# A New Kind of Fuzzyn-ary Hypergroups in the Framework of Soft Set Theory

Authors: Hongjie Li; Yunqiang Yin
Journal: The Scientific World Journal
 (2014)

---

## Abstract

Maji et al. introduced the concept of fuzzy soft sets as a generalization
of the standard soft sets and presented an application of fuzzy soft sets in a decision
making problem. The aim of this paper is to apply the concept of fuzzy soft sets ton-ary hypergroup theory. The concepts of (
∈
γ
,
∈
γ
∨
q
δ
)-fuzzy soft (invertible) n-ary subhypergroups over a commutative n-ary hypergroup are introduced and some related properties and characterizations are obtained. The homomorphism properties of (
∈
γ
,
∈
γ
∨
q
δ
)-fuzzy soft (invertible) n-ary subhypergroups are also derived.


---

## Body

## 1. Introduction

To solve complicated problems in economics, engineering, and environment, we cannot successfully use classical methods because of various uncertainties typical for those problems. There are three theories: theory of probability, theory of fuzzy sets, and the interval mathematics, which we can consider as mathematical tools for dealing with uncertainties. However, all of these have their advantages as well as inherent limitations in dealing with uncertainties. One major problem shared by those theories is their incompatibility with the parameterization tools. To overcome these difficulties, Molodtsov [1] introduced the concept of soft set as a new mathematical tool for dealing with uncertainties that is free from the difficulties that have troubled the usual theoretical approaches. Molodtsov pointed out several directions for the applications of soft sets. This theory has proven useful in many different fields such as decision making [2–6], data analysis [7, 8], and forecasting [9].Up to the present, research on soft sets has been very active and many important results have been achieved in the theoretical aspect. Maji et al. [10] introduced several algebraic operations in soft set theory and published a detailed theoretical study on soft sets. Ali et al. [11] further presented and investigated some new algebraic operations for soft sets. Qin and Hong [12] further investigated the algebraic structure of soft sets and established the soft quotient algebra. Maji et al. [13] and Majumdar and Samanta [14] extended (classical) soft sets to fuzzy soft sets, respectively. Maji et al. [15] extended (classical) soft sets to intuitionistic fuzzy soft sets, which were further discussed by Maji et al. [16] and Jiang et al. [17]. Aktaş and Çağman [18] compared soft sets to the related concepts of fuzzy sets and rough sets. They also defined the notion of soft groups and derived some related properties. Aygünoğlu and Aygün [19] discussed the applications of fuzzy soft sets to group theory and investigated (normal) fuzzy soft groups. Feng et al. [20] investigated soft semirings by using the soft set theory. Jun [21] introduced and investigated the notion of soft BCK/BCI-algebras. Jun and Park [22] and Jun et al. [23] discussed the applications of soft sets in ideal theory of BCK/BCI-algebras and in d-algebras, respectively. Acar et al. [24] introduced and studied soft rings. Zhan and Jun [25] characterized the (implicative, positive implicative, and fantastic) filteristic soft B
L-algebras based on ∈-soft sets and q-soft sets.On the other hand, the fuzzy sets and hyperstructures introduced by Zadeh and Marty, respectively, are now extensively applied to many disciplines. The study of fuzzy hyperstructures is an interesting research topic of fuzzy sets. There is a considerable amount of work on the connections between fuzzy sets and hyperstructures. The reader is referred to [26–38]. As a generalization of the concepts of “belongingness (∈)” and “quasi-coincidence (q)” of a fuzzy point with a fuzzy set introduced by Pu and Liu [39], Yin and Zhan [40] introduced the concepts of “γ-belongingness (∈
γ)” and “δ-quasi-coincidence (q
δ)” of a fuzzy point with a fuzzy set. Using these new concepts, Yin and Zhan [40] introduced the concepts of (
α
,
β
)-fuzzy (implicative, positive implicative, and fantastic) filters and (
β
¯
,
α
¯
)-fuzzy (implicative, positive implicative, and fantastic) filters of B
L-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. Special attention is paid to (
∈
γ
¯
,
∈
γ
¯
∨
q
δ
¯
)-fuzzy (implicative, positive implicative, and fantastic) filters and (
∈
γ
¯
,
∈
γ
¯
∨
q
δ
¯
)-fuzzy (implicative, positive implicative, and fantastic) filters which are generalizations of (
∈
,
∈
∨
q
)-fuzzy (implicative, positive implicative, and fantastic) filters and (
∈
¯
,
∈
¯
∨
q
¯
)-fuzzy (implicative, positive implicative, and fantastic) filters studied by Ma et al. [41, 42]. This idea is continued and studied by Ma et al. [43], Yin and Zhan [44, 45], Yin et al. [46], and so on. The purpose of this paper is to deal with the algebraic structure of n-ary hypergroups by applying fuzzy soft set theory. We introduce the concepts of (
∈
γ
,
∈
γ
∨
q
δ
)-fuzzy soft (invertible) n-ary subhypergroups over a commutative n-ary hypergroup and focus on investigating their characterization and algebraic properties. We also discuss the homomorphism properties of (
∈
γ
,
∈
γ
∨
q
δ
)-fuzzy soft (invertible) n-ary subhypergroups.


## 2. Preliminaries

In this section, we recall some basic notions and results aboutn-ary hypergroups, fuzzy sets, and fuzzy soft sets (see [13, 28, 30, 47]).


### 2.1.n-Ary Hypergroups

We will be concerned primarily with a basic nonempty setH. Denote by H
n the cartesian product H
×
⋯
×
H, where H appears n times. An element of H
n will be denoted by (
x
1
,
…
,
x
n
), where x
i
∈
H for all 1
≤
i
≤
n. In general, a mapping f
:
H
n
→
P
*
(
H
) is called an n
-ary hyperoperation and n is called thearity of the hyperoperation f. Let f be an n-ary hyperoperation on H and A
1
,
…
,
A
n subsets in H. Define
(1)
f
(
A
1
,
…
,
A
n
)
=
∪
{
f
(
x
1
,
…
,
x
n
)
∣
x
i
∈
A
i
,
1
≤
i
≤
n
}
.
In the sequel, we will denote the sequence x
i
,
x
i
+
1
,
…
,
x
j by x
i
j. For j
<
i, x
i
j
=
∅. Thus,
(2)
f
(
x
1
,
…
,
x
i
,
y
i
+
1
,
…
,
y
j
,
z
j
+
1
,
…
,
z
n
)
will be written as f
(
x
1
i
,
y
i
+
1
j
,
z
j
+
1
n
). Also f
(
a
1
i
,
x
*
) means f
(
a
1
i
,
x
,
…
,
x
︸
n
-
i
) for a
1
i
,
x
∈
H and 1
≤
i
≤
n.H with an n-ary hyperoperation f
:
H
n
→
P
*
(
H
) is called an n
-ary hypergroupoid and will be denoted by (
H
,
f
). An n-ary hypergroupoid (
H
,
f
) is said to be an n
-ary semihypergroup if the following associative axiom holds:
(3)
f
(
x
1
i
-
1
,
f
(
x
i
n
+
i
-
1
)
,
x
n
+
i
2
n
-
1
)
=
f
(
x
1
j
-
1
,
f
(
x
j
n
+
j
-
1
)
,
x
n
+
j
2
n
-
1
)
for all i
,
j
∈
{
1
,
…
,
n
} and x
1
2
n
-
1
∈
H. An n-ary semihypergroup is said to be an n
-ary hypergroup if, for all y
,
y
1
n
∈
H and fixed i
∈
{
1
,
…
,
n
}, there exists x
∈
H such that y
∈
f
(
y
1
i
-
1
,
x
,
y
i
+
1
n
). An n-ary hypergroup is said to becommutative if, for all x
1
n
∈
H and any permutation ρ of {
1
,
…
,
n
}, we have f
(
x
1
n
)
=
f
(
x
ρ
1
,
…
,
x
ρ
n
). An element e of H is called anidentity of H if, for any x
∈
H and i
∈
{
1
,
…
,
n
}, we have x
∈
f
(
e
,
…
,
e
︸
i
-
1
,
x
,
e
,
…
,
e
︸
n
-
i
).Definition 1.
Let(
H
,
f
) be an n-ary hypergroup and K a nonempty subset of H. One says that K is an n-ary subhypergroup of H if the following conditions hold:(i)
Kis closed under the n-ary hyperoperation f;
(ii)
for ally
,
y
1
n
∈
K and fixed i
∈
{
1
,
…
,
n
}, there exists x
∈
K such that y
∈
f
(
y
1
i
-
1
,
x
,
y
i
+
1
n
).In what follows,(
H
,
f
) denotes a commutative n-ary hypergroup with an identity e unless otherwise stated.Ann-ary subhypergroup K of H is said to beinvertible if, for any x
,
y
∈
H, x
∈
f
(
y
,
K
,
…
,
K
︸
n
-
1
) implies y
∈
f
(
x
,
K
,
…
,
K
︸
n
-
1
).


### 2.2. Fuzzy Sets

LetX be a nonempty set. A fuzzy subset μ of X is defined as a mapping from X into [
0,1
], where [
0,1
] is the usual interval of real numbers. We denote by F
(
X
) the set of all fuzzy subsets of X. For μ
,
ν
∈
F
(
X
), by μ
⊆
ν, we mean μ
(
x
)
≤
ν
(
x
) for all x
∈
X. And theunion andintersection of μ and ν, denoted by μ
∪
ν and μ
∩
ν, are defined as the fuzzy subsets of X by (
μ
∪
ν
)
(
x
)
=
max
⁡
{
μ
(
x
)
,
ν
(
x
)
} and (
μ
∩
ν
)
(
x
)
=
min
⁡
{
μ
(
x
)
,
ν
(
x
)
} for all x
∈
X.Forx
i
,
x
i
+
1
,
…
,
x
j
∈
X and μ
∈
F
(
X
), the sequence μ
(
x
i
)
,
μ
(
x
i
+
1
)
,
…
,
μ
(
x
j
) is denoted by μ
x
i
x
j, and μ
x
i
x
j
=
0 if j
<
i. Moreover, for μ
i
,
μ
i
+
1
,
…
,
μ
j
∈
F
(
X
), the sequence μ
i
,
μ
i
+
1
,
…
,
μ
j is denoted by μ
i
j, and μ
i
j denotes the zero fuzzy set if j
<
i.For anyY
⊆
X and r
∈
(
0,1
], define a fuzzy subset r
Y of X by
(4)
r
Y
(
x
)
=
{
r
if
x
∈
Y
,
0
otherwise
,
for all x
∈
X. In particular, when r
=
1, r
Y is said to be thecharacteristic function of Y, denoted by χ
Y, and whenX
=
{
x
}, r
X is said to be afuzzy point with support x and value r and is denoted by r
x.In what follows, letγ
,
δ
∈
[
0,1
] be such that γ
<
δ. For a fuzzy point r
x and a fuzzy subset μ of X, we say that(1)
r
x
∈
γ
μ if μ
(
x
)
≥
r
>
γ;
(2)
r
x
q
δ
μ if μ
(
x
)
+
r
>
2
δ;
(3)
r
x
∈
γ
∨
q
δ
μ if r
x
∈
γ
μ or r
x
q
δ
μ [40].Let us now introduce a new relation onF
(
X
), denoted by “⊆
∨
q
(
γ
,
δ
),” as follows.For anyμ
,
ν
∈
F
(
X
), by μ
⊆
∨
q
(
γ
,
δ
)
ν, we mean that r
x
∈
γ
μ implies r
x
∈
γ
∨
q
δ
ν for all x
∈
X and r
∈
(
γ
,
1
]. Moreover, μ and ν are said to be (
γ
,
δ
)-equal, denoted by μ
=
(
γ
,
δ
)
ν, if μ
⊆
∨
q
(
γ
,
δ
)
ν and ν
⊆
∨
q
(
γ
,
δ
)
μ.In the sequel, unless otherwise stated,α
¯ means that α does not hold, where α
∈
{
∈
γ
,
q
δ
,
∈
γ
∨
q
δ
,
⊆
∨
q
(
γ
,
δ
)
}.Lemma 2 (see [40]).
Letμ
,
ν
∈
F
(
X
). Then μ
⊆
∨
q
(
γ
,
δ
)
ν if and only if max
⁡
{
ν
(
x
)
,
γ
}
≥
min
⁡
{
μ
(
x
)
,
δ
} for all x
∈
X.Lemma 3 (see [40]).
Letμ
,
ν
,
ω
∈
F
(
X
). If μ
⊆
∨
q
(
γ
,
δ
)
ν and  ν
⊆
∨
q
(
γ
,
δ
)
ω, then μ
⊆
∨
q
(
γ
,
δ
)
ω.Proof.
It is straightforward by Lemma2Lemmas2 and 3 give that “=
(
γ
,
δ
)” is an equivalence relation on F
(
X
). It is also worth noticing that μ
=
(
γ
,
δ
)
ν if and only if max
⁡
{
min
⁡
{
μ
(
x
)
,
δ
}
,
γ
}
=
max
⁡
{
min
⁡
{
ν
(
x
)
,
δ
}
,
γ
} for all x
∈
X by Lemma 2


### 2.3. Fuzzy Soft Sets

LetU be an initial universe set and E the set of all possible parameters under consideration with respect to U. As a generalization of soft set introduced by Molodtsov [1], Maji et al. [13] defined fuzzy soft set in the following way.Definition 4.
A pair〈
F
,
A
〉 is called afuzzy soft set over U, where A
⊆
E and F is a mapping given by F
:
A
→
F
(
U
).In general, for everyε
∈
A, F
(
ε
) is a fuzzy set of U and it is calledfuzzy value set of parameter ε. The set of all fuzzy soft sets over U with parameters from E is calledfuzzy soft classes, and it is denoted by F
S
(
U
,
E
). A fuzzy soft set 〈
F
,
A
〉 in F
S
(
U
,
E
) is said tohave sup-property if, for any V
∈
P
*
(
U
) and ε
∈
A, there exists x
0
∈
V such that F
(
ε
)
(
x
0
)
=
sup
⁡
x
∈
V
⁡
F
(
ε
)
(
x
).LetA
⊆
E and μ
∈
F
(
U
). Define 〈
μ
~
,
A
〉to be a fuzzy soft set by μ
~
(
ε
)
=
μ for all ε
∈
A. For any fuzzy point r
x in U, define (
r
x
~
,
A
)to be a fuzzy soft set by r
x
~
(
ε
)
=
r
x for all ε
∈
A.Definition 5 (see [6]).
Let〈
F
,
A
〉 and 〈
G
,
B
〉 be two fuzzy soft sets overU. One says that 〈
F
,
A
〉 is afuzzy soft subset of 〈
G
,
B
〉 and writes 〈
F
,
A
〉
⋐
〈
G
,
B
〉 if(i)
A
⊆
B;
(ii)
for anyε
∈
A, F
(
ε
)
⊆
G
(
ε
).
〈
F
,
A
〉 and 〈
G
,
B
〉 are said to befuzzy soft equal and write 〈
F
,
A
〉
=
〈
G
,
B
〉 if 〈
F
,
A
〉
⋐
〈
G
,
B
〉 and 〈
G
,
B
〉
⋐
〈
F
,
A
〉.Let us now introduce some new concepts on fuzzy soft sets analogous to the concepts introduced by Ali et al. [11].Definition 6.
Theextended intersection of two fuzzy soft sets 〈
F
,
A
〉 and 〈
G
,
B
〉 over U is a fuzzy soft set denoted by 〈
H
,
C
〉, where C
=
A
∪
B and
(5)
H
(
ε
)
=
{
F
(
ε
)
if
ε
∈
A
-
B
,
G
(
ε
)
if
ε
∈
B
-
A
,
F
(
ε
)
∩
G
(
ε
)
if
ε
∈
A
∩
B
,
for all ε
∈
C. This is denoted by 〈
H
,
C
〉
=
〈
F
,
A
〉
∩
~
〈
G
,
B
〉.Definition 7.
Let〈
F
,
A
〉 and 〈
G
,
B
〉 be two fuzzy soft sets over U such that A
∩
B
≠
∅. Therestricted intersection of 〈
F
,
A
〉 and 〈
G
,
B
〉 is defined to be the fuzzy soft set 〈
H
,
C
〉, where C
=
A
∩
B and H
(
ε
)
=
F
(
ε
)
∩
G
(
ε
) for all ε
∈
C. This is denoted by 〈
H
,
C
〉
=
〈
F
,
A
〉
⋒
〈
G
,
B
〉.Definition 8.
LetV
⊆
U and r
∈
(
γ
,
1
]. A fuzzy soft set 〈
F
,
A
〉 over V is said to be arelative whole
(
γ
,
δ
)-fuzzy soft set (with respect to universe set V, parameter set A, and value r), denoted by Σ
(
V
,
A
,
r
), if F
(
ε
)
=
r
V for all ε
∈
A.Definition 9.
Let〈
F
,
A
〉 and 〈
G
,
B
〉 be two fuzzy soft sets over U. One says that 〈
F
,
A
〉 is (
γ
,
δ
)-fuzzy soft subset of 〈
G
,
B
〉and writes 〈
F
,
A
〉
⋐
(
γ
,
δ
)
〈
G
,
B
〉 if(i)
A
⊆
B;
(ii)
for anyε
∈
A, F
(
ε
)
⊆
∨
q
(
γ
,
δ
)
G
(
ε
).
〈
F
,
A
〉 and 〈
G
,
B
〉 are said to be (
γ
,
δ
)-fuzzy soft equal and write 〈
F
,
A
〉
≍
(
γ
,
δ
)
〈
G
,
B
〉 if 〈
F
,
A
〉
⋐
(
γ
,
δ
)
〈
G
,
B
〉 and 〈
G
,
B
〉
⋐
(
γ
,
δ
)
〈
F
,
A
〉.Clearly,〈
F
,
A
〉
⋐
〈
G
,
B
〉 implies 〈
F
,
A
〉
⊆
∨
q
(
γ
,
δ
)
〈
G
,
B
〉 by Lemma 2 and Definition 9.Lemma 10.
Let〈
F
,
A
〉, 〈
G
,
B
〉, and 〈
H
,
C
〉 be fuzzy soft sets over U. If 〈
F
,
A
〉
⋐
(
γ
,
δ
)
〈
G
,
B
〉 and 〈
G
,
B
〉
⋐
(
γ
,
δ
)
〈
H
,
C
〉, then 〈
F
,
A
〉
⋐
(
γ
,
δ
)
〈
F
,
C
〉.Proof.
It is straightforward by Lemma3 and Definition 9.Lemmas2 and 10 and Definition 9 give that “≍
(
γ
,
δ
)” is an equivalence relation on F
S
(
U
,
E
).


## 2.1.n-Ary Hypergroups

We will be concerned primarily with a basic nonempty setH. Denote by H
n the cartesian product H
×
⋯
×
H, where H appears n times. An element of H
n will be denoted by (
x
1
,
…
,
x
n
), where x
i
∈
H for all 1
≤
i
≤
n. In general, a mapping f
:
H
n
→
P
*
(
H
) is called an n
-ary hyperoperation and n is called thearity of the hyperoperation f. Let f be an n-ary hyperoperation on H and A
1
,
…
,
A
n subsets in H. Define
(1)
f
(
A
1
,
…
,
A
n
)
=
∪
{
f
(
x
1
,
…
,
x
n
)
∣
x
i
∈
A
i
,
1
≤
i
≤
n
}
.
In the sequel, we will denote the sequence x
i
,
x
i
+
1
,
…
,
x
j by x
i
j. For j
<
i, x
i
j
=
∅. Thus,
(2)
f
(
x
1
,
…
,
x
i
,
y
i
+
1
,
…
,
y
j
,
z
j
+
1
,
…
,
z
n
)
will be written as f
(
x
1
i
,
y
i
+
1
j
,
z
j
+
1
n
). Also f
(
a
1
i
,
x
*
) means f
(
a
1
i
,
x
,
…
,
x
︸
n
-
i
) for a
1
i
,
x
∈
H and 1
≤
i
≤
n.H with an n-ary hyperoperation f
:
H
n
→
P
*
(
H
) is called an n
-ary hypergroupoid and will be denoted by (
H
,
f
). An n-ary hypergroupoid (
H
,
f
) is said to be an n
-ary semihypergroup if the following associative axiom holds:
(3)
f
(
x
1
i
-
1
,
f
(
x
i
n
+
i
-
1
)
,
x
n
+
i
2
n
-
1
)
=
f
(
x
1
j
-
1
,
f
(
x
j
n
+
j
-
1
)
,
x
n
+
j
2
n
-
1
)
for all i
,
j
∈
{
1
,
…
,
n
} and x
1
2
n
-
1
∈
H. An n-ary semihypergroup is said to be an n
-ary hypergroup if, for all y
,
y
1
n
∈
H and fixed i
∈
{
1
,
…
,
n
}, there exists x
∈
H such that y
∈
f
(
y
1
i
-
1
,
x
,
y
i
+
1
n
). An n-ary hypergroup is said to becommutative if, for all x
1
n
∈
H and any permutation ρ of {
1
,
…
,
n
}, we have f
(
x
1
n
)
=
f
(
x
ρ
1
,
…
,
x
ρ
n
). An element e of H is called anidentity of H if, for any x
∈
H and i
∈
{
1
,
…
,
n
}, we have x
∈
f
(
e
,
…
,
e
︸
i
-
1
,
x
,
e
,
…
,
e
︸
n
-
i
).Definition 1.
Let(
H
,
f
) be an n-ary hypergroup and K a nonempty subset of H. One says that K is an n-ary subhypergroup of H if the following conditions hold:(i)
Kis closed under the n-ary hyperoperation f;
(ii)
for ally
,
y
1
n
∈
K and fixed i
∈
{
1
,
…
,
n
}, there exists x
∈
K such that y
∈
f
(
y
1
i
-
1
,
x
,
y
i
+
1
n
).In what follows,(
H
,
f
) denotes a commutative n-ary hypergroup with an identity e unless otherwise stated.Ann-ary subhypergroup K of H is said to beinvertible if, for any x
,
y
∈
H, x
∈
f
(
y
,
K
,
…
,
K
︸
n
-
1
) implies y
∈
f
(
x
,
K
,
…
,
K
︸
n
-
1
).


## 2.2. Fuzzy Sets

LetX be a nonempty set. A fuzzy subset μ of X is defined as a mapping from X into [
0,1
], where [
0,1
] is the usual interval of real numbers. We denote by F
(
X
) the set of all fuzzy subsets of X. For μ
,
ν
∈
F
(
X
), by μ
⊆
ν, we mean μ
(
x
)
≤
ν
(
x
) for all x
∈
X. And theunion andintersection of μ and ν, denoted by μ
∪
ν and μ
∩
ν, are defined as the fuzzy subsets of X by (
μ
∪
ν
)
(
x
)
=
max
⁡
{
μ
(
x
)
,
ν
(
x
)
} and (
μ
∩
ν
)
(
x
)
=
min
⁡
{
μ
(
x
)
,
ν
(
x
)
} for all x
∈
X.Forx
i
,
x
i
+
1
,
…
,
x
j
∈
X and μ
∈
F
(
X
), the sequence μ
(
x
i
)
,
μ
(
x
i
+
1
)
,
…
,
μ
(
x
j
) is denoted by μ
x
i
x
j, and μ
x
i
x
j
=
0 if j
<
i. Moreover, for μ
i
,
μ
i
+
1
,
…
,
μ
j
∈
F
(
X
), the sequence μ
i
,
μ
i
+
1
,
…
,
μ
j is denoted by μ
i
j, and μ
i
j denotes the zero fuzzy set if j
<
i.For anyY
⊆
X and r
∈
(
0,1
], define a fuzzy subset r
Y of X by
(4)
r
Y
(
x
)
=
{
r
if
x
∈
Y
,
0
otherwise
,
for all x
∈
X. In particular, when r
=
1, r
Y is said to be thecharacteristic function of Y, denoted by χ
Y, and whenX
=
{
x
}, r
X is said to be afuzzy point with support x and value r and is denoted by r
x.In what follows, letγ
,
δ
∈
[
0,1
] be such that γ
<
δ. For a fuzzy point r
x and a fuzzy subset μ of X, we say that(1)
r
x
∈
γ
μ if μ
(
x
)
≥
r
>
γ;
(2)
r
x
q
δ
μ if μ
(
x
)
+
r
>
2
δ;
(3)
r
x
∈
γ
∨
q
δ
μ if r
x
∈
γ
μ or r
x
q
δ
μ [40].Let us now introduce a new relation onF
(
X
), denoted by “⊆
∨
q
(
γ
,
δ
),” as follows.For anyμ
,
ν
∈
F
(
X
), by μ
⊆
∨
q
(
γ
,
δ
)
ν, we mean that r
x
∈
γ
μ implies r
x
∈
γ
∨
q
δ
ν for all x
∈
X and r
∈
(
γ
,
1
]. Moreover, μ and ν are said to be (
γ
,
δ
)-equal, denoted by μ
=
(
γ
,
δ
)
ν, if μ
⊆
∨
q
(
γ
,
δ
)
ν and ν
⊆
∨
q
(
γ
,
δ
)
μ.In the sequel, unless otherwise stated,α
¯ means that α does not hold, where α
∈
{
∈
γ
,
q
δ
,
∈
γ
∨
q
δ
,
⊆
∨
q
(
γ
,
δ
)
}.Lemma 2 (see [40]).
Letμ
,
ν
∈
F
(
X
). Then μ
⊆
∨
q
(
γ
,
δ
)
ν if and only if max
⁡
{
ν
(
x
)
,
γ
}
≥
min
⁡
{
μ
(
x
)
,
δ
} for all x
∈
X.Lemma 3 (see [40]).
Letμ
,
ν
,
ω
∈
F
(
X
). If μ
⊆
∨
q
(
γ
,
δ
)
ν and  ν
⊆
∨
q
(
γ
,
δ
)
ω, then μ
⊆
∨
q
(
γ
,
δ
)
ω.Proof.
It is straightforward by Lemma2Lemmas2 and 3 give that “=
(
γ
,
δ
)” is an equivalence relation on F
(
X
). It is also worth noticing that μ
=
(
γ
,
δ
)
ν if and only if max
⁡
{
min
⁡
{
μ
(
x
)
,
δ
}
,
γ
}
=
max
⁡
{
min
⁡
{
ν
(
x
)
,
δ
}
,
γ
} for all x
∈
X by Lemma 2


## 2.3. Fuzzy Soft Sets

LetU be an initial universe set and E the set of all possible parameters under consideration with respect to U. As a generalization of soft set introduced by Molodtsov [1], Maji et al. [13] defined fuzzy soft set in the following way.Definition 4.
A pair〈
F
,
A
〉 is called afuzzy soft set over U, where A
⊆
E and F is a mapping given by F
:
A
→
F
(
U
).In general, for everyε
∈
A, F
(
ε
) is a fuzzy set of U and it is calledfuzzy value set of parameter ε. The set of all fuzzy soft sets over U with parameters from E is calledfuzzy soft classes, and it is denoted by F
S
(
U
,
E
). A fuzzy soft set 〈
F
,
A
〉 in F
S
(
U
,
E
) is said tohave sup-property if, for any V
∈
P
*
(
U
) and ε
∈
A, there exists x
0
∈
V such that F
(
ε
)
(
x
0
)
=
sup
⁡
x
∈
V
⁡
F
(
ε
)
(
x
).LetA
⊆
E and μ
∈
F
(
U
). Define 〈
μ
~
,
A
〉to be a fuzzy soft set by μ
~
(
ε
)
=
μ for all ε
∈
A. For any fuzzy point r
x in U, define (
r
x
~
,
A
)to be a fuzzy soft set by r
x
~
(
ε
)
=
r
x for all ε
∈
A.Definition 5 (see [6]).
Let〈
F
,
A
〉 and 〈
G
,
B
〉 be two fuzzy soft sets overU. One says that 〈
F
,
A
〉 is afuzzy soft subset of 〈
G
,
B
〉 and writes 〈
F
,
A
〉
⋐
〈
G
,
B
〉 if(i)
A
⊆
B;
(ii)
for anyε
∈
A, F
(
ε
)
⊆
G
(
ε
).
〈
F
,
A
〉 and 〈
G
,
B
〉 are said to befuzzy soft equal and write 〈
F
,
A
〉
=
〈
G
,
B
〉 if 〈
F
,
A
〉
⋐
〈
G
,
B
〉 and 〈
G
,
B
〉
⋐
〈
F
,
A
〉.Let us now introduce some new concepts on fuzzy soft sets analogous to the concepts introduced by Ali et al. [11].Definition 6.
Theextended intersection of two fuzzy soft sets 〈
F
,
A
〉 and 〈
G
,
B
〉 over U is a fuzzy soft set denoted by 〈
H
,
C
〉, where C
=
A
∪
B and
(5)
H
(
ε
)
=
{
F
(
ε
)
if
ε
∈
A
-
B
,
G
(
ε
)
if
ε
∈
B
-
A
,
F
(
ε
)
∩
G
(
ε
)
if
ε
∈
A
∩
B
,
for all ε
∈
C. This is denoted by 〈
H
,
C
〉
=
〈
F
,
A
〉
∩
~
〈
G
,
B
〉.Definition 7.
Let〈
F
,
A
〉 and 〈
G
,
B
〉 be two fuzzy soft sets over U such that A
∩
B
≠
∅. Therestricted intersection of 〈
F
,
A
〉 and 〈
G
,
B
〉 is defined to be the fuzzy soft set 〈
H
,
C
〉, where C
=
A
∩
B and H
(
ε
)
=
F
(
ε
)
∩
G
(
ε
) for all ε
∈
C. This is denoted by 〈
H
,
C
〉
=
〈
F
,
A
〉
⋒
〈
G
,
B
〉.Definition 8.
LetV
⊆
U and r
∈
(
γ
,
1
]. A fuzzy soft set 〈
F
,
A
〉 over V is said to be arelative whole
(
γ
,
δ
)-fuzzy soft set (with respect to universe set V, parameter set A, and value r), denoted by Σ
(
V
,
A
,
r
), if F
(
ε
)
=
r
V for all ε
∈
A.Definition 9.
Let〈
F
,
A
〉 and 〈
G
,
B
〉 be two fuzzy soft sets over U. One says that 〈
F
,
A
〉 is (
γ
,
δ
)-fuzzy soft subset of 〈
G
,
B
〉and writes 〈
F
,
A
〉
⋐
(
γ
,
δ
)
〈
G
,
B
〉 if(i)
A
⊆
B;
(ii)
for anyε
∈
A, F
(
ε
)
⊆
∨
q
(
γ
,
δ
)
G
(
ε
).
〈
F
,
A
〉 and 〈
G
,
B
〉 are said to be (
γ
,
δ
)-fuzzy soft equal and write 〈
F
,
A
〉
≍
(
γ
,
δ
)
〈
G
,
B
〉 if 〈
F
,
A
〉
⋐
(
γ
,
δ
)
〈
G
,
B
〉 and 〈
G
,
B
〉
⋐
(
γ
,
δ
)
〈
F
,
A
〉.Clearly,〈
F
,
A
〉
⋐
〈
G
,
B
〉 implies 〈
F
,
A
〉
⊆
∨
q
(
γ
,
δ
)
〈
G
,
B
〉 by Lemma 2 and Definition 9.Lemma 10.
Let〈
F
,
A
〉, 〈
G
,
B
〉, and 〈
H
,
C
〉 be fuzzy soft sets over U. If 〈
F
,
A
〉
⋐
(
γ
,
δ
)
〈
G
,
B
〉 and 〈
G
,
B
〉
⋐
(
γ
,
δ
)
〈
H
,
C
〉, then 〈
F
,
A
〉
⋐
(
γ
,
δ
)
〈
F
,
C
〉.Proof.
It is straightforward by Lemma3 and Definition 9.Lemmas2 and 10 and Definition 9 give that “≍
(
γ
,
δ
)” is an equivalence relation on F
S
(
U
,
E
).


## 3. A New Operation on Fuzzy Soft Sets over ann-Ary Hypergroupoid

We first introduce a fuzzyn-ary hyperoperation on an n-ary hypergroupoid as follows.Definition 11.
Let(
H
,
f
) be an n-ary hypergroupoid and μ
1
,
…
,
μ
n
∈
F
(
H
). Define a fuzzy n-ary hyperoperation F
:
F
(
H
)
×
⋯
×
F
(
H
)
︸
n
→
F
(
H
) by
(6)
F
(
μ
1
,
…
,
μ
n
)
(
x
)
=
sup
⁡
x
∈
f
(
y
1
n
)
min
⁡
{
μ
1
(
y
1
)
,
…
,
μ
n
(
y
n
)
}
for all x
∈
H. In particular, for any x
1
n
∈
H and i
∈
{
1
,
…
,
n
}, define
(7)
F
(
μ
1
,
…
,
μ
i
,
x
i
+
1
n
)
(
x
)
=
sup
⁡
x
∈
f
(
y
1
i
,
x
i
+
1
n
)
min
⁡
{
μ
1
(
y
1
)
,
…
,
μ
i
(
y
i
)
}
for all x
∈
H.Clearly, for anyx
1
n
∈
H and 1
≤
i
≤
n, F
(
μ
1
,
…
,
μ
i
,
x
i
+
1
n
)
=
F
(
μ
1
,
…
,
μ
i
,
1
x
i
+
1
,
…
,
1
x
n
). Note that, for any r
1
n
∈
(
0,1
] and x
1
n
∈
H, by Definition 11, F
(
r
1
x
1
,
r
2
x
2
,
…
,
r
n
x
n
)
=
min
⁡
{
r
1
n
}
f
(
x
1
n
).Definition 12.
Let(
H
,
f
) be an n-ary hypergroupoid and 〈
F
i
,
A
i
〉
∈
F
S
(
H
,
E
), where i
∈
I and I is an index set, such that ⋂
i
∈
I
‍
A
i
≠
∅. One defines a fuzzy soft set over H, denoted by 〈
F
(
F
1
,
…
,
F
n
)
,
C
〉, where C
=
⋂
i
∈
I
‍
A
i and
(8)
F
(
F
1
,
…
,
F
n
)
(
ε
)
=
F
(
F
1
(
ε
)
,
…
,
F
n
(
ε
)
)
,
for all ε
∈
C. This is denoted by 〈
F
(
F
1
,
…
,
F
n
)
,
C
〉
=
F
(
〈
F
1
,
A
1
〉
,
…
,
〈
F
n
,
A
n
〉
).In the following, we will denote the sequence〈
F
i
,
A
i
〉
,
〈
F
i
+
1
,
A
i
+
1
〉
,
…
,
〈
F
j
,
A
j
〉 by 〈
F
,
A
〉
i
j. For j
<
i, 〈
F
,
A
〉
i
j is the empty set. The following elementary facts follow easily from the definition.Lemma 13.
Let(
H
,
f
) be an n-ary hypergroupoid and 〈
F
,
A
〉
1
2
n
-
1
,
〈
G
,
B
〉
1
n
∈
F
S
(
H
,
E
). Then(1)
if(
H
,
f
) is an n-ary semihypergroup and ⋂
i
=
1
2
n
‍
A
i
≠
∅, then
(9)
F
(
〈
F
,
A
〉
1
i
-
1
,
F
(
〈
F
,
A
〉
i
n
+
i
-
1
)
,
〈
F
,
A
〉
n
+
i
2
n
-
1
)
=
F
(
〈
F
,
A
〉
1
j
-
1
,
F
(
〈
F
,
A
〉
j
n
+
j
-
1
)
,
〈
F
,
A
〉
n
+
j
2
n
-
1
)
for all 1
≤
i
,
j
≤
n;
(2)
if(
H
,
f
) is commutative and ⋂
i
=
1
n
‍
A
i
≠
∅, then the value of F
(
〈
F
,
A
〉
1
n
) does not depend on the permutation of 〈
F
1
,
A
1
〉
,
…
,
〈
F
n
,
A
n
〉;
(3)
if〈
F
i
,
A
i
〉
⋐
(
γ
,
δ
)
〈
G
i
,
B
i
〉 for all 1
≤
i
≤
n and ⋂
i
=
1
n
‍
A
i
≠
∅, then F
(
〈
F
,
A
〉
1
n
)
⋐
(
γ
,
δ
)
F
(
〈
G
,
B
〉
1
n
);
(4)
if〈
F
1
,
A
1
〉
⋐
(
γ
,
δ
)
〈
G
1
,
A
1
〉 and〈
F
2
,
A
2
〉
⋐
(
γ
,
δ
)
〈
G
2
,
A
2
〉, then 〈
F
1
,
A
1
〉
∩
~
〈
F
2
,
A
2
〉
⋐
(
γ
,
δ
)
〈
G
1
,
A
1
〉
∩
~
〈
G
2
,
A
2
〉 and 〈
F
1
,
A
1
〉
⋒
〈
F
2
,
A
2
〉
⋐
(
γ
,
δ
)
〈
G
1
,
A
1
〉
⋒
〈
G
2
,
A
2
〉 if A
1
∩
A
2
≠
∅.


## 4.(
∈
γ
,
∈
γ
∨
q
δ
)-Fuzzy Soft (Invertible) n-Ary Subhypergroups over a Commutative n-Ary Hypergroup

In this section, we will introduce and investigate the concept of(
∈
γ
,
∈
γ
∨
q
δ
)-fuzzy soft n-ary subhypergroups over a commutative n-ary hypergroup. Let us start by giving the following concept.Definition 14.
Let〈
F
,
A
〉
∈
F
S
(
H
,
E
). Then 〈
F
,
A
〉 is called (
∈
γ
,
∈
γ
∨
q
δ
)-fuzzy soft
n-ary subhypergroup over H if it satisfies the following conditions:(F1a)
for allx
∈
H and ε
∈
A, max
⁡
{
F
(
ε
)
(
e
)
,
γ
}
≥
min
⁡
{
F
(
ε
)
(
x
)
,
δ
};
(F2a)
for allx
1
n
∈
H and ε
∈
A, max
⁡
{
inf
⁡
x
∈
f
(
x
1
n
)
⁡
F
(
ε
)
(
x
)
,
γ
}
≥
min
⁡
{
F
(
ε
)
x
1
x
n
,
δ
};
(F3a)
for ally
,
y
1
n
-
1
∈
H and ε
∈
A, there exists x
∈
H such that
(10)
y
∈
f
(
x
,
y
1
n
-
1
)
,
max
⁡
{
F
(
ε
)
(
x
)
,
γ
}
≥
min
⁡
{
F
(
ε
)
y
1
y
n
-
1
,
F
(
ε
)
(
y
)
,
δ
}
.An (
∈
γ
,
∈
γ
∨
q
δ
)-fuzzy n-ary subhypergroup 〈
F
,
A
〉 over H is said to beinvertible if it satisfies the following:(F4a)
for allx
,
y
∈
H and ε
∈
A,
(11)
max
⁡
{
min
⁡
{
F
(
y
,
F
(
ε
)
,
…
,
F
(
ε
)
︸
n
-
1
)
(
x
)
,
δ
}
,
γ
}
=
max
⁡
{
min
⁡
{
F
(
x
,
F
(
ε
)
,
…
,
F
(
ε
)
︸
n
-
1
)
(
y
)
,
δ
}
,
γ
}
.Example 15.
LetZ
8 be the residue class of (
Z
,
+
) modulo 8 with an n-ary hyperoperation f as follows:
(12)
f
:
Z
8
n
⟶
P
*
(
Z
8
)
,
f
(
x
1
n
)
=
{
m
1
x
1
+
⋯
+
m
n
x
n
∣
m
1
,
…
,
m
n
∈
Z
}
.
Then, (
Z
8
,
f
) is a commutative n-ary hypergroup. Let E
=
(
0.2,0.6
] and r
≥
0.6. Define a fuzzy soft set 〈
F
,
A
〉 over Z
8 by
(13)
F
(
ε
)
(
x
)
=
{
r
if
x
=
0
,
ε
if
x
∈
{
2,4
}
,
0.2
otherwise
,
for all ε
∈
E and x
∈
Z
8. Then 〈
F
,
E
〉 is an (
∈
0.2
,
∈
0.2
∨
q
0.6
)-fuzzy soft invertible n-ary subhypergroup over Z
8.Next let us first provide some auxiliary lemmas as follows.Lemma 16.
Let〈
F
,
A
〉
∈
F
S
(
H
,
E
). Then (F2a) holds if and only if one of the following conditions holds:(F2b)
F
(
〈
F
,
A
〉
,
…
,
〈
F
,
A
〉
︸
n
)
⋐
(
γ
,
δ
)
〈
F
,
A
〉;
(F2c)
for allx
1
n
∈
H and ε
∈
A, F
(
F
(
ε
)
(
x
1
)
x
1,…
,
F
(
ε
)
(
x
n
)
x
n
)
⊆
∨
q
(
γ
,
δ
)
F
(
ε
);
(F2d)
for allε
∈
A and r
i
x
i
∈
γ
F
(
ε
)
(
1
≤
i
≤
n
), F
(
r
1
x
1
,
…
,
r
n
x
n
)
⊆
∨
q
(
γ
,
δ
)
F
(
ε
).Proof.
(F2a)⇒(F2b) Assume that (F2a) holds. Then, for any x
∈
H and ε
∈
A, we have
(14)
min
⁡
{
F
(
F
(
ε
)
,
…
,
F
(
ε
)
︸
n
)
(
x
)
,
δ
}
=
min
⁡
{
sup
⁡
x
∈
f
(
y
1
n
)
min
⁡
{
F
(
ε
)
y
1
y
n
}
,
δ
}
=
sup
⁡
x
∈
f
(
y
1
n
)
min
⁡
{
F
(
ε
)
y
1
y
n
,
δ
}
≤
max
⁡
{
F
(
ε
)
(
x
)
,
γ
}
and so F
(
F
(
ε
)
,
…
,
F
(
ε
)
︸
n
)
⊆
∨
q
(
γ
,
δ
)
F
(
ε
) by Lemma 2. Therefore, F
(
〈
F
,
A
〉
,
…
,
〈
F
,
A
〉
︸
n
)
⋐
(
γ
,
δ
)
〈
F
,
A
〉.
(F2b)⇒(F2c) Let x
1
n
∈
H and ε
∈
A. Since F
(
ε
)
(
x
i
)
x
i
∈
γ
F
(
ε
), for all 1
≤
i
≤
n, we have
(15)
F
(
F
(
ε
)
(
x
1
)
x
1
,
…
,
F
(
ε
)
(
x
n
)
x
n
)
⊆
F
(
F
(
ε
)
,
…
,
F
(
ε
)
︸
n
)
⊆
∨
q
(
γ
,
δ
)
F
(
ε
)
,
and so F
(
F
(
ε
)
(
x
1
)
x
1
,
…
,
F
(
ε
)
(
x
n
)
x
n
)
⊆
∨
q
(
γ
,
δ
)
F
(
ε
).
(F2c)⇔(F2d) This is straightforward.
(F2d)⇒(F2a) Let ε
∈
A, x
∈
H, and x
1
n
∈
H be such that x
∈
f
(
x
1
n
). Then, by Lemma 2 and (F2d), we have
(16)
max
⁡
{
F
(
ε
)
(
x
)
,
γ
}
≥
min
⁡
{
F
(
F
(
ε
)
(
x
1
)
x
1
,
…
,
F
(
ε
)
(
x
n
)
x
n
)
(
x
)
,
δ
}
=
min
⁡
{
(
min
⁡
{
F
(
ε
)
x
1
x
n
}
f
(
x
1
n
)
)
(
x
)
,
δ
}
=
min
⁡
{
F
(
ε
)
x
1
x
n
,
δ
}
.
This implies max
⁡
{
inf
⁡
x
∈
f
(
x
1
n
)
⁡
F
(
ε
)
(
x
)
,
γ
}
≥
min
⁡
{
F
(
ε
)
x
1
x
n
,
δ
} for all x
1
n
∈
H and ε
∈
A. Hence, (F2a) holds.Lemma 17.
Let〈
F
,
A
〉
∈
F
S
(
H
,
E
). Then, one has the following.(1)
(F3a) holds if and only if the following condition holds: (F3b) for ally
,
y
1
n
-
1
∈
H and ε
∈
A, there exists x
∈
H such that
(17)
min
⁡
{
F
(
ε
)
y
1
y
n
-
1
,
F
(
ε
)
(
y
)
}
y
⊆
∨
q
(
γ
,
δ
)
F
(
F
(
ε
)
(
x
)
x
,
F
(
ε
)
(
y
1
)
y
1
,
…
,
F
(
ε
)
(
y
n
-
1
)
y
n
-
1
)
.
(2)
If (F3a) holds, then the following condition holds:
(F3c)
for ally
1
n
-
1
∈
H and ε
∈
A,
(18)
F
(
ε
)
∩
min
⁡
{
F
(
ε
)
y
1
y
n
-
1
}
H
⊆
∨
q
(
γ
,
δ
)
F
(
F
(
ε
)
,
F
(
ε
)
(
y
1
)
y
1
,
…
,
F
(
ε
)
(
y
n
-
1
)
y
n
-
1
)
.Moreover, (F3c) implies (F3a) if 〈
F
,
A
〉 has the sup-property or H is finite.Proof.
(F3a)⇒(F3b) Let y
,
y
1
n
-
1
∈
H and ε
∈
A. By (F3a), there exists x
∈
H such that
(19)
y
∈
f
(
x
,
y
1
n
-
1
)
,
max
⁡
{
F
(
ε
)
(
x
)
,
γ
}
≥
min
⁡
{
F
(
ε
)
y
1
y
n
-
1
,
F
(
ε
)
(
y
)
,
δ
}
.
This implies min
⁡
{
max
⁡
{
F
(
ε
)
y
1
y
n
-
1
,
γ
},max
⁡
{
F
(
ε
)
(
x
)
,
γ
}
}
≥
min
⁡
{
F
(
ε
)
y
1
y
n
-
1
,
F
(
ε
)
(
y
)
,
δ
}, and so
(20)
max
⁡
{
F
(
F
(
ε
)
(
x
)
x
,
F
(
ε
)
(
y
1
)
y
1
,
…
,
F
(
ε
)
(
y
n
-
1
)
y
n
-
1
)
(
y
)
,
γ
}
=
max
⁡
{
(
min
⁡
{
F
(
ε
)
y
1
y
n
-
1
,
F
(
ε
)
(
x
)
}
f
(
x
,
y
1
n
-
1
)
)
(
y
)
,
γ
}
≥
min
⁡
{
F
(
ε
)
y
1
y
n
-
1
,
F
(
ε
)
(
y
)
,
δ
}
=
min
⁡
{
(
min
⁡
{
F
(
ε
)
y
1
y
n
-
1
,
F
(
ε
)
(
y
)
}
)
y
(
y
)
,
δ
}
.
It follows that min
⁡
{
F
(
ε
)
y
1
y
n
-
1
,
F
(
ε
)
(
y
)
}
y
⊆
∨
q
(
γ
,
δ
)
F
(
F
(
ε
)
(
x
)
x, F
(
ε
)
(
y
1
)
y
1,…,F
(
ε
)
(
y
n
-
1
)
y
n
-
1
) and so (F3b) holds.
(F3b)⇒(F3a) Let y
,
y
1
n
-
1
∈
H. By (F3b), there exists x
∈
H such that
(21)
min
⁡
{
F
(
ε
)
y
1
y
n
-
1
,
F
(
ε
)
(
y
)
}
y
⊆
∨
q
(
γ
,
δ
)
F
(
F
(
ε
)
(
x
)
x
,
F
(
ε
)
(
y
1
)
y
1
,
…
,
F
(
ε
)
(
y
n
-
1
)
y
n
-
1
)
.
It follows from Lemma 2 that
(22)
max
⁡
{
(
min
⁡
{
F
(
ε
)
y
1
y
n
-
1
,
F
(
ε
)
(
x
)
}
f
(
x
,
y
1
n
-
1
)
)
(
y
)
,
γ
}
=
max
⁡
{
F
(
F
(
ε
)
(
x
)
x
,
F
(
ε
)
(
y
1
)
y
1
,
…
,
F
(
ε
)
(
y
n
-
1
)
y
n
-
1
)
v
v
v
v
v
v
v
v
v
v
×
(
y
)
,
γ
(
F
(
ε
)
(
x
)
x
,
F
(
ε
)
(
y
1
)
y
1
,
…
,
F
(
ε
)
(
y
n
-
1
)
y
n
-
1
)
}
≥
min
⁡
{
(
min
⁡
⁡
{
F
(
ε
)
y
1
y
n
-
1
,
F
(
ε
)
(
y
)
}
)
y
(
y
)
,
δ
}
=
min
⁡
{
F
(
ε
)
y
1
y
n
-
1
,
F
(
ε
)
(
y
)
,
δ
}
.
This implies y
∈
f
(
x
,
y
1
n
-
1
) and max
⁡
⁡
{
min
⁡
⁡
{
F
(
ε
)
y
1
y
n
-
1,F
(
ε
)
(
x
)
},γ
}
≥
min
⁡
⁡
{
F
(
ε
)
y
1
y
n
-
1
,
F
(
ε
)
(
y
),δ
}. Hence,
(23)
max
⁡
{
F
(
ε
)
(
x
)
,
γ
}
≥
max
⁡
{
min
⁡
{
F
(
ε
)
y
1
y
n
-
1
,
F
(
ε
)
(
x
)
}
,
γ
}
≥
min
⁡
{
F
(
ε
)
y
1
y
n
-
1
,
F
(
ε
)
(
y
)
,
δ
}
and so (F3a) holds.
(F3a)⇒(F3c) This is straightforward by (F3b).
In the following, assume that〈
F
,
A
〉 has the sup-property or H is finite. We prove (F3c)⇒(F3a). Let y
1
n
-
1
∈
H. If, for any x
∈
H such that y
∈
f
(
x
,
y
1
n
-
1
), we have
(24)
F
(
ε
)
(
x
)
≤
max
⁡
{
F
(
ε
)
(
x
)
,
γ
}
<
min
⁡
{
F
(
ε
)
y
1
y
n
-
1
,
F
(
ε
)
(
y
)
,
δ
}
≤
F
(
ε
)
y
1
y
n
-
1
,
then since〈
F
,
A
〉 has the sup-property or H is finite, we have
(25)
max
⁡
{
F
(
F
(
ε
)
,
F
(
ε
)
(
y
1
)
y
1
,
…
,
F
(
ε
)
(
y
n
-
1
)
y
n
-
1
)
(
y
)
,
γ
}
=
max
⁡
{
sup
⁡
y
∈
f
(
x
,
y
1
n
-
1
)
min
⁡
{
F
(
ε
)
(
x
)
,
F
(
ε
)
y
1
y
n
-
1
}
,
γ
}
=
max
⁡
{
sup
⁡
y
∈
f
(
x
,
y
1
n
-
1
)
F
(
ε
)
(
x
)
,
γ
}
=
sup
⁡
y
∈
f
(
x
,
y
1
n
-
1
)
max
⁡
{
F
(
ε
)
(
x
)
,
γ
}
<
min
⁡
{
F
(
ε
)
y
1
y
n
-
1
,
F
(
ε
)
(
y
)
,
δ
}
=
min
⁡
{
(
F
(
ε
)
∩
min
⁡
{
F
(
ε
)
y
1
y
n
-
1
}
H
)
(
y
)
,
δ
}
,
which contradicts F
(
ε
)
∩
min
⁡
{
F
(
ε
)
y
1
y
n
-
1
}
H
⊆
∨
q
(
γ
,
δ
)
F
(
F
(
ε
), F
(
ε
)
(
y
1
)
y
1,…,F
(
ε
)
(
y
n
-
1
)
y
n
-
1
) by Lemma 2. Hence (F3a) holds.Lemma 18.
Let〈
F
,
A
〉
∈
F
S
(
H
,
E
). Then (F4a) holds if and only if the following condition holds:
(F
4
c) for all x
,
y
∈
H, ε
∈
A, and r
∈
(
γ
,
1
], r
x
∈
γ
F
(
r
y
,
F
(
ε
)
,
…
,
F
(
ε
)
︸
n
-
1
) implies r
y
∈
γ
∨
q
δ
F
(
r
x
,
F
(
ε
)
,
…
,
F
(
ε
)
︸
n
-
1
).Proof.
(F4a)⇒(F4c) Let x
,
y
∈
H and r
∈
(
γ
,
1
]. If r
y
∈
γ
F
(
r
x
,
F
(
ε
)
,
…
,
F
(
ε
)
︸
n
-
1
) but r
x
∈
γ
∨
q
δ
¯
F
(
r
y
,
F
(
ε
)
,
…
,
F
(
ε
)
︸
n
-
1
). Then
(26)
F
(
x
,
F
(
ε
)
,
…
,
F
(
ε
)
︸
n
-
1
)
(
y
)
≥
F
(
r
x
,
F
(
ε
)
,
…
,
F
(
ε
)
︸
n
-
1
)
≥
r
>
γ
,
but F
(
r
y,F
(
ε
)
,
…
,
F
(
ε
)
︸
n
-
1
)
(
x
)
<
r and F
(
r
y,F
(
ε
)
,
…
,
F
(
ε
)
︸
n
-
1
)
(
x
)
+
r
≤
2
δ. Then
(27)
min
⁡
{
F
(
y
,
F
(
ε
)
,
…
,
F
(
ε
)
︸
n
-
1
)
(
x
)
,
r
}
=
F
(
r
y
,
F
(
ε
)
,
…
,
F
(
ε
)
︸
n
-
1
)
(
x
)
<
r
gives F
(
y
,
μ
,
…
,
μ
︸
n
-
1
)
(
x
)
<
r and so 2
F
(
y
,
F
(
ε
)
,
…
,
F
(
ε
)
︸
n
-
1
)
(
x
)
<
min
⁡
⁡
{
F
(
y
,
F
(
ε
)
,
…
,
F
(
ε
)
︸
n
-
1
)
(
x
)
,
r
}
+
r
=
F
(
r
y,F
(
ε
)
,
…
,
F
(
ε
)
︸
n
-
1
)
(
x
)
+
r
≤
2
δ; that is, F
(
y, F
(
ε
)
,
…
,
F
(
ε
)
︸
n
-
1
)
(
x
)
<
δ. Thus we have max
⁡
⁡
{
F
(
y
,
F
(
ε
)
,
…
,
F
(
ε
)
︸
n
-
1
)
(
x
)
,
γ
}
<
min
⁡
⁡
{
F
(
x
,
F
(
ε
)
,
…
,
F
(
ε
)
︸
n
-
1
)
(
y
)
,
δ
}, a contradiction. Hence (F4c) holds.
(F4c)⇒(F4a) Let x
,
y
∈
H. If there exists r such that max
⁡
{
F
(
y
,
F
(
ε
)
,
…
,
F
(
ε
)
︸
n
-
1
)
(
x
)
,
γ
}
<
r
<
min
⁡
{
F
(
x
,
F
(
ε
)
,
…
,
F
(
ε
)
︸
n
-
1
)
(
y
)
,
δ
}, then
(28)
F
(
r
y
,
F
(
ε
)
,
…
,
F
(
ε
)
︸
n
-
1
)
(
x
)
=
min
⁡
{
F
(
y
,
F
(
ε
)
,
…
,
F
(
ε
)
︸
n
-
1
)
(
x
)
,
r
}
<
r
≤
min
⁡
{
F
(
x
,
F
(
ε
)
,
…
,
F
(
ε
)
︸
n
-
1
)
(
y
)
,
r
}
=
F
(
r
x
,
F
(
ε
)
,
…
,
F
(
ε
)
︸
n
-
1
)
(
y
)
and γ
<
r
<
δ. This implies that r
y
∈
γ
F
(
r
x,F
(
ε
)
,
…
,
F
(
ε
)
︸
n
-
1
) but r
x
∈
γ
∨
q
δ
¯
F
(
r
y,F
(
ε
)
,
…
,
F
(
ε
)
︸
n
-
1
), a contradiction. Hence max
⁡
⁡
{
F
(
y,F
(
ε
)
,
…
,
F
(
ε
)
︸
n
-
1
)
(
x
),γ
}
≥
min
⁡
⁡
{
F
(
x, F
(
ε
)
,
…
,
F
(
ε
)
︸
n
-
1
)
(
y
),δ
} and so max
⁡
⁡
{
min
⁡
⁡
{
F
(
y, F
(
ε
)
,
…
,
F
(
ε
)
︸
n
-
1
)
(
x
),δ
},γ
}
≥
max
⁡
⁡
{
min
⁡
⁡
{
F
(
x, F
(
ε
)
,
…
,
F
(
ε
)
︸
n
-
1
)
(
y
),δ
},γ
}. In a similar way, we have max
⁡
⁡
{
min
⁡
⁡
{
F
(
x,F
(
ε
)
,
…
,
F
(
ε
)
︸
n
-
1
)
(
y
),δ
},γ
}
≥
max
⁡
⁡
{
min
⁡
⁡
{
F
(
y,F
(
ε
)
,
…
,
F
(
ε
)
︸
n
-
1
)
(
x
),δ
},γ
}. Thus (F4a) holds.For any fuzzy soft set〈
F
,
A
〉 over H, ε
∈
A, andr
∈
(
γ
,
1
], denoteF
(
ε
)
r
=
{
x
∈
H
∣
x
r
∈
γ
F
(
ε
)
}, 〈
F
(
ε
)
〉
r
=
{
x
∈
H
∣
x
r
q
δ
F
(
ε
)
}, and [
F
(
ε
)
]
r
=
{
x
∈
H
∣
x
r
∈
γ
∨
q
δ
F
(
ε
)
}. The next theorem presents the relationships between (
∈
γ
,
∈
γ
∨
q
δ
)-fuzzy soft (invertible) n-ary subhypergroups over H and crisp (invertible) n-ary subhypergroups of H.Theorem 19.
Let〈
F
,
A
〉
∈
F
S
(
H
,
E
). Then,(1)
〈
F
,
A
〉 is an (
∈
γ
,
∈
γ
∨
q
δ
)-fuzzy soft (invertible) n-ary subhypergroup over H if and only if nonempty subset F
(
ε
)
r is an (invertible) n-ary subhypergroup of H for all ε
∈
A and r
∈
(
γ
,
δ
];
(2)
if2
δ
=
1
+
γ, then 〈
F
,
A
〉 is an (
∈
γ
,
∈
γ
∨
q
δ
)-fuzzy soft (invertible) n-ary subhypergroup over H if and only if nonempty subset 〈
F
(
ε
)
〉
r is an (invertible) n-ary subhypergroup of H for all ε
∈
A and r
∈
(
δ
,
1
];
(3)
〈
F
,
A
〉 is an (
∈
γ
,
∈
γ
∨
q
δ
)-fuzzy soft (invertible) n-ary subhypergroup over H if and only if nonempty subset [
F
(
ε
)
]
r is an (invertible) n-ary subhypergroup of H for all ε
∈
A and r
∈
(
γ
,
min
⁡
{
2
δ
-
γ
,
1
}
].Proof.
We only prove (2) and (3). (1) can be easily proved.
(2) Assume that2
δ
=
1
+
γ. Let 〈
F
,
A
〉 be an (
∈
γ
,
∈
γ
∨
q
δ
)-fuzzy soft n-ary subhypergroup over H and assume that 〈
F
(
ε
)
〉
r
≠
∅ for some ε
∈
A and r
∈
(
δ
,
1
]. Let x
1
n
∈
〈
F
(
ε
)
〉
r. Then r
x
1
q
δ
F
(
ε
)
,
…
,
r
x
n
q
δ
F
(
ε
); that is, F
(
ε
)
(
x
1
)
+
r
>
2
δ
,
…
,
F
(
ε
)
(
x
n
)
+
r
>
2
δ. Since 〈
F
,
A
〉 is an (
∈
γ
,
∈
γ
∨
q
δ
)-fuzzy n-ary subhypergroup over H, we have max
⁡
{
F
(
ε
)
(
x
)
,
γ
}
≥
min
⁡
{
F
(
ε
)
x
1
x
n
,
δ
} for all x
∈
f
(
x
1
n
). Hence, by r
>
δ,
(29)
max
⁡
{
F
(
ε
)
(
x
)
+
r
,
γ
+
r
}
=
max
⁡
{
F
(
ε
)
(
x
)
,
γ
}
+
r
≥
min
⁡
{
F
(
ε
)
x
1
x
n
,
δ
}
+
r
=
min
⁡
{
F
(
ε
)
(
x
1
)
+
r
,
…
,
F
(
ε
)
(
x
n
)
+
r
,
δ
+
r
}
>
2
δ
.
From r
≤
1
=
2
δ
-
γ, that is, r
+
γ
≤
2
δ, we have F
(
ε
)
(
x
)
+
r
>
2
δ and so x
∈
〈
F
(
ε
)
〉
r. Similarly we can show that e
∈
〈
F
(
ε
)
〉
r and, for all y
,
y
1
n
-
1
∈
〈
F
(
ε
)
〉
r, there exists x
∈
〈
F
(
ε
)
〉
r such that y
∈
f
(
x
,
y
1
n
-
1
). Therefore, 〈
F
(
ε
)
〉
r is an n-ary hypergroup of H.
Conversely, assume that the given condition holds. Letx
1
n
∈
H and ε
∈
A. If there exists x
∈
f
(
x
1
n
) such that max
⁡
{
F
(
ε
)
(
x
)
,
γ
}
<
min
⁡
{
F
(
ε
)
x
1
x
n
,
δ
}, take r
=
2
δ
-
max
⁡
{
F
(
ε
)
(
x
)
,
γ
}. Then r
∈
(
δ
,
1
], F
(
ε
)
(
x
)
≤
2
δ
-
r, F
(
ε
)
(
x
1
)
>
max
⁡
{
G
(
ε
)
(
x
)
,
γ
}
=
2
δ
-
r
,
…
,
F
(
ε
)
(
x
n
)
>
max
⁡
{
G
(
ε
)
(
x
)
,
γ
}
=
2
δ
-
r, that is, x
1
n
∈
〈
F
(
ε
)
〉
r, butx
∉
〈
F
(
ε
)
〉
r, a contradiction. Hence max
⁡
{
F
(
ε
)
(
x
)
,
γ
}
≥
min
⁡
{
F
(
ε
)
x
1
x
n
,
δ
} and so max
⁡
{
inf
⁡
x
∈
f
(
x
1
n
)
⁡
F
(
ε
)
(
x
)
,
γ
}
≥
min
⁡
{
F
(
ε
)
x
1
x
n
,
δ
}. Similarly we can show that conditions (F1a) and (F3a) hold. Therefore, 〈
F
,
A
〉 is an (
∈
γ
,
∈
γ
∨
q
δ
)-fuzzy soft n-ary subhypergroup over H.
(3) Let〈
F
,
A
〉 be an (
∈
γ
,
∈
γ
∨
q
δ
)-fuzzy soft n-ary subhypergroup over H and assume that [
F
(
ε
)
]
r
≠
∅ for some ε
∈
A and r
∈
(
γ
,
min
⁡
{
2
δ
-
γ
,
1
}
]. Let x
1
n
∈
[
F
(
ε
)
]
r. Then r
x
1
∈
γ
∨
q
δ
F
(
ε
)
,
…
,
r
x
n
∈
γ
∨
q
δ
F
(
ε
); that is, F
(
ε
)
(
x
1
)
≥
r
>
γ or F
(
ε
)
(
x
1
)
>
2
δ
-
r
≥
2
δ
-
(
2
δ
-
γ
)
=
γ
,
…, F
(
ε
)
(
x
n
)
≥
r
>
γ or F
(
ε
)
(
x
n
)
>
2
δ
-
r
≥
2
δ
-
(
2
δ
-
γ
)
=
γ. Since 〈
F
,
A
〉 is an (
∈
γ
,
∈
γ
∨
q
δ
)-fuzzy n-ary subhypergroup over H, we have max
⁡
{
F
(
ε
)
(
x
)
,
γ
}
≥
min
⁡
{
F
(
ε
)
x
1
x
n
,
δ
} for all x
∈
f
(
x
1
n
) and so F
(
ε
)
(
x
)
≥
min
⁡
{
F
(
ε
)
x
1
x
n
,
δ
} since γ
<
min
⁡
{
F
(
ε
)
x
1
x
n
,
δ
} in any case. Now we consider the following cases.
Case  1
(
r
∈
(
γ
,
δ
]
). In this case, 2
δ
-
r
≥
δ
≥
r. (1)
IfF
(
ε
)
(
x
1
)
≥
r or ⋯ or F
(
ε
)
(
x
n
)
≥
r, then F
(
ε
)
(
x
)
≥
min
⁡
{
F
(
ε
)
x
1
x
n
,
δ
}
≥
r. Hence z
r
∈
γ
F
(
ε
).
(2)
IfF
(
ε
)
(
x
i
)
+
r
>
2
δ, for all 1
≤
i
≤
n, then F
(
ε
)
(
x
)
≥
min
⁡
{
F
(
ε
)
x
1
x
n
,
δ
}
=
δ
≥
r. Hence z
r
∈
γ
F
(
ε
).
Case  2
(
r
∈
(
δ
,
min
⁡
{
2
δ
-
γ
,
1
}
]
). In this case, r
>
δ
>
2
δ
-
r. (1)
IfF
(
ε
)
(
x
i
)
≥
r, for all 1
≤
i
≤
n, then F
(
ε
)
(
x
)
≥
min
⁡
{
F
(
ε
)
x
1
x
n
,
δ
}
=
δ
>
2
δ
-
r. Hence z
r
q
δ
F
(
ε
).
(2)
IfF
(
ε
)
(
x
1
)
+
r
>
2
δ or ⋯ or F
(
ε
)
(
x
n
)
+
r
>
2
δ, then F
(
ε
)
(
x
)
≥
min
⁡
{
F
(
ε
)
x
1
x
n
,
δ
}
>
2
δ
-
r. Hence z
r
q
δ
F
(
ε
).
Thus, in any case,x
r
∈
γ
∨
q
δ
F
(
ε
); that is, x
∈
[
F
(
ε
)
]
r. Similarly, we can show that e
∈
[
F
(
ε
)
]
r and, for all y
,
y
1
n
-
1
∈
[
F
(
ε
)
]
r, there exists x
∈
[
F
(
ε
)
]
r such that y
∈
f
(
x
,
y
1
n
-
1
). Therefore, [
F
(
ε
)
]
r is an n-ary hypergroup of H.
Conversely, assume that the given condition holds. Letx
1
n
∈
H and ε
∈
A. If there exists x
∈
f
(
x
1
n
) such that max
⁡
{
F
(
ε
)
(
x
)
,
γ
}
<
r
=
min
⁡
{
F
(
ε
)
x
1
x
n
,
δ
}, then F
(
ε
)
(
x
1
)
≥
r
>
γ
,
…
,
F
(
ε
)
(
x
n
)
≥
r
>
γ
,
F
(
ε
)
(
x
)
<
r, and F
(
ε
)
(
x
)
+
r
<
2
r
≤
2
δ, that is, (
x
1
)
r
∈
γ
F
(
ε
)
,
…
,
(
x
n
)
r
∈
γ
F
(
ε
), but x
r
∈
γ
∨
q
δ
¯
F
(
ε
), that is, x
1
n
∈
[
F
(
ε
)
]
r, but x
∉
[
F
(
ε
)
]
r, a contradiction. Hence max
⁡
{
F
(
ε
)
(
x
)
,
γ
}
≥
min
⁡
{
F
(
ε
)
x
1
x
n
,
δ
} and so max
⁡
{
inf
⁡
x
∈
f
(
x
1
n
)
⁡
F
(
ε
)
(
x
)
,
γ
}
≥
min
⁡
{
F
(
ε
)
x
1
x
n
,
δ
}. Similarly, we can show that conditions (F1a) and (F3a) hold. Therefore, 〈
F
,
A
〉 is an (
∈
γ
,
∈
γ
∨
q
δ
)-fuzzy soft n-ary subhypergroup over H.As a direct consequence of Theorem19, we have the following results.Corollary 20.
Letγ
,
γ
′
,
δ
,
δ
′
∈
[
0,1
] be such that γ
<
δ, γ
′
<
δ
′, γ
<
γ
′, andδ
′
<
δ. Then every (
∈
γ
,
∈
γ
∨
q
δ
)-fuzzy (invertible) n-ary soft subhypergroup over H is an (
∈
γ
′
,
∈
γ
′
∨
q
δ
′
)-fuzzy soft (invertible) n-ary subhypergroup over H.Corollary 21.
LetP
∈
P
*
(
H
). Then P is an (invertible) n-ary subhypergroup of H if and only if Σ
(
P
,
A
,
r
) is an (
∈
γ
,
∈
γ
∨
q
δ
)-fuzzy soft (invertible) n-ary subhypergroup over H for any A
⊆
E and r
≥
δ.Lemma 22.
Let〈
F
,
A
〉
∈
F
S
(
H
,
E
). Then
(30)
〈
F
,
A
〉
⋐
(
γ
,
δ
)
F
(
〈
1
e
~
,
A
〉
,
…
,
〈
1
e
~
,
A
〉
︸
i
-
1
,
〈
F
,
A
〉
,
〈
1
e
~
,
A
〉
,
…
,
〈
1
e
~
,
A
〉
︸
n
-
i
)
for all 1
≤
i
≤
n.Proof.
It is straightforward.Lemma 23.
Let〈
F
,
A
〉 be an (
∈
γ
,
∈
γ
∨
q
δ
)-fuzzy soft n-ary subhypergroup over H. Then(1)
F
(
〈
F
,
A
〉
,
…
,
〈
F
,
A
〉
︸
i
,
〈
1
e
~
,
A
〉
,
…
,
〈
1
e
~
,
A
〉
︸
n
-
i
)
≍
(
γ
,
δ
)
〈
F
,
A
〉 for all 1
≤
i
≤
n;
(2)
F
(
〈
1
x
~
,
A
〉
,
〈
F
,
A
〉
,
…
,
〈
F
,
A
〉
︸
i, 〈
1
e
~
,
A
〉
,
…
,
〈
1
e
~
,
A
〉
︸
n
-
i
-
1
)
≍
(
γ
,
δ
)
F
(
〈
F
,
A
〉
,
…
,
〈
F
,
A
〉
︸
n
-
1
〈
1
x
~
,
A
〉, 〈
F
,
A
〉
,
…
,
〈
F
,
A
〉
︸
n
-
1
) for all x
∈
H and 1
≤
i
≤
n.Proof.
(1) Let〈
F
,
A
〉 be an (
∈
γ
,
∈
γ
∨
q
δ
)-fuzzy soft n-ary subhypergroup over H, ε
∈
A, and 1
≤
i
≤
n. Since max
⁡
{
F
(
ε
)
(
e
)
,
γ
}
≥
min
⁡
{
F
(
ε
)
(
x
)
,
δ
}, for all x
∈
H, we have
(31)
F
(
ε
)
⊆
∨
q
(
γ
,
δ
)
F
(
F
(
ε
)
,
1
e
,
…
,
1
e
︸
n
-
1
)
⊆
∨
q
(
γ
,
δ
)
F
(
F
(
ε
)
,
…
,
F
(
ε
)
︸
i
,
1
e
,
…
,
1
e
︸
n
-
i
)
⊆
∨
q
(
γ
,
δ
)
F
(
F
(
ε
)
,
…
,
F
(
ε
)
︸
n
)
⊆
∨
q
(
γ
,
δ
)
F
(
ε
)
.
Therefore, F
(
F
(
ε
)
,
1
e
,
…
,
1
e
︸
n
-
1
)
=
(
γ
,
δ
)
F
(
ε
) and so F
(
〈
F
,
A
〉
,
…
,
〈
F
,
A
〉
︸
i
,
〈
1
e
~
,
A
〉
,
…
,
〈
1
e
~
,
A
〉
︸
n
-
i
)
≍
(
γ
,
δ
)
〈
F
,
A
〉.
(2) Letx
∈
H and 1
≤
i
≤
n. By (1), we have
(32)
F
(
〈
1
x
~
,
A
〉
,
〈
F
,
A
〉
,
…
,
〈
F
,
A
〉
︸
n
-
1
)
≍
(
γ
,
δ
)
F
(
〈
1
x
~
,
A
〉
,
〈
F
,
A
〉
,
…
,
〈
F
,
A
〉
︸
n
-
2
)
,
F
(
〈
F
,
A
〉
,
〈
1
e
~
,
A
〉
,
…
,
〈
1
e
~
,
A
〉
︸
n
-
1
)
≍
(
γ
,
δ
)
F
(
〈
1
x
~
,
A
〉
,
〈
F
,
A
〉
,
…
,
〈
F
,
A
〉
︸
i
-
1
,
vvvvv
F
(
〈
F
,
A
〉
,
…
,
〈
F
,
A
〉
︸
n
-
i
,
〈
1
e
~
,
A
〉
,
…
,
〈
1
e
~
,
A
〉
︸
i
)
,
v
v
v
v
v
v
v
v
v
v
v
〈
1
e
~
,
A
〉
,
…
,
〈
1
e
~
,
A
〉
︸
n
-
i
-
1
)
≍
(
γ
,
δ
)
F
(
〈
1
x
~
,
A
〉
,
〈
F
,
A
〉
,
…
,
〈
F
,
A
〉
︸
i
-
1
,
v
v
v
v
v
v
〈
F
,
A
〉
,
〈
1
e
~
,
A
〉
,
…
,
〈
1
e
~
,
A
〉
︸
n
-
i
-
1
)
≍
(
γ
,
δ
)
F
(
〈
1
x
~
,
A
〉
,
〈
F
,
A
〉
,
…
,
〈
F
,
A
〉
︸
i
,
v
v
v
v
v
v
v
v
〈
1
e
~
,
A
〉
,
…
,
〈
1
e
~
,
A
〉
︸
n
-
i
-
1
)
.
This completes the proof.Lemma 24.
Let〈
F
,
A
〉 be an (
∈
γ
,
∈
γ
∨
q
δ
)-fuzzy soft n-ary subhypergroup over H. Then 〈
F
,
A
〉 is invertible if and only if
(33)
max
⁡
{
min
⁡
{
F
(
x
,
F
(
ε
)
,
…
,
F
(
ε
)
︸
i
,
e
*
)
(
y
)
,
δ
}
,
γ
}
=
max
⁡
{
min
⁡
{
F
(
y
,
F
(
ε
)
,
…
,
F
(
ε
)
︸
i
,
e
*
)
(
x
)
,
δ
}
,
γ
}
for all x
,
y
∈
H, ε
∈
A, and 1
≤
i
≤
n. Proof
It is straightforward by Lemmas18 and 23.Lemma 25.
Let〈
F
,
A
〉,〈
F
1
,
A
〉
,
…
,
〈
F
i
,
A
〉
(
2
≤
i
≤
n
) be (
∈
γ
,
∈
γ
∨
q
δ
)-fuzzy soft invertible n-ary subhypergroups over H. Then(1)
max
⁡
{
min
⁡
{
F
(
x
,
F
1
i
(
ε
)
,
e
*
)
(
y
)
,
δ
}
,
γ
}
=
max
⁡
⁡
{
min
⁡
⁡
{
F
(
y,F
1
i
(
ε
)
,
e
*
)
(
x
),δ
},γ
}, for all x
,
y
∈
H and ε
∈
A;
(2)
for allx
,
y
,
y
1
n
-
1
∈
H such that y
∈
f
(
x
,
y
1
n
-
1
) and ε
∈
A, one has max
⁡
{
F
(
ε
)
(
x
)
,
γ
}
≥
min
⁡
{
F
(
ε
)
y
1
y
n
-
1
,
F
(
ε
)
(
y
)
,
δ
}.Proof.
(1) Letx
,
y
∈
H and ε
∈
A. Then, by Lemma 23, we have
(34)
F
(
x
,
F
1
i
(
ε
)
,
e
*
)
=
F
(
x
,
F
1
(
ε
)
,
F
2
(
ε
)
,
F
3
i
(
ε
)
,
e
*
)
≍
(
γ
,
δ
)
F
(
x
,
F
1
(
ε
)
,
F
(
e
,
…
,
e
︸
n
-
2
,
F
2
(
ε
)
,
e
)
,
F
3
i
(
ε
)
,
e
*
)
≍
(
γ
,
δ
)
F
(
F
(
x
,
F
1
(
ε
)
,
e
*
)
,
F
2
(
ε
)
,
e
,
F
3
i
(
ε
)
,
e
*
)
≍
(
γ
,
δ
)
F
(
F
(
x
,
F
1
(
ε
)
,
e
*
)
,
F
2
i
(
ε
)
,
e
*
)
≍
(
γ
,
δ
)
F
(
F
(
x
,
F
1
(
ε
)
,
e
*
)
,
F
(
F
2
i
(
ε
)
,
e
*
)
,
e
*
)
.
It follows that
(35)
max
⁡
{
min
⁡
⁡
{
F
(
x
,
F
1
i
(
ε
)
,
e
*
)
(
y
)
,
δ
}
,
γ
}
≥
min
⁡
⁡
{
(
F
2
i
(
ε
)
,
e
*
)
max
⁡
{
F
2
i
(
ε
)
F
(
F
(
F
2
i
(
ε
)
x
,
F
1
(
ε
)
,
e
*
)
,
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
F
(
F
2
i
(
ε
)
,
e
*
)
,
e
*
)
(
y
)
,
γ
F
2
i
(
ε
)
}
,
δ
F
2
i
(
ε
)
}
=
min
⁡
{
max
⁡
{
sup
⁡
y
∈
f
(
a
1
,
a
2
,
e
*
)
min
⁡
{
(
F
2
i
(
ε
)
,
e
*
)
F
(
x
,
F
1
(
ε
)
,
e
*
)
(
a
1
)
,
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
F
(
F
2
i
(
ε
)
,
e
*
)
(
a
2
)
}
,
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
γ
sup
⁡
y
∈
f
(
a
1
,
a
2
,
e
*
)
}
,
δ
sup
⁡
y
∈
f
(
a
1
,
a
2
,
e
*
)
}
=
sup
⁡
z
1
∈
H
min
⁡
{
(
z
1
,
F
2
i
(
ε
)
,
e
*
)
max
⁡
{
(
z
1
,
F
2
i
(
ε
)
,
e
*
)
min
⁡
{
(
z
1
,
F
2
i
(
ε
)
,
e
*
)
F
(
x
,
F
1
(
ε
)
,
e
*
)
(
z
1
)
,
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
F
(
z
1
,
F
2
i
(
ε
)
,
e
*
)
(
y
)
}
,
γ
(
z
1
,
F
2
i
(
ε
)
,
e
*
)
}
,
δ
(
z
1
,
F
2
i
(
ε
)
,
e
*
)
}
=
sup
⁡
z
1
,
z
2
∈
H
min
⁡
⁡
{
(
z
2
,
F
3
i
(
ε
)
,
e
*
)
max
⁡
{
(
z
2
,
F
3
i
(
ε
)
,
e
*
)
min
⁡
{
(
z
2
,
F
3
i
(
ε
)
,
e
*
)
F
(
x
,
F
1
(
ε
)
,
e
*
)
(
z
1
)
,
vvvvvvvvvvvvvvvvvvvvvvv
F
(
z
1
,
F
2
(
ε
)
,
e
*
)
(
z
2
)
,
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
i
v
v
v
v
v
v
v
v
F
(
z
2
,
F
3
i
(
ε
)
,
e
*
)
(
y
)
}
,
γ
(
z
2
,
F
3
i
(
ε
)
,
e
*
)
}
,
δ
(
z
2
,
F
3
i
(
ε
)
,
e
*
)
}
=
⋯
=
sup
⁡
z
1
i
∈
H
min
⁡
{
F
3
i
max
⁡
{
F
3
i
min
⁡
{
F
3
i
F
(
x
,
F
1
(
ε
)
,
e
*
)
(
z
1
)
,
vvvvvvvvvvvvvvvvvvvvv
F
(
z
1
,
F
2
(
ε
)
,
e
*
)
(
z
2
)
,
…
,
v
v
v
v
v
v
v
v
v
i
v
v
v
v
v
v
v
v
v
v
v
v
v
v
F
(
z
i
-
1
,
F
i
(
ε
)
,
e
*
)
(
y
)
F
3
i
}
,
γ
F
3
i
}
,
δ
F
3
i
}
≥
sup
⁡
z
1
i
∈
H
min
⁡
{
F
(
z
1
,
F
1
(
ε
)
,
e
*
)
(
x
)
,
vvvvvvvvvvv
F
(
z
2
,
F
2
(
ε
)
,
e
*
)
(
z
1
)
,
…
,
v
v
v
v
v
v
v
v
v
v
v
v
F
(
y
,
F
i
(
ε
)
,
e
*
)
(
z
i
-
1
)
,
δ
}
=
min
⁡
{
F
(
y
,
F
i
(
ε
)
,
F
i
-
1
(
ε
)
,
…
,
F
1
(
ε
)
,
e
*
)
(
x
)
,
δ
}
=
min
⁡
⁡
{
F
(
y
,
F
1
i
(
ε
)
,
e
*
)
(
x
)
,
δ
}
.
This implies max
⁡
{
min
⁡
{
F
(
x
,
F
1
i
(
ε
)
,
e
*
)
(
y
)
,
δ
}
,
γ
}
≥
max
⁡
{
min
⁡
⁡
{
F
(
y
,
F
1
i
(
ε
)
,
e
*
)
(
x
)
,
δ
}
,
γ
}. In a similar way, we have max
⁡
⁡
{
min
⁡
⁡
{
F
(
y
,
F
1
i
(
ε
), e
*
)
(
x
)
,
δ
}
,
γ
}
≥
max
⁡
⁡
{
min
⁡
⁡
{
F
(
x
,
F
1
i
(
ε
)
,
e
*
)
(
y
)
,
δ
}
,
γ
}. Hence max
⁡
⁡
{
min
⁡
⁡
{
F
(
x
,
F
1
i
(
ε
)
,
e
*
)
(
y
)
,
δ
}
,
γ
}
=
max
⁡
⁡
{
min
⁡
⁡
{
F
1
i
(
y
,
F
(
ε
)
,
e
*
)
(
x
)
,
δ
}
,
γ
}.
(2) Letx
,
y
,
y
1
n
-
1
∈
H be such that y
∈
f
(
x
,
y
1
n
-
1
) and ε
∈
A. Then F
(
x
,
F
(
ε
)
,
…
,
F
(
ε
)
︸
n
-
1
)
(
y
)
≥
min
⁡
{
F
(
ε
)
y
1
y
n
-
1
}. Thus, we have
(36)
max
⁡
{
F
(
ε
)
(
x
)
,
γ
}
≥
max
⁡
{
min
⁡
{
F
(
F
(
ε
)
,
…
,
F
(
ε
)
︸
n
)
(
x
)
,
δ
}
,
γ
}
≥
max
⁡
{
min
⁡
{
F
(
ε
)
(
y
)
,
F
(
y
,
F
(
ε
)
,
…
,
F
(
ε
)
︸
n
-
1
)
(
x
)
,
δ
(
y
,
F
(
ε
)
,
…
,
F
(
ε
)
︸
n
-
1
)
}
,
γ
(
y
,
F
(
ε
)
,
…
,
F
(
ε
)
︸
n
-
1
)
}
≥
min
⁡
{
F
(
ε
)
(
y
)
,
max
⁡
{
F
(
y
,
F
(
ε
)
,
…
,
F
(
ε
)
︸
n
-
1
)
(
x
)
,
γ
(
y
,
F
(
ε
)
,
…
,
F
(
ε
)
︸
n
-
1
)
}
,
δ
(
y
,
F
(
ε
)
,
…
,
F
(
ε
)
︸
n
-
1
)
}
≥
min
⁡
{
F
(
ε
)
(
y
)
,
F
(
x
,
F
(
ε
)
,
…
,
F
(
ε
)
︸
n
-
1
)
(
y
)
,
δ
}
≥
min
⁡
{
F
(
ε
)
(
y
)
,
F
(
ε
)
y
1
y
n
-
1
,
δ
}
.
This completes the proof.Theorem 26.
Let〈
F
k
,
A
〉 be (
∈
γ
,
∈
γ
∨
q
δ
)-fuzzy soft invertible n-ary subhypergroup over H for all 1
≤
k
≤
n and 2
≤
i
≤
n. Then F
(
〈
F
1
,
A
〉
,
…
,
〈
F
i
,
A
〉
,
〈
1
e
~
,
A
〉
,
…
,
〈
1
e
~
,
A
〉
︸
n
-
i
) is an (
∈
γ
,
∈
γ
∨
q
δ
)-fuzzy soft invertible n-ary subhypergroup over H.Proof.
Since each〈
F
k
,
A
〉 is an (
∈
γ
,
∈
γ
∨
q
δ
)-fuzzy soft invertible n-ary subhypergroup over H for all1
≤
k
≤
n, let ε
∈
A. It is clear that max
⁡
⁡
{
F
(
F
1
i
(
ε
),e
*
)
(
e
),γ
}
≥
min
⁡
⁡
{
F
(
F
1
i
(
ε
),e
*
)
(
x
),δ
} for all x
∈
H. From Lemma 23, for any 2
≤
j
≤
n, we have
(37)
F
(
F
(
F
1
i
(
ε
)
,
e
*
)
,
…
,
F
(
F
1
i
(
ε
)
,
e
*
)
︸
j
,
e
*
)
=
F
(
F
(
F
1
(
ε
)
,
…
,
F
1
(
ε
)
︸
j
,
e
*
)
,
…
,
v
v
v
v
v
v
v
F
(
F
i
(
ε
)
,
…
,
F
i
(
ε
)
︸
j
,
e
*
)
,
e
*
)
≍
(
γ
,
δ
)
F
(
F
1
i
(
ε
)
,
e
*
)
.
In particular, we have
(38)
F
(
F
(
F
1
i
(
ε
)
,
e
*
)
,
…
,
F
(
F
1
i
(
ε
)
,
e
*
)
︸
n
)
≍
(
γ
,
δ
)
F
(
F
1
i
(
ε
)
,
e
*
)
.
Now we show that(39)
max
⁡
{
min
⁡
{
F
(
x
,
F
(
F
1
i
(
ε
)
,
e
*
)
,
…
,
F
(
F
1
i
(
ε
)
,
e
*
)
︸
n
-
1
)
(
y
)
,
δ
}
,
γ
}
=
max
⁡
{
min
⁡
{
F
(
y
,
F
(
F
1
i
(
ε
)
,
e
*
)
,
…
,
F
(
F
1
i
(
ε
)
,
e
*
)
︸
n
-
1
)
(
x
)
,
δ
}
,
γ
}for all x
,
y
∈
H; that is, F
(
F
1
i
(
ε
)
,
e
*
) is invertible. In fact, by Lemmas 23 and 25(1), we have(40)
max
⁡
{
min
⁡
{
F
(
x
,
F
(
F
1
i
(
ε
)
,
e
*
)
,
…
,
F
(
F
1
i
(
ε
)
,
e
*
)
︸
n
-
1
)
(
y
)
,
δ
}
,
γ
}
=
max
⁡
{
min
⁡
{
F
(
x
,
F
1
i
(
ε
)
,
e
*
)
(
y
)
,
δ
}
,
γ
}
=
max
⁡
{
min
⁡
{
F
(
y
,
F
1
i
(
ε
)
,
e
*
)
(
x
)
,
δ
}
,
γ
}
=
max
⁡
{
min
⁡
{
F
(
y
,
F
(
F
1
i
(
ε
)
,
e
*
)
,
…
,
F
(
F
1
i
(
ε
)
,
e
*
)
︸
n
-
1
)
(
x
)
,
δ
}
,
γ
}
.
Next, letx
,
y
,
y
1
n
-
1
∈
H be such that y
∈
f
(
x
,
y
1
n
-
1
). Then by Lemma 25(2), we have max
⁡
{
F
(
F
1
i
(
ε
)
,
e
*
)
(
x
)
,
γ
}
≥
min
⁡
{
F
(
F
1
i
(
ε
)
,
e
*
)
y
1
y
n
-
1
,
F
(
F
1
i
(
ε
)
,
e
*
)
(
y
)
,
δ
}.
Summing up the above arguments,F
(
〈
F
1
,
A
〉
,
…
,
〈
F
i
,
A
〉
,
〈
1
e
~
,
A
〉
,
…
,
〈
1
e
~
,
A
〉
︸
n
-
i
) is an (
∈
γ
,
∈
γ
∨
q
δ
)-fuzzy soft invertible n-ary subhypergroup over H.Theorem 27.
Let〈
F
,
A
〉 and 〈
G
,
B
〉 be two (
∈
γ
,
∈
γ
∨
q
δ
)-fuzzy soft invertible n-ary subhypergroups over H. Then both 〈
F
,
A
〉
∩
~
〈
G
,
B
〉 and 〈
F
,
A
〉
⋒
〈
G
,
B
〉 are (
∈
γ
,
∈
γ
∨
q
δ
)-fuzzy soft n-ary subhypergroups over H.Proof.
Let〈
F
,
A
〉 and 〈
G
,
B
〉 be (
∈
γ
,
∈
γ
∨
q
δ
)-fuzzy soft invertible n-ary subhypergroups over H and 〈
F
,
A
〉
∩
~
〈
G
,
B
〉
=
〈
L
,
A
∪
B
〉. Then, for any ε
∈
A
∪
B, we consider the following cases.
Case  1
(
ε
∈
A
-
B
). In this case, L
(
ε
)
=
F
(
ε
). It follows that L
(
ε
) satisfies conditions (F1a), (F2a), and (F3a) since 〈
F
,
A
〉 is an (
∈
γ
,
∈
γ
∨
q
δ
)-fuzzy soft invertible n-ary subhypergroup over S.
Case  2
(
ε
∈
B
-
A
). In this case, L
(
ε
)
=
G
(
ε
). It follows that L
(
ε
) satisfies conditions (F1a), (F2a), and (F3a) since 〈
G
,
B
〉 is an (
∈
γ
,
∈
γ
∨
q
δ
)-fuzzy soft invertible n-ary subhypergroup over S.
Case  3
(
ε
∈
A
∩
B
). In this case, L
(
ε
)
=
F
(
ε
)
∩
G
(
ε
). Now we show that F
(
ε
)
∩
G
(
ε
) satisfies conditions (F1a), (F2a), and (F3a). (1)
For anyx
∈
H, we have
(41)
max
⁡
{
(
F
(
ε
)
∩
G
(
ε
)
)
(
e
)
,
γ
}
=
max
⁡
{
min
⁡
{
F
(
ε
)
(
e
)
,
G
(
ε
)
(
e
)
}
,
γ
}
=
min
⁡
{
max
⁡
{
F
(
ε
)
(
e
)
,
γ
}
,
max
⁡
{
G
(
ε
)
(
e
)
,
γ
}
}
≥
min
⁡
{
F
(
ε
)
(
x
)
,
G
(
ε
)
(
x
)
,
δ
}
=
min
⁡
{
(
F
(
ε
)
∩
G
(
ε
)
)
(
x
)
,
δ
}
.
(2)
Letx
,
x
1
n
∈
H be such that x
∈
f
(
x
1
n
). Then
(42)
max
⁡
{
(
F
(
ε
)
∩
G
(
ε
)
)
(
x
)
,
γ
}
=
max
⁡
{
min
⁡
{
F
(
ε
)
(
x
)
,
G
(
ε
)
(
x
)
}
,
γ
}
=
min
⁡
{
max
⁡
{
F
(
ε
)
(
x
)
,
γ
}
,
max
⁡
{
G
(
ε
)
(
x
)
,
γ
}
}
≥
min
⁡
{
F
(
ε
)
x
1
x
n
,
G
(
ε
)
x
1
x
n
,
δ
}
=
min
⁡
{
(
F
(
ε
)
∩
G
(
ε
)
)
x
1
x
n
,
δ
}
.It follows that max
⁡
{
inf
⁡
x
∈
f
(
x
1
n
)
⁡
(
F
(
ε
)
∩
G
(
ε
)
)
(
x
)
,
γ
}
≥
min
⁡
{
(
F
(
ε
)
∩
G
(
ε
)
)
x
1
x
n
,
δ
}.(3)
Letx
,
y
,
y
1
n
-
1
∈
H be such that y
∈
f
(
x
,
y
1
n
-
1
). Analogous to (2), we have
(43)
max
⁡
{
(
F
(
ε
)
∩
G
(
ε
)
)
(
x
)
,
γ
}
≥
min
⁡
{
(
F
(
ε
)
∩
G
(
ε
)
)
y
1
y
n
-
1
,
(
F
(
ε
)
∩
G
(
ε
)
)
(
y
)
,
δ
}
.
Therefore,〈
F
,
A
〉
∩
~
〈
G
,
B
〉 is an (
∈
γ
,
∈
γ
∨
q
δ
)-fuzzy soft n-ary subhypergroup over H. The case for 〈
F
,
A
〉
⋒
〈
G
,
B
〉 can be similarly proved.


## 5. The Homomorphism of(
∈
γ
,
∈
γ
∨
q
δ
)-Fuzzy Soft (Invertible) Subhypergroups

In this section, we consider the homomorphism properties of(
∈
γ
,
∈
γ
∨
q
δ
)-fuzzy soft (invertible) subhypergroups. Let us first introduce the following definition.Definition 28.
LetF
S
(
U
,
E
) and F
S
(
U
′
,
E
′
) be two fuzzy soft classes, and let φ
:
U
→
U
′ and ϕ
:
E
→
E
′ be mappings. Define a mapping (
φ
,
ϕ
)
:
F
S
(
U
,
E
)
→
F
S
(
U
′
,
E
′
) by the following: for 〈
F
,
A
〉
∈
F
S
(
U
,
E
), theimage of 〈
F
,
A
〉 under (
φ
,
ϕ
), denoted by (
φ
,
ϕ
)
〈
F
,
A
〉
=
〈
φ
(
F
)
,
ϕ
(
A
)
〉, is a fuzzy soft set in F
S
(
U
′
,
E
′
) given by
(44)
φ
(
F
)
(
ε
′
)
(
x
′
)
=
{
sup
⁡
ε
∈
ϕ
-
1
(
ε
′
)
∩
A
,
x
∈
φ
-
1
(
x
′
)
F
(
ε
)
(
x
)
if
φ
-
1
(
x
′
)
≠
∅
,
0
otherwise
,
for all ε
′
∈
ϕ
(
A
) and x
′
∈
U
′. For 〈
F
′
,
A
′
〉
∈
F
S
(
U
′
,
E
′
), theinverse image of 〈
F
′
,
A
′
〉 under (
φ
,
ϕ
), denoted by (
φ
,
ϕ
)
-
1
〈
F
′
,
A
′
〉
=
〈
φ
-
1
(
F
′
)
,
ϕ
-
1
(
A
′
)
〉, is a fuzzy soft set in F
S
(
U
,
E
) given by
(45)
φ
-
1
(
F
′
)
(
ε
)
(
x
)
=
F
′
(
ϕ
(
ε
)
)
(
φ
(
x
)
)
for all ε
∈
ϕ
-
1
(
B
) and x
∈
U.The following results can be easily deduced.Lemma 29.
Let〈
F
,
A
〉
,
〈
G
,
B
〉
∈
F
S
(
U
,
E
) and φ
:
U
→
U
′ and ϕ
:
E
→
E
′ be two mappings. Then(1)
〈
F
,
A
〉
⋐
(
φ
,
ϕ
)
-
1
(
(
φ
,
ϕ
)
〈
F
,
A
〉
) and 〈
F
,
A
〉
=
(
φ
,
ϕ
)
-
1
(
(
φ
,
ϕ
)
〈
F
,
A
〉
) if both φ and ϕ are injective;
(2)
(
φ
,
ϕ
)
(
〈
F
,
A
〉
⋒
〈
G
,
B
〉
)
⋐
(
φ
,
ϕ
)
〈
F
,
A
〉
⋒
(
φ
,
ϕ
)
〈
G
,
B
〉;
(3)
(
φ
,
ϕ
)
(
〈
F
,
A
〉
∩
~
〈
G
,
B
〉
)
⋐
(
φ
,
ϕ
)
〈
F
,
A
〉
∩
~
(
φ
,
ϕ
)
〈
G
,
B
〉 if ϕ is injective;
(4)
if〈
F
,
A
〉
⋐
(
γ
,
δ
)
〈
G
,
B
〉, then (
φ
,
ϕ
)
〈
F
,
A
〉
⋐
(
γ
,
δ
)
(
φ
,
ϕ
)
〈
G
,
B
〉.Lemma 30.
Let〈
F
′
,
A
′
〉
,
〈
G
′
,
B
′
〉
∈
F
S
(
U
′
,
E
′
) and φ
:
U
→
U
′ and ϕ
:
E
→
E
′ be two mappings. Then(1)
(
φ
,
ϕ
)
(
(
φ
,
ϕ
)
-
1
〈
F
′
,
A
′
〉
)
⋐
〈
F
′
,
A
′
〉 and (
φ
,
ϕ
)
(
(
φ
,
ϕ
)
-
1
〈
F
′
,
A
′
〉
)
=
〈
F
′
,
A
′
〉 if both φ and ϕ are surjective;
(2)
(
φ
,
ϕ
)
-
1
(
〈
F
′
,
A
′
〉
⋒
〈
G
′
,
B
′
〉
)
=
(
φ
,
ϕ
)
-
1
〈
F
′
,
A
′
〉
⋒
(
φ
,
ϕ
)
-
1
〈
G
′
,
B
′
〉;
(3)
(
φ
,
ϕ
)
-
1
(
〈
F
′
,
A
′
〉
∩
~
〈
G
′
,
B
′
〉
)
=
(
φ
,
ϕ
)
-
1
〈
F
′
,
A
′
〉
∩
~
(
φ
,
ϕ
)
-
1
〈
G
′
,
B
′
〉;
(4)
if〈
F
′
,
A
′
〉
⋐
(
γ
,
δ
)
〈
G
′
,
B
′
〉, then (
φ
,
ϕ
)
-
1
〈
F
′
,
A
′
〉
⋐
(
γ
,
δ
)
(
φ
,
ϕ
)
-
1
〈
G
′
,
B
′
〉.Definition 31.
Let(
H
,
f
) and (
H
′
,
f
′
) be two n-ary hypergroups with identities e and e
′, respectively, and φ a mapping from H to H
′. Then, φ is called ahomomorphism if φ
(
e
)
=
e
′ and φ
(
f
(
x
1
n
)
)
=
f
′
(
φ
(
x
1
)
,
…
,
φ
(
x
n
)
) for all x
1
n
∈
H. If such a homomorphism is surjective, injective, or bijective, it is called anepimorphism, amonomorphism, or anisomorphism.Lemma 32.
Let(
H
,
f
) and (
H
′
,
f
′
) be two n-ary hypergroups with identities e and e
′, respectively, φ
:
H
→
H
′ a homomorphism, and ϕ
:
E
→
E
′ a mapping. Then (
φ
,
ϕ
)
(
F
(
〈
F
,
A
〉
1
n
)
)
⋐
F
′
(
(
φ
,
ϕ
)
〈
F
1
,
A
1
〉
,
…
,
(
φ
,
ϕ
)
〈
F
n
,
A
n
〉
) for all 〈
F
,
A
〉
1
n
∈
F
S
(
H
,
E
) such that ⋂
i
=
1
n
‍
A
i
≠
∅ and (
φ
,
ϕ
)
(
F
(
〈
F
,
A
〉
1
n
)
)
=
F
′
(
(
φ
,
ϕ
)
〈
F
1
,
A
1
〉
,
…
,
(
φ
,
ϕ
)
〈
F
n
,
A
n
〉
) if ϕ is injective.Proof.
Let〈
F
,
A
〉
1
n
∈
F
S
(
H
,
E
), ε
′
∈
ϕ
(
⋂
i
=
1
n
‍
A
i
)
⊆
⋂
i
=
1
n
‍
ϕ
(
A
i
), and x
′
∈
H
′. If φ
-
1
(
x
′
)
=
∅, then F
′
(
φ
(
F
1
)
,
…
,
φ
(
F
n
)
)
(
ε
′
)
(
x
′
)
=
0
=
φ
(
F
(
F
1
n
)
(
ε
′
)
)
(
x
′
). Otherwise, we have(46)
F
′
(
φ
(
F
1
)
,
…
,
φ
(
F
n
)
)
(
ε
′
)
(
x
′
)
=
F
′
(
φ
(
F
1
)
(
ε
′
)
,
…
,
φ
(
F
n
)
(
ε
′
)
)
(
x
′
)
=
sup
⁡
x
′
∈
f
′
(
x
1
′
,
…
,
x
n
′
)
min
⁡
{
φ
(
F
1
)
(
ε
′
)
(
x
1
′
)
,
…
,
φ
(
F
n
)
(
ε
′
)
(
x
n
′
)
}
=
sup
⁡
x
1
′
,
…
,
x
n
′
∈
Im
⁡
(
φ
)
,
x
′
∈
f
′
(
x
1
′
,
…
,
x
n
′
)
min
⁡
{
sup
⁡
ε
1
∈
ϕ
-
1
(
ε
′
)
∩
A
1
,
φ
(
x
1
)
=
x
1
′
F
1
(
ε
1
)
(
x
1
)
,
…
,
sup
⁡
ε
n
∈
ϕ
-
1
(
ε
′
)
∩
A
n
,
φ
(
x
n
)
=
x
n
′
F
n
(
ε
n
)
(
x
n
)
}
=
sup
⁡
x
1
′
,
…
,
x
n
′
∈
Im
⁡
(
φ
)
,
x
′
∈
f
′
(
x
1
′
,
…
,
x
n
′
)
,
ε
1
∈
ϕ
-
1
(
ε
′
)
∩
A
1
,
…
,
ε
n
∈
ϕ
-
1
(
ε
′
)
∩
A
n
min
⁡
{
F
1
(
ε
1
)
(
x
1
)
,
…
,
F
n
(
ε
n
)
(
x
n
)
}
≥
sup
⁡
x
′
∈
f
′
(
φ
(
x
1
)
,
…
,
φ
(
x
n
)
)
,
ε
∈
ϕ
-
1
(
ε
′
)
∩
∩
i
=
1
n
A
i
min
⁡
{
F
1
(
ε
)
(
x
1
)
,
…
,
F
n
(
ε
)
(
x
n
)
}
=
sup
⁡
x
′
∈
φ
(
f
(
x
1
,
…
,
x
n
)
)
,
ε
∈
ϕ
-
1
(
ε
′
)
∩
∩
i
=
1
n
A
i
min
⁡
{
F
1
(
ε
)
(
x
1
)
,
…
,
F
n
(
ε
)
(
x
n
)
}
=
sup
⁡
x
∈
φ
-
1
(
x
′
)
,
x
∈
f
(
x
1
,
…
,
x
n
)
,
ε
∈
ϕ
-
1
(
ε
′
)
∩
∩
i
=
1
n
A
i
min
⁡
{
F
1
(
ε
)
(
x
1
)
,
…
,
F
n
(
ε
)
(
x
n
)
}
=
sup
⁡
x
∈
φ
-
1
(
x
′
)
,
ε
∈
ϕ
-
1
(
ε
′
)
∩
∩
i
=
1
n
A
i
F
(
F
1
n
(
ε
)
)
(
x
)
=
sup
⁡
x
∈
φ
-
1
(
x
′
)
,
ε
∈
ϕ
-
1
(
ε
′
)
∩
∩
i
=
1
n
A
i
F
(
F
1
n
)
(
ε
)
(
x
)
=
φ
(
F
(
F
1
n
)
(
ε
′
)
)
(
x
′
)
.Hence (
φ
,
ϕ
)
(
F
(
〈
F
,
A
〉
1
n
)
)
⋐
F
'
(
(
φ
,
ϕ
)
〈
F
1
,
A
1
〉
,
…
,
(
φ
,
ϕ
)
〈
F
n
,
A
n
〉
). If ϕ is injective, then the symbol “≥” in the above proof can be replaced with “=”; hence (
φ
,
ϕ
)
(
F
(
〈
F
,
A
〉
1
n
)
)
=
F
'
(
(
φ
,
ϕ
)
〈
F
1
,
A
1
〉,…
,
(
φ
,
ϕ
)
〈
F
n
,
A
n
〉
).Theorem 33.
Let(
H
,
f
) and (
H
′
,
f
′
) be two commutative n-ary hypergroups with identities e and e
′, respectively, φ
:
H
→
H
′ a homomorphism, and ϕ
:
E
→
E
′ an injective mapping. Let 〈
F
,
A
〉 be an (
∈
γ
,
∈
γ
∨
q
δ
)-fuzzy n-ary soft subhypergroup over H which has the sup-property. Then, (
φ
,
ϕ
)
〈
F
,
A
〉 is an (
∈
γ
,
∈
γ
∨
q
δ
)-fuzzy n-ary subhypergroup over H
′. If φ is an epimorphism from H onto H
′ and 〈
F
,
A
〉 is invertible, then (
φ
,
ϕ
)
〈
F
,
A
〉 is also invertible.Proof.
Assume that〈
F
,
A
〉 is an (
∈
γ
,
∈
γ
∨
q
δ
)-fuzzy n-ary soft subhypergroup over H which has the sup-property and ε
′
∈
ϕ
(
A
). Then there exists a unique ε
∈
A such that ϕ
(
ε
)
=
ε
′ and we have the following.(1)
It is clear thatmax
⁡
{
φ
(
F
)
(
ε
′
)
(
e
′
)
,
γ
}
≥
min
⁡
{
φ
(
F
)
(
ε
′
)
(
x
′
)
,
δ
} for all x
′
∈
H.
(2)
By Lemma32, we have
(47)
F
′
(
φ
(
F
)
(
ε
′
)
,
…
,
φ
(
F
)
(
ε
′
)
︸
n
)
=
φ
(
F
(
F
,
…
,
F
︸
n
)
)
(
ε
′
)
⊆
∨
q
(
γ
,
δ
)
φ
(
F
)
(
ε
′
)
.
(3)
Lety
′
,
y
1
′
,
…
,
y
n
-
1
′
∈
H
′. If φ
-
1
(
y
j
′
)
=
∅
(
1
≤
j
≤
n
-
1
) or φ
-
1
(
y
′
)
=
∅, then(48)
max
⁡
{
φ
(
F
)
(
ε
′
)
(
x
′
)
,
γ
}
≥
0
=
min
⁡
{
φ
(
F
)
(
ε
′
)
y
1
′
y
n
-
1
′
,
φ
(
F
)
(
ε
′
)
(
y
′
)
,
δ
}for y
′
∈
f
′
(
x
′
,
y
1
′
,
…
,
y
n
-
1
′
). Otherwise, there exist y
,
y
1
n
-
1
∈
H such that φ
(
y
)
=
y
′
,
φ
(
y
i
)
=
y
i
′, φ
(
F
)
(
ε
′
)
(
y
′
)
=
F
(
ε
)
(
y
), and φ
(
F
)
(
ε
′
)
(
y
i
′
)
=
F
(
ε
)
(
y
i
) for all 1
≤
i
≤
n
-
1 since 〈
F
,
A
〉 has the sup-property. Now, for y
,
y
1
n
-
1
∈
H, since 〈
F
,
A
〉 is an (
∈
γ
,
∈
γ
∨
q
δ
)-fuzzy soft n-ary subhypergroup over H, there exists x
∈
H such that y
∈
f
(
x
,
y
1
n
-
1
) and max
⁡
{
F
(
ε
)
(
x
)
,
γ
}
≥
min
⁡
{
F
(
ε
)
y
1
y
n
-
1
,
F
(
ε
)
(
y
)
,
δ
}. Hence we have
(49)
y
′
=
φ
(
y
)
∈
φ
(
f
(
x
,
y
1
n
-
1
)
)
=
f
′
(
φ
(
x
)
,
φ
(
y
1
)
,
…
,
φ
(
y
n
-
1
)
)
=
f
′
(
φ
(
x
)
,
y
1
′
,
…
,
y
n
-
1
′
)
,
max
⁡
{
φ
(
F
)
(
ε
′
)
(
φ
(
x
)
)
,
γ
}
=
max
⁡
{
sup
⁡
φ
(
a
)
=
φ
(
x
)
F
(
ε
)
(
a
)
,
γ
}
≥
max
⁡
{
F
(
ε
)
(
x
)
,
γ
}
≥
min
⁡
{
F
(
ε
)
y
1
y
n
-
1
,
F
(
ε
)
(
y
)
,
δ
}
=
min
⁡
{
φ
(
F
)
(
ε
′
)
y
1
′
y
n
-
1
′
,
φ
(
F
)
(
ε
′
)
(
y
′
)
,
δ
}
.
Summing up the above analysis, (
φ
,
ϕ
)
(
〈
F
,
A
〉
) is an (
∈
γ
,
∈
γ
∨
q
δ
)-fuzzy soft n-ary subhypergroup over H
′.
Next assume thatφ is an epimorphism from H onto H
′ and that 〈
F
,
A
〉 is an invertible (
∈
γ
,
∈
γ
∨
q
δ
)-fuzzy soft n-ary subhypergroup over H. Then, for any x
′
,
y
′
∈
H
′, we have(50)
max
⁡
{
min
⁡
{
F
′
(
x
′
,
φ
(
F
)
(
ε
′
)
,
…
,
φ
(
F
)
(
ε
′
)
︸
n
-
1
)
(
y
′
)
,
δ
}
,
γ
}
=
max
⁡
{
min
⁡
{
sup
⁡
φ
(
x
)
=
x
′
F
′
(
φ
(
x
)
,
φ
(
F
)
(
ε
′
)
,
…
,
φ
(
F
)
(
ε
′
)
︸
n
-
1
)
(
y
′
)
,
δ
}
,
γ
}
=
max
⁡
{
min
⁡
{
sup
⁡
φ
(
x
)
=
x
′
φ
(
F
(
1
x
~
,
F
,
…
,
F
︸
n
-
1
)
)
(
ε
′
)
(
y
′
)
,
δ
}
,
γ
}
=
max
⁡
{
min
⁡
{
sup
⁡
φ
(
x
)
=
x
′
,
φ
(
y
)
=
y
′
F
(
x
,
F
(
ε
)
,
…
,
F
(
ε
)
︸
n
-
1
)
(
y
)
,
δ
}
,
γ
}
=
sup
⁡
φ
(
x
)
=
x
′
,
φ
(
y
)
=
y
′
max
⁡
{
min
⁡
{
F
(
x
,
F
(
ε
)
,
…
,
F
(
ε
)
︸
n
-
1
)
(
y
)
,
δ
}
,
γ
}
.Similarly, we have(51)
max
⁡
{
min
⁡
{
F
′
(
y
′
,
φ
(
F
)
(
ε
′
)
,
…
,
φ
(
F
)
(
ε
′
)
︸
n
-
1
)
(
x
′
)
,
δ
}
,
γ
}
=
sup
⁡
φ
(
x
)
=
x
′
,
φ
(
y
)
=
y
′
max
⁡
{
min
⁡
{
F
(
y
,
F
(
ε
)
,
…
,
F
(
ε
)
︸
n
-
1
)
(
x
)
,
δ
}
,
γ
}
.It follows that(52)
max
⁡
{
min
⁡
{
F
′
(
x
′
,
φ
(
F
)
(
ε
′
)
,
…
,
φ
(
F
)
(
ε
′
)
︸
n
-
1
)
(
y
′
)
,
δ
}
,
γ
}
=
max
⁡
{
min
⁡
{
F
′
(
y
′
,
φ
(
F
)
(
ε
′
)
,
…
,
φ
(
F
)
(
ε
′
)
︸
n
-
1
)
(
x
′
)
,
δ
}
,
γ
}since max
⁡
{
min
⁡
{
F
(
x
,
F
(
ε
)
,
…
,
F
(
ε
)
︸
n
-
1
)
(
y
)
,
δ
}
,
γ
}
=
max
⁡
{
min
⁡
{
F
(
y
,
F
(
ε
)
,
…
,
F
(
ε
)
︸
n
-
1
)
(
x
)
,
δ
}
,
γ
}.
Therefore,(
φ
,
ϕ
)
〈
F
,
A
〉 is invertible.Theorem 34.
Let(
H
,
f
) and (
H
′
,
f
′
) be two commutative n-ary hypergroups with identities e and e
′, respectively, φ
:
H
→
H
′ a homomorphism, andϕ
:
E
→
E
′ a mapping. Let 〈
F
′
,
A
′
〉 be an (
∈
γ
,
∈
γ
∨
q
δ
)-fuzzy soft n-ary subhypergroup over H
′. Then (
φ
,
ϕ
)
-
1
〈
F
′
,
A
′
〉 is an (
∈
γ
,
∈
γ
∨
q
δ
)-fuzzy soft n-ary subhypergroup over H. If both φ and ϕ are injective and 〈
F
′
,
A
′
〉 is an (
∈
γ
,
∈
γ
∨
q
δ
)-fuzzy soft (invertible) n-ary subhypergroup over H
′, then (
φ
,
ϕ
)
-
1
〈
F
′
,
A
′
〉 is an (
∈
γ
,
∈
γ
∨
q
δ
)-fuzzy soft (invertible) n-ary subhypergroup over H.Proof.
Assume that〈
F
′
,
A
′
〉 is an (
∈
γ
,
∈
γ
∨
q
δ
)-fuzzy soft n-ary subhypergroup over H
′. Let ε
∈
ϕ
-
1
(
A
′
). Then we have the following.(1)
For anyx
∈
H, max
⁡
{
φ
-
1
(
F
′
)
(
ε
)
(
e
)
,
γ
}
=
max
⁡
{
F
′
(
ϕ
(
ε
)
)
(
φ
(
e
)
)
,
γ
}
≥
min
⁡
{
F
′
(
ϕ
(
ε
)
)
(
φ
(
x
)
)
,
δ
}
=
min
⁡
{
φ
-
1
(
F
′
)
(
ε
)
(
x
)
,
δ
}.
(2)
Letx
,
x
1
n
∈
H be such that x
∈
f
(
x
1
n
). Then φ
(
x
)
∈
φ
(
f
(
x
1
n
)
)
=
f
′
(
φ
(
x
)
1
n
) and so
(53)
max
⁡
{
φ
-
1
(
F
′
)
(
ε
)
(
x
)
,
γ
}
=
max
⁡
{
F
′
(
ϕ
(
ε
)
)
(
φ
(
x
)
)
,
γ
}
≥
min
⁡
{
F
′
(
ϕ
(
ε
)
)
(
φ
(
x
1
)
)
,
…
,
F
′
(
ϕ
(
ε
)
)
(
φ
(
x
n
)
)
,
δ
}
=
min
⁡
{
φ
-
1
(
F
′
)
(
ε
)
x
1
x
n
,
δ
}
.It follows that max
⁡
{
inf
⁡
x
∈
f
(
x
1
n
)
⁡
φ
-
1
(
F
′
)
(
ε
)
(
z
)
,
γ
}
≥
min
⁡
{
φ
-
1
(
F
′
)
(
ε
)
x
1
x
n
,
δ
}.(3)
Letx
,
y
,
y
1
n
-
1
∈
H be such that y
∈
f
(
x
,
y
1
n
-
1
). Then φ
(
y
)
∈
f
′
(
φ
(
x
)
,
φ
(
y
1
)
,
…
,
φ
(
y
n
-
1
)
). Since 〈
F
′
,
A
′
〉 is invertible, by Lemma 25(2), we have
(54)
max
⁡
{
φ
-
1
(
F
′
)
(
ε
)
(
x
)
,
γ
}
=
max
⁡
{
F
′
(
ϕ
(
ε
)
)
(
φ
(
x
)
)
,
γ
}
≥
min
⁡
{
F
′
(
ϕ
(
ε
)
)
(
φ
(
y
1
)
)
,
…
,
F
′
(
ϕ
(
ε
)
)
(
φ
(
y
n
-
1
)
)
,
F
′
(
ϕ
(
ε
)
)
(
φ
(
y
)
)
,
δ
}
=
min
⁡
{
φ
-
1
(
F
′
)
(
ε
)
y
1
y
n
-
1
,
φ
-
1
(
F
′
)
(
ε
)
(
y
)
,
δ
}
.Summing up the above analysis, (
φ
,
ϕ
)
-
1
〈
F
′
,
A
′
〉 is an (
∈
γ
,
∈
γ
∨
q
δ
)-fuzzy n-ary soft subhypergroup over H. If both φ and ϕ are injective and 〈
F
′
,
A
′
〉 is an (
∈
γ
,
∈
γ
∨
q
δ
)-fuzzy soft invertible n-ary subhypergroup over H
′, it is easy to check that (
φ
,
ϕ
)
-
1
〈
F
′
,
A
′
〉 is an (
∈
γ
,
∈
γ
∨
q
δ
)-fuzzy soft invertible n-ary subhypergroup over H.


## 6. Conclusions

In this paper, our aim is to promote the research and development of fuzzy soft technology by studying fuzzy softn-ary hypergroups which generalize some algebra structures: soft groups, fuzzy n-ary hypergroups, fuzzy hypergroups, and so on. The goal is to explain new methodological development in n-ary hypergroups which will also be of growing importance in the future. The results presented in this paper can hopefully provide more insight into and a full understanding of fuzzy soft set theory and algebraic hyperstructures. Our future work on this topic will focus on studying some other classes of fuzzy soft n-ary hypergroups.



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*Source: 957391-2014-07-14.xml*