rmin
{
[
0.7,0.8
]
,
[
1,1
]
}
)
=
U
(
c
;
[
0.7,0.8
]
)
∈
[
0.3,0.5
]
∨
q
[
0.6,0.7
]
¯
F.If we takeγ
~
=
[
0,0
] and δ
~
=
[
0.5,0.5
] in Propositions 9, 10, and 22, then we have the following corollary.Corollary 24.
LetF
∈
I
F
(
R
). Then(1)
any interval valued(
∈
∨
q
,
∈
∨
q
)-fuzzy hyperideal of R is an interval valued (
∈
,
∈
∨
q
)-fuzzy hyperideal of R [21];
(2)
any interval valued(
q
,
∈
∨
q
)-fuzzy hyperideal of R is an interval valued (
∈
,
∈
∨
q
)-fuzzy hyperideal of R;
(3)
any interval valued(
∈
,
∈
)-fuzzy hyperideal of R is an interval valued (
∈
∨
q
,
∈
∨
q
)-fuzzy hyperideal of R [21].For anyF
∈
I
F
(
R
), we define F
r
~
=
{
x
∈
R
∣
U
(
x
;
r
~
)
∈
γ
~
F
}, F
r
~
δ
~
=
{
x
∈
R
∣
U
(
x
;
r
~
)
q
δ
~
F
}, and [
F
]
r
~
δ
~
=
{
x
∈
R
∣
U
(
x
;
r
~
)
∈
γ
~
∨
q
δ
~
F
} for all r
~
∈
D
′
[
0,1
]. It is clear that [
F
]
r
~
δ
~
=
F
r
~
∪
F
r
~
δ
~.The next theorem provides the relationship between the interval valued(
∈
γ
~
,
∈
γ
~
∨
q
δ
~
)-fuzzy hyperideals of R and the crisp hyperideals of R.Theorem 25.
LetF
∈
I
F
(
R
).(1)
F is an interval valued (
∈
γ
~
,
∈
γ
~
∨
q
δ
~
)-fuzzy hyperideal of R if and only if F
r
~
(
F
r
~
≠
∅
) is an hyperideal of R for all γ
~
<
r
~
≤
δ
~.
(2)
If2
δ
=
1
+
γ, then F is an interval valued (
∈
γ
~
,
∈
γ
~
∨
q
δ
~
)-fuzzy hyperideal of R if and only if F
r
~
δ
~
(
F
r
~
δ
~
≠
∅
) is a hyperideal of R for all δ
~
<
r
~
≤
[
1,1
].
(3)
If2
δ
=
1
+
γ, then F is an interval valued (
∈
γ
~
,
∈
γ
~
∨
q
δ
~
)-fuzzy hyperideal of R if and only if [
F
]
r
~
δ
~
(
[
F
]
r
~
δ
~
≠
∅
) is a hyperideal of R for all γ
~
<
r
~
≤
[
1,1
].Proof.
We only show (3). LetF be an interval valued (
∈
γ
~
,
∈
γ
~
∨
q
δ
~
)-fuzzy hyperideal of R and x
,
y
∈
[
F
]
r
~
δ
~ for some γ
~
<
r
~
≤
[
1,1
]. Then U
(
x
;
r
~
)
∈
γ
~
∨
q
δ
~
F and U
(
y
;
r
~
)
∈
γ
~
∨
q
δ
~
F; that is, μ
F
~
(
x
)
≥
r
~ or μ
F
~
(
x
)
>
2
δ
~
-
r
~
>
2
δ
~
-
[
1,1
]
=
γ
~, and μ
F
~
(
y
)
≥
r
~ or μ
F
~
(
y
)
>
2
δ
~
-
r
~
>
2
δ
~
-
[
1,1
]
=
γ
~. Since F is an interval valued (
∈
γ
~
,
∈
γ
~
∨
q
δ
~
)-fuzzy hyperideal of R, we have μ
F
~
(
z
)
≥
rmin
{
μ
F
~
(
x
)
,
μ
F
~
(
y
)
,
δ
~
} for all z
∈
x
+
y since γ
~
<
rmin
{
μ
F
~
(
x
)
,
μ
F
~
(
y
)
,
δ
~
}. We consider the following cases.
Case
1. Consider γ
~
<
r
~
≤
δ
~. Since γ
~
<
r
~
≤
δ
~, we have 2
δ
~
-
r
~
≥
δ
~
≥
r
~ and so μ
F
~
(
z
)
≥
rmin
{
r
~
,
r
~
,
δ
~
}
=
r
~ or μ
F
~
(
z
)
≥
rmin
{
r
~
,
2
δ
~
-
r
~
,
δ
~
}
=
r
~, or μ
F
~
(
z
)
≥
rmin
{
2
δ
~
-
r
~
,
2
δ
~
-
r
~
,
δ
~
}
=
r
~
=
δ
~
≥
r
~. Hence U
(
z
;
r
~
)
∈
γ
~
F.
Case
2. Consider δ
~
<
r
~
≤
[
1,1
]. Since δ
~
<
r
~
≤
[
1,1
], we have 2
δ
~
-
r
~
<
δ
~
<
r
~ and so μ
F
~
(
z
)
≥
rmin
{
r
~
,
r
~
,
δ
~
}
=
δ
~
>
2
δ
~
-
r
~ or μ
F
~
(
z
)
>
rmin
{
r
~
,
2
δ
~
-
r
~
,
δ
~
}
=
2
δ
~
-
r
~, or μ
F
~
(
z
)
>
rmin
{
2
δ
~
-
r
~
,
2
δ
~
-
r
~
,
δ
~
}
=
2
δ
~
-
r
~. Hence U
(
z
;
r
~
)
q
δ
~
F.
Thus, in any case,U
(
z
;
r
~
)
∈
γ
~
∨
q
δ
~
F; that is, z
∈
[
F
]
r
~
δ
~ for all z
∈
x
+
y. Similarly we can show that -
x
∈
[
F
]
r
~
δ
~ and x
y
∈
[
F
]
r
~
δ
~. Therefore, [
F
]
r
~
δ
~ is a hyperideal of R.
Conversely, assume that the given condition holds. Letx
,
y
∈
R. If there exists z
∈
R such that z
∈
x
+
y and rmax
{
μ
F
~
(
z
)
,
γ
~
}
<
r
~
=
rmin
{
μ
F
~
(
x
)
,
μ
F
~
(
y
)
,
δ
~
}, then x
,
y
∈
[
F
]
r
~
δ
~ but z
∉
[
F
]
r
~
δ
~, a contradiction. Therefore, rmax
{
μ
F
~
(
z
)
,
γ
~
}
≥
rmin
{
μ
F
~
(
x
)
,
μ
F
~
(
y
)
,
δ
~
} for all z
∈
x
+
y. Similarly we can show that conditions (F2b) and (F3b) are valid. Therefore, F is an interval valued (
∈
γ
~
,
∈
γ
~
∨
q
δ
~
)-fuzzy hyperideal of R by Theorem 20.As a direct consequence of Theorem25, we have the following result.Corollary 26.
Letγ
~
,
γ
~
′
,
δ
~
,
δ
~
′
∈
D
[
0,1
] be such that γ
~
<
δ
~, γ
~
′
<
δ
~
′, γ
~
<
γ
~
′, and δ
~
′
<
δ
~. Then every interval valued (
∈
γ
~
,
∈
γ
~
∨
q
δ
~
)-fuzzy hyperideal of R is an interval valued (
∈
γ
~
′
,
∈
γ
~
′
∨
q
δ
~
′
)-fuzzy hyperideal of R.The following example shows that the converse of Corollary26 is not true in general.Example 27.
Consider Example8. Define interval valued fuzzy subset F of R by
(14)
μ
F
~
(
a
)
=
[
0.6,0.7
]
,
μ
F
~
(
b
)
=
[
0.2,0.3
]
,
μ
F
~
(
c
)
=
[
0.3,0.4
]
,
μ
F
~
(
d
)
=
[
0.3,0.4
]
.
Then F is an interval valued (
∈
[
0.3,0.4
]
,
∈
[
0.3,0.4
]
∨
q
[
0.6,0.7
]
)-fuzzy hyperideal of R but is not an interval valued (
∈
[
0.2,0.3
]
,
∈
[
0.2,0.3
]
∨
q
[
0.6,0.7
]
)-fuzzy hyperideal of R.If we takeγ
~
=
[
0,0
] and δ
~
=
[
0.5,0.5
] in Theorem 25, then we have the following corollary.Corollary 28.
LetF
∈
I
F
(
R
).(1)
F is an interval valued (
∈
,
∈
∨
q
)-fuzzy hyperideal of R if and only if F
r
~
(
F
r
~
≠
∅
) is a hyperideal of R for all [
0,0
]
<
r
~
≤
[
0.5,0.5
].
(2)
F is an interval valued (
∈
,
∈
∨
q
)-fuzzy hyperideal of R if and only if F
r
~
[
0.5,0.5
]
(
F
r
~
[
0.5,0.5
]
≠
∅
) is a hyperideal of R for all [
0.5,0.5
]
<
r
~
≤
[
1,1
].
(3)
F is an interval valued (
∈
,
∈
∨
q
)-fuzzy hyperideal of R if and only if [
F
]
r
~
[
0.5,0.5
]
(
[
F
]
r
~
[
0.5,0.5
]
≠
∅
) is a hyperideal of R for all [
0,0
]
<
r
~
≤
[
1,1
].Theorem 29.
LetF and G be two interval valued (
∈
γ
~
,
∈
γ
~
∨
q
δ
~
)-fuzzy hyperideals of R. Then so are F
⊕
G and F
∩
G.Proof.
We only show thatF
⊕
G is an interval valued (
∈
γ
~
,
∈
γ
~
∨
q
δ
~
)-fuzzy hyperideal of R. The verification is as follows.(1)
By Lemmas5, 6, and 17, (
F
⊕
G
)
⊕
(
F
⊕
G
)
=
(
F
⊕
F
)
⊕
(
G
⊕
G
)
⊆
∨
q
(
γ
~
,
δ
~
)
F
⊕
G.
(2)
Letx
∈
R. Then
(15)
rmax
{
μ
F
⊕
G
~
(
-
x
)
,
γ
~
}
=
rmax
{
⋁
-
x
∈
y
+
z
rmin
{
μ
F
~
(
y
)
,
μ
G
~
(
z
)
}
,
γ
~
}
=
⋁
-
x
∈
y
+
z
rmin
{
rmax
{
μ
F
~
(
y
)
,
γ
~
}
,
rmax
{
μ
G
~
(
z
)
,
γ
~
}
}
=
⋁
x
∈
-
y
-
z
rmin
{
rmax
{
μ
F
~
(
y
)
,
γ
~
}
,
rmax
{
μ
G
~
(
z
)
,
γ
~
}
}
≥
⋁
x
∈
-
y
-
z
rmin
{
rmin
{
μ
F
~
(
-
y
)
,
δ
~
}
,
rmin
{
μ
G
~
(
-
z
)
,
δ
~
}
}
=
rmin
{
⋁
x
∈
-
y
-
z
rmin
{
μ
F
~
(
-
y
)
,
μ
G
~
(
-
z
)
}
,
δ
~
}
=
rmin
{
μ
F
⊕
G
~
(
x
)
,
δ
~
}
.
(3)
Letx
,
y
∈
R. We have
(16)
rmin
{
μ
F
⊕
G
~
(
x
)
,
δ
~
}
=
rmin
{
⋁
x
∈
a
+
b
rmin
{
μ
F
~
(
a
)
,
μ
G
~
(
b
)
}
,
δ
~
}
=
⋁
x
∈
a
+
b
rmin
{
rmin
{
μ
F
~
(
a
)
,
δ
~
}
,
rmin
{
μ
G
~
(
b
)
,
δ
~
}
}
≤
⋁
x
∈
a
+
b
rmin
{
rmax
{
μ
F
~
(
a
y
)
,
γ
~
}
,
rmax
{
μ
G
~
(
b
y
)
,
γ
~
}
}
=
rmax
{
⋁
x
∈
a
+
b
rmin
{
μ
F
~
(
a
y
)
,
μ
G
~
(
b
y
)
}
,
γ
~
}
≤
rmax
{
μ
F
⊕
G
~
(
x
y
)
,
γ
~
}
.In a similar way, we have rmax
{
μ
F
⊕
G
~
(
x
y
)
,
γ
~
}
≥
rmin
{
μ
F
⊕
G
~
(
y
)
,
δ
~
}. Hence, rmax
{
μ
F
⊕
G
~
(
x
y
)
,
γ
~
}
≥
rmin
{
μ
F
⊕
G
~
(
x
)
,
μ
F
⊕
G
~
(
y
)
,
δ
~
}.Theorem 30.
LetI
F
I
(
R
) be the set of all interval valued (
∈
γ
~
,
∈
γ
~
∨
q
δ
~
)-fuzzy hyperideals of R with the same tip “t” (i.e., μ
F
~
(
0
)
=
μ
G
~
(
0
) for all F
,
G
∈
I
F
I
(
R
)
). Then (
I
F
I
(
R
)
,
∩
,
⊕
,
⊆
∨
q
(
γ
~
,
δ
~
)
) is a modular lattice.Proof.
LetF
,
G
∈
I
F
I
(
R
). Clearly, F
∩
G is the greatest lower bound of F and G. By Theorem 29, we have F
⊕
G
∈
I
F
I
(
R
). Now, we prove that F
⊕
G
=
F
∨
G. Let x
∈
R. Since x
=
0
+
x and rmax
{
μ
F
~
(
0
)
,
γ
~
}
=
rmax
{
μ
G
~
(
0
)
,
γ
~
}
≥
rmin
{
μ
G
~
(
y
)
,
δ
~
} for all y
∈
R, we have
(17)
rmax
{
μ
F
⊕
G
~
(
x
)
,
γ
~
}
=
rmax
{
⋁
x
∈
y
+
z
rmin
{
μ
F
~
(
y
)
,
μ
G
~
(
z
)
}
,
γ
~
}
≥
rmax
{
rmin
{
μ
F
~
(
0
)
,
μ
G
~
(
x
)
}
,
γ
~
}
=
rmin
{
rmax
{
μ
F
~
(
0
)
,
γ
~
}
,
rmax
{
μ
G
~
(
x
)
,
γ
~
}
}
≥
rmin
{
rmin
{
μ
G
~
(
x
)
,
δ
~
}
,
rmin
{
μ
G
~
(
x
)
,
δ
~
}
}
=
rmin
{
μ
G
~
(
x
)
,
δ
~
}
;
it follows that F
⊆
∨
q
(
γ
~
,
δ
~
)
F
⊕
G. Similarly, G
⊆
∨
q
(
γ
~
,
δ
~
)
F
⊕
G and so F
⊕
G is an upper bound of F and G. Now, let H
∈
I
F
I
(
R
) be such that F
⊆
∨
q
(
γ
~
,
δ
~
)
H and G
⊆
∨
q
(
γ
~
,
δ
~
)
H. It is easy to see F
⊕
G
⊆
∨
q
(
γ
~
,
δ
~
)
H. So, F
⊕
G
=
F
∨
G. Therefore, (
I
F
I
(
R
)
,
∩
,
⊕
,
⊆
∨
q
(
γ
~
,
δ
~
)
) is a lattice.
For modularity, we first letF
,
G
,
H
∈
I
F
I
(
R
) be such that F
⊆
∨
q
(
γ
~
,
δ
~
)
H. We need to show that F
∩
(
G
⊕
H
)
≈
(
F
∩
G
)
⊕
H. But, the relation (
F
∩
G
)
⊕
H
⊆
∨
q
(
γ
~
,
δ
~
)
F
∩
(
G
⊕
H
) is clear. It remains to show that F
∩
(
G
⊕
H
)
⊆
∨
q
(
γ
~
,
δ
~
)
(
F
∩
G
)
⊕
H. Now let x
∈
R. Then we have
(18)
rmax
{
μ
(
F
∩
G
)
⊕
H
~
(
x
)
,
γ
~
}
=
rmax
{
⋁
x
∈
y
+
z
rmin
{
μ
F
∩
G
~
(
y
)
,
μ
H
~
(
z
)
}
,
γ
~
}
=
rmax
{
⋁
x
∈
y
+
z
rmin
{
μ
F
~
(
y
)
,
μ
G
~
(
y
)
,
μ
H
~
(
z
)
}
,
γ
~
}
=
⋁
x
∈
y
+
z
rmin
{
rmax
{
μ
F
~
(
y
)
,
γ
~
}
,
rmax
{
μ
G
~
(
y
)
,
γ
~
}
,
m
m
m
m
m
m
m
rmax
{
μ
H
~
(
z
)
,
γ
~
}
}
≥
⋁
x
∈
y
+
z
rmin
{
rmin
{
μ
F
~
(
x
)
,
μ
F
~
(
-
z
)
,
δ
~
}
,
mmmmmmmi
rmax
{
μ
G
~
(
y
)
,
γ
~
}
,
rmax
{
μ
H
~
(
z
)
,
γ
~
}
}
=
⋁
x
∈
y
+
z
rmin
{
rmin
{
μ
F
~
(
x
)
,
δ
~
}
,
rmin
{
μ
F
~
(
-
z
)
,
δ
~
}
,
m
m
m
m
m
m
m
i
rmax
{
μ
G
~
(
y
)
,
γ
~
}
,
rmax
{
μ
H
~
(
z
)
,
γ
~
}
}
=
⋁
x
∈
y
+
z
rmin
{
rmin
{
μ
F
~
(
x
)
,
δ
~
}
,
rmax
{
μ
G
~
(
y
)
,
γ
~
}
,
m
m
m
m
m
m
m
m
rmax
{
μ
H
~
(
z
)
,
γ
~
}
}
(
it
follows
from
F
⊆
∨
q
(
γ
~
,
δ
~
)
H
that
m
m
m
rmin
{
μ
F
~
(
-
z
)
,
δ
~
}
≤
rmax
{
μ
H
~
(
z
)
,
γ
~
}
)
≥
⋁
x
∈
y
+
z
rmin
{
rmin
{
μ
F
~
(
x
)
,
δ
~
}
,
rmin
{
μ
G
~
(
y
)
,
δ
~
}
,
m
m
m
m
m
m
m
m
rmin
{
μ
H
~
(
z
)
,
δ
~
}
}
=
rmin
{
rmin
{
μ
F
~
(
x
)
,
⋁
x
∈
y
+
z
rmin
{
μ
G
~
(
y
)
,
μ
H
~
(
z
)
}
}
,
δ
~
}
=
rmin
{
rmin
{
μ
F
~
(
x
)
,
μ
G
⊕
H
~
(
x
)
}
,
δ
~
}
=
rmin
{
μ
F
∩
(
G
⊕
H
)
~
(
x
)
,
δ
~
}
.
It follows that F
∩
(
G
⊕
H
)
⊆
∨
q
(
γ
~
,
δ
~
)
(
F
∩
G
)
⊕
H. Thus F
∩
(
G
⊕
H
)
≈
(
F
∩
G
)
⊕
H and so (
I
F
I
(
R
)
,
∩
,
⊕
,
⊆
∨
q
(
γ
~
,
δ
~
)
) is a modular lattice.


## 6. Conclusions

In this paper, we introduced and studied the generalized interval valued(
α
,
β
)-fuzzy hyperideals of a hyperring, with α
,
β
∈
{
∈
,
q
,
∈
∧
q
,
∈
∨
q
} and α
≠
∈
∧
q, in which special attention was concentrated on the interval valued (
∈
γ
~
,
∈
γ
~
∨
q
δ
~
)-fuzzy hyperideals. As a consequence, some characterizations theorems of interval valued (
∈
γ
~
,
∈
γ
~
∨
q
δ
~
)-fuzzy hyperideals which generalize many results obtained in [21] were provided. Our future work on this topic will focus on studying some other classes of fuzzy interval valued hyperstructure.



---

*Source: 173267-2014-06-12.xml*