# On Fuzzy Corsini's Hyperoperations

Authors: Yuming Feng; P. Corsini
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

We generalize the concept of C-hyperoperation and introduce the concept of F-C-hyperoperation. We list some basic properties of F-C-hyperoperation and the relationship between the concept of C-hyperoperation and the concept of F-C-hyperoperation. We also research F-C-hyperoperations associated with special fuzzy relations.


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## Body

## 1. Introduction and Preliminaries

Hyperstructures and binary relations have been studied by many researchers, for instance, Chvalina [1, 2], Corsini and Leoreanu [3], Feng [4], Hort [5], Rosenberg [6], Spartalis [7], and so on.A partial hypergroupoid〈H,*〉 is a nonempty set H with a function from H×H to the set of subsets of H.A hypergroupoid is a nonempty setH, endowed with a hyperoperation, that is, a function from H×H to P(H), the set of nonempty subsets of H.IfA,B∈P(H)-{∅}, then we define A*B=∪{a*b∣a∈A,b∈B}, x*B={x}*B and A*y=A*{y}.A Corsini's hyperoperation was first introduced by Corsini [8] and studied by many researchers; for example, see [3, 8–15].Definition 1.1 (see [8]).
Let〈H,R〉 be a a pair of sets where H is a nonempty set and R is a binary relation on H. Corsini's hyperoperation (briefly, C-hyperoperation) *R associated with R is defined in the following way:
(1.1)*R:H×H⟶P(H):x*Ry={z∈H∣xRz,zRy},
where P(H) denotes the family of all the subsets of H.A fuzzy subsetA of a nonempty set H is a function A:H→[0,1]. The family of all the fuzzy subsets of H is denoted by F(H).We use∅ to denote a special fuzzy subset of H which is defined by ∅(x)=0, for all x∈H.For a fuzzy subsetA of a nonempty set H, the p-cut of A is denoted Ap, for any p∈(0,1], and defined by Ap≐{x∈H∣A(x)≥p}.A fuzzy binary relationR on a nonempty set H is a function R:H×H→[0,1]. In the following, sometimes we use fuzzy relation to refer to fuzzy binary relation.For anya,b∈[0,1], we use a∧b to stand for the minimum of a and b and a∨b to denote the maximum of a and b.GivenA,B∈F(H), we will use the following definitions:
(1.2)A⊆B≐A(x)≤B(x),∀x∈H,A=B≐A(x)=B(x),∀x∈H,(A∪B)(x)≐A(x)∨B(x),∀x∈H,(A∩B)(x)≐A(x)∧B(x),∀x∈H.A partial fuzzy hypergroupoid〈H,*〉 is a nonempty set endowed with a fuzzy hyperoperation *:H×H→F(H). Moreover, 〈H,*〉 is called a fuzzy hypergroupoid if for all x,y∈H, there exists at least one z∈H, such that (x*y)(z)≠0 holds.Given a fuzzy hyperoperation*:H×H→F(H), for all a∈H, B∈F(H), the fuzzy subset a*B of H is defined by
(1.3)(a*B)(x)≐∨B(b)>0(a*b)(x).B
*
a, A*B can be defined similarly. When B is a crisp subset of H, we treat B as a fuzzy subset by treating it as B(x)=1, for all x∈B and B(x)=0, for all x∈H-B.


## 2. Fuzzy Corsini's Hyperoperation

In this section, we will generalize the concept of Corsini's hyperoperation and introduce the fuzzy version of Corsini's hyperoperation.Definition 2.1.
Let〈H,R〉 be a pair of sets where H is a non-empty set and R is a fuzzy relation on H. We define a fuzzy hyperoperation *R:H×H→F(H), for any x,y,z∈H, as follows:
(2.1)(x*Ry)(z)≐R(x,z)∧R(z,y).*R is called a fuzzy Corsini's hyperoperation (briefly, F-C-hyperoperation) associated with R. The fuzzy hyperstructure 〈H,*R〉 is called a partial F-C-hypergroupoid.Remark 2.2.
It is obvious that the concept of F-C-hyperoperation is a generalization of the concept of C-hyperoperation.Example 2.3.
LettingH={a,b} be a non-empty set, R is a fuzzy relation on H as described in Table 1.Table 1
R
a
b
a
0.1
0.2
b
0.3
0.4From the previous definition, by calculating, for example,(a*Ra)(a)=R(a,a)∧R(a,a)=0.1∧0.1=0.1, R(a*b)(a)=R(a,a)∧R(a,b)=0.1∧0.2=0.1, we can obtain Table 2 which is a partial F-C-hypergroupoid.Table 2
*
R
a
b
a
0.1
/
a
+
0.2
/
b
0.1
/
a
+
0.2
/
b
b
0.1
/
a
+
0.3
/
b
0.2
/
a
+
0.4
/
bDefinition 2.4.
SupposingR, S are two fuzzy relations on a non-empty set H, the composition of R and S is a fuzzy relation on H and is defined by (R∘S)(x,y)≐⋁z∈H(R(x,z)∧S(z,y)), for all x,y∈H.Proposition 2.5.
A partial F-C-hypergroupoid〈H,*R〉 is a F-C-hypergroupoid if and only if supp(R∘R)=H×H, where supp(R∘R)={(x,y)∣(R∘R)(x,y)≠0}.Proof.
Suppose that〈H,*R〉 is a hypergroupoid. For any x,y∈H, there exists at least one z∈H, such that (x*Ry)(z)≠0 holds.
So(R∘R)(x,y)=⋁z∈H(R(x,z)∧R(z,y))≠0. Thus (x,y)∈supp(R∘R). And we conclude that H×H⊆supp(R∘R).
supp
(
R
∘
R
)
⊆
H
×
H is obvious. And so supp(R∘R)=H×H.
Conversely, ifsupp(R∘R)=H×H, then for any x,y∈H, (x,y)∈H×H=supp(R∘R). So (R∘R)(x,y)=⋁z∈H(R(x,z)∧R(z,y))≠0. That is, there exists at least one z∈H such that (x*Ry)(z)≠0 holds. And so 〈H,*R〉 is a hypergroupoid.
Thus we complete the proof.Definition 2.6.
LettingH be a non-empty set, * is a fuzzy hyperoperation of H, the hyperoperation *p is defined by x*py=(x*y)p, for all x,y∈H, p∈[0,1]. *p is called the p-cut of *.Definition 2.7.
LettingR be a fuzzy relation on a non-empty set H, we define a binary relation Rp on H, for all p∈(0,1], as follows:
(2.2)xRpy≐R(x,y)≥p.Rp is called the p-cut of the fuzzy relation R.Proposition 2.8.
Let〈H,*R〉 be a partial F-C-hypergroupoid. Then (*R)p is a C-hyperoperation associated with Rp, for all 0<p≤1.Proof.
For any0<p≤1 and for any x,y∈H, we have
(2.3)x(*R)py=(x*Ry)p={z∈H∣(x*Ry)(z)≥p}={z∈H∣R(x,z)∧R(z,y)≥p}={z∈H∣R(x,z)≥p,R(z,y)≥p}={z∈H∣xRpz,zRpy}.
From the definition of C-hyperoperation, we conclude that(*R)p is a C-hyperoperation associated with Rp.
Thus we complete the proof.From the previous proposition and the construction of the F-C-hyperoperation, we can easily conclude that a fuzzy hyperoperation is a F-C-hyperoperation if and only if every p-cut of the F-C-hyperoperation is a C-hyperoperation. That is, consider the following.Proposition 2.9.
LetH be a non-empty set and let * be a fuzzy hyperoperation of H, then the fuzzy hyperoperation * is an F-C-hyperoperation associated with a fuzzy relation R on H if and only if *p is a C-hyperoperation associated with Rp, for any 0<p≤1.


## 3. Basic Properties of F-C-Hyperoperations

In this section, we list some basic properties of F-C-hyperoperations.Proposition 3.1.
Let〈H,*R〉 be a partial or nonpartial F-C-hypergroupoid defined on H≠∅. Then, for all x,y,a,b∈H, we have
(3.1)x*Ry∩a*Rb=x*Rb∩a*Ry.Proof.
For anyx,y,a,b,z∈H, we have that (x*Ry∩a*Rb)(z)=(x*Ry)(z)∧(a*Rb)(z)=R(x,z)∧R(z,y)∧R(a,z)∧R(z,b)=R(x,z)∧R(z,b)∧R(a,z)∧R(z,y)=(x*Rb∩a*Ry)(z).
So(3.2)x*Ry∩a*Rb=x*Rb∩a*Ry,
for all x,y,a,b∈H.Proposition 3.2.
Let〈H,*R〉 be a partial F-C-hypergroupoid and x,y∈H, x*Ry=∅. Then,(1)
x*RH∩H*Ry=∅;(2)
IfH=x*RH then H*Ry=∅;(3)
IfH=H*Rx then y*RH=∅.Proof.
(1) Supposingx*RH∩H*Ry≠∅, then there exist a,b∈H, such that x*Ra∩b*Ry≠∅. So from the previous proposition, we have x*Ry∩b*Ra≠∅. This is a contradiction.
(2) FromH=x*RH and x*RH∩H*Ry=∅, we have that H∩H*Ry=∅, and so, H*Ry=∅.
(3) is proved similar to (2).Proposition 3.3.
Letting*R be the F-C-hyperoperation defined on the non-empty set H, p∈(0,1], then the following are equivalent:(1)
for somea∈H, (a*Ra)p=H;(2)
for allx,y∈H, a∈(x*Ry)p.Proof.
Let(a*Ra)p=H. Then, for all x,y∈H, we have that (a*Ra)(x)≥p,(a*Ra)(y)≥p, that is R(a,x)≥p,R(x,a)≥p,R(a,y)≥p,R(y,a)≥p and so R(x,a)∧R(a,y)≥p. Thus a∈(x*Ry)p, for all x,y∈H.
Conversely, leta∈(x*Ry)p, for all x,y∈H. Specially, we have a∈(a*Rx)p and a∈(x*Ra)p. Thus, R(a,x)≥p and R(x,a)≥p. And so x∈(a*Ra)p.Proposition 3.4.
Let〈H,*R〉 be a partial or nonpartial F-C-hypergroupoid defined on H≠∅. Then, for all a,b∈H, p∈(0,1], we have
(3.3)a∈(b*Rb)p⟺b∈(a*Ra)p.Proof.
For anya,b∈H, we have that
(3.4)a∈(b*Rb)p⟹(b*Rb)(a)≥p⟹R(b,a)∧R(a,b)≥p⟹R(a,b)∧R(b,a)≥p⟹(a*Ra)(b)≥p⟹b∈(a*Ra)p.
The remaining part can be proved similarly.


## 4. F-C-Hyperoperations Associated with p-Fuzzy Reflexive Relations

In this section, we will assume thatR is a p-fuzzy reflexive relation on a non-empty set.Definition 4.1.
A fuzzy relationR on a non-empty set H is called p-fuzzy reflexive if for any x∈H,
(4.1)R(x,x)≥p.Example 4.2.
The fuzzy relationR introduced in Example 2.3 is 0.1-fuzzy reflexive. Of course, it is p-fuzzy reflexive, where 0≤p≤0.1.Proposition 4.3.
Letting〈H,*R〉 be a partial F-C-hypergroupoid defined on H≠∅, R is p-fuzzy reflexive. Then, for all a,b∈H, p∈(0,1], the following are equivalent:(1)
R(a,b)≥p;(2)
a∈(a*Rb)p;(3)
b∈(a*Rb)p.Proof.
“(1)⇒(2)”
FromR(a,a)≥p and R(a,b)≥p we have that R(a,a)∧R(a,b)≥p which shows that a∈(a*Rb)p.
“(2)⇒(3)”
Froma∈(a*Rb)p we have that R(a,b)≥p. Since R(b,b)≥p, so R(a,b)∧R(b,b)≥p which implies that b∈(a*Rb)p.
“(3)⇒(1)”
It is obvious.Proposition 4.4.
Letting〈H,*R〉 be a partial F-C-hypergroupoid defined on H≠∅, R is p-fuzzy reflexive. Then, for any a∈H, we have that
(4.2)a∈(a*Ra)p.Proof.
FromR(a,a)≥p we have R(a,a)∧R(a,a)≥p. That is a∈(a*Ra)p.Proposition 4.5.
Letting〈H,*R〉 be a partial F-C-hypergroupoid defined on H≠∅, R is p-fuzzy reflexive. Then, for any a,b∈H, p∈(0,1], we have that
(4.3)b∈(a*Ra)p⟺a∈(a*Rb∩b*Ra)p.Proof.
Fromb∈(a*Ra)p we have that R(a,b)∧R(b,a)≥p. So R(a,b)≥p and R(b,a)≥p. Thus R(a,a)∧R(a,b)≥p and R(b,a)∧R(a,a)≥p. That is (a*Rb)(a)≥p and (b*Ra)(a)≥p. So (a*Rb∩b*Ra)(a)≥p. Thus a∈(a*Rb∩b*Ra)p.
Conversely, suppose thata∈(a*Rb∩b*Ra)p. Then (a*Rb)(a)∧(b*Ra)(a)≥p. Thus R(a,a)∧R(a,b)∧R(b,a)∧R(a,a)≥p. So R(a,b)∧R(b,a)≥p. That is b∈(a*Ra)p.Corollary 4.6.
Letting〈H,*R〉 be a partial F-C-hypergroupoid defined on H≠∅, R is p-fuzzy reflexive. Then, for any a,b∈H, p∈(0,1], we have that
(4.4)b∈(a*Ra)p⟺a∈(b*Rb)p⟺a∈(a*Rb∩b*Ra)p⟺b∈(a*Rb∩b*Ra)p.Proposition 4.7.
Letting〈H,*R〉 be a partial F-C-hypergroupoid defined on H≠∅, R is p-fuzzy reflexive. Then, for any a,b∈H, we have that
(4.5)c∈(a*Rb)p⟺c∈(a*Rc∩c*Rb)p.Proof.
Ifc∈(a*Rb)p, then R(a,c)≥p and R(c,b)≥p. Thus c∈(a*Rc)p and c∈(c*Rb)p. So c∈(a*Rc∩c*Rb)p.
Conversely, ifc∈(a*Rc∩c*Rb)p, then (a*Rc)(c)∧(c*Rb)(c)≥p. Thus R(a,c)∧R(c,c)∧R(c,c)∧R(c,b)≥p. And so R(a,c)∧R(c,b)≥p. Thus c∈(a*Rb)p.Proposition 4.8.
Letting〈H,*R〉 be a partial F-C-hypergroupoid defined on H≠∅, R is p-fuzzy reflexive. Then, for any a,b,c∈H, p∈(0,1], the following are equivalent:(1)
c∈(a*Rb)p;(2)
a∈(a*Rc)p and b∈(c*Rb)p;(3)
a∈(a*Rc)p and c∈(c*Rb)p.Proof.
“(1)⇒(2)”
Suppose thatc∈(a*Rb)p. Then R(a,c)≥p and R(c,b)≥p. So R(a,a)∧R(a,c)≥p and R(c,b)∧R(b,b)≥p. Thus a∈(a*Rc)p and b∈(c*Rb)p.
“(2)⇒(3)”
Suppose thatb∈(c*Rb)p. Then R(c,b)≥p. Thus R(c,c)∧R(c,b)≥p. And so c∈(c*Rb)p.
“(3)⇒(1)”
Froma∈(a*Rc)p and c∈(c*Rb)p, we have that R(a,c)≥p and R(c,b)≥p. Thus R(a,c)∧R(c,b)≥p. So c∈(a*Rb)p.


## 5. F-C-Hyperoperations Associated with p-Fuzzy Symmetric Relations

In this section, we will assume thatR is a p-fuzzy symmetric relation on a non-empty set.Definition 5.1.
A fuzzy binary relationR on a non-empty set H is called  p-fuzzy symmetric if for any x,y∈H,
(5.1)R(x,y)≥p⟹R(y,x)≥p.Example 5.2.
The fuzzy relationR introduced in Example 2.3 is 0.2-fuzzy symmetric. Of course, it is p-fuzzy reflexive, where 0≤p≤0.2.Proposition 5.3.
Letting〈H,*R〉 be a partial F-C-hypergroupoid defined on H≠∅, R is p-fuzzy symmetric relation. Then, for all a,b∈H, we have that
(5.2)(a*Rb)p=(b*Ra)p.Proof.
For alla,b∈H, two cases are possible.(1)
If(a*Rb)p=∅, then (a*Rb)p⊆(b*Ra)p.(2)
If(a*Rb)p≠∅, let x∈(a*Rb)p. Then R(a,x)≥p and R(x,b)≥p.Since R is p-fuzzy symmetric, so R(x,a)≥p and R(b,x)≥p. Thus (b*Ra)(x)=R(b,x)∧R(x,a)≥p. So x∈(b*Ra)p. And in this case, we also have that (a*Rb)p⊆(b*Ra)p.
The remaining part can be proved by exchanginga and b.Proposition 5.4.
Let〈H,*R〉 be a partial F-C-hypergroupoid defined on H≠∅, p∈(0,1], if(1)
for alla,b∈H, (a*Rb)p=(b*Ra)p,(2)
for anyx∈H, there exists a y∈H, such that R(x,y)≥p.Then R is a p-fuzzy symmetric binary relation on H.Proof.
For alla,b∈H, suppose that R(a,b)≥p. We need to show that R(b,a)≥p.
Since forb∈H, there exists a x∈H, such that R(b,x)≥p. So R(a,b)∧R(b,x)≥p. That is, b∈(a*Rx)p=(x*Ra)p. And so R(x,b)∧R(b,a)≥p. And finally we have that R(b,a)≥p.


## 6. F-C-Hyperoperations Associated with p-Fuzzy Transitive Relations

In this section, we will assume thatR is a p-fuzzy transitive relation on a non-empty set.Definition 6.1.
A fuzzy binary relationR on a non-empty set H is called p-fuzzy transitive if for any x,y,z∈H,
(6.1)R(x,y)≥p,R(y,z)≥p⟹R(x,z)≥p.Example 6.2.
The fuzzy relationR introduced in Example 2.3 is 0.1-fuzzy transitive. Of course, it is p-fuzzy transitive, where 0≤p≤0.1.Proposition 6.3.
Letting〈H,*R〉 be a partial F-C-hypergroupoid defined on H≠∅, R is a p-fuzzy transitive relation on H, p∈(0,1]. Then for all x,y∈H, we have that
(6.2)R(x,y)≥p⟹(x*Rx∪y*Ry)p⊆(x*Ry)p.Proof.
(1) If(x*Rx)p=∅, then obviously (x*Rx)p⊆(x*Ry)p.
Supposing that(x*Rx)p≠∅, then for any w∈(x*Rx)p, we have that R(x,w)∧R(w,x)≥p, that is, R(x,w)≥p and R(w,x)≥p. From R(w,x)≥p and R(x,y)≥p we have that R(w,y)≥p. From R(x,w)≥p and R(w,y)≥p we conclude that w∈(x*Ry)p.
So(x*Rx)p⊆(x*Ry)p.
(2) If(y*Ry)p=∅, then obviously (y*Ry)p⊆(x*Ry)p.
Supposing that(y*Ry)p≠∅, then for any w∈(y*Ry)p, we have that R(y,w)∧R(w,y)≥p, that is, R(y,w)≥p and R(w,y)≥p. From R(y,w)≥p and R(x,y)≥p we have that R(x,w)≥p. From R(x,w)≥p and R(w,y)≥p we conclude that w∈(x*Ry)p.
So(y*Ry)p⊆(x*Ry)p.Proposition 6.4.
Letting〈H,*R〉 be a partial F-C-hypergroupoid defined on H≠∅, R is a p-fuzzy transitive binary relation. For any a,b,c∈H, we have that(1)
((a*Rb)p*Rc)p⊆(a*Rc)p;(2)
(a*R(b*Rc)p)p⊆(a*Rc)p.Proof.
(1) If((a*Rb)p*Rc)p=∅, then it is obvious that ((a*Rb)p*Rc)p⊆(a*Rc)p.
Suppose that((a*Rb)p*Rc)p≠∅. Then for any w∈((a*Rb)p*Rc)p, there exists a w1∈(a*Rb)p such that w∈(w1*Rc)p. That is R(a,w1)≥p, R(w1,b)≥p, R(w1,w)≥p and R(w,c)≥p. From R(a,w1)≥p and R(w1,w)≥p, we have that R(a,w)≥p. Thus R(a,w)∧R(w,c)≥p∧p=p. That is, w∈(a*Rc)p. So ((a*Rb)p*Rc)p⊆(a*Rc)p.
(2) Can be proved similarly.



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*Source: 685681-2012-07-03.xml*