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.
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 set H, endowed with a hyperoperation, that is, a function from H×H to P(H), the set of nonempty subsets of H.
If A,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 [<xref ref-type="bibr" rid="B3">8</xref>]).
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 subset A 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 subset A 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 relation R 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 any a,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.
Given A,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.
Letting H={a,b} be a non-empty set, R is a fuzzy relation on H as described in Table 1.
R
a
b
a
0.1
0.2
b
0.3
0.4
From 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.
*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/b
Definition 2.4.
Supposing R, 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, if supp(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.
Letting H 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.
Letting R 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 any 0<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.
Let H 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 any x,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,
x*RH∩H*Ry=∅;
If H=x*RH then H*Ry=∅;
If H=H*Rx then y*RH=∅.
Proof.
(1) Supposing x*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) From H=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:
for some a∈H, (a*Ra)p=H;
for all x,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, let a∈(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 any a,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 that R is a p-fuzzy reflexive relation on a non-empty set.
Definition 4.1.
A fuzzy relation R 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 relation R 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:
R(a,b)≥p;
a∈(a*Rb)p;
b∈(a*Rb)p.
Proof.
“(1)⇒(2)”
From R(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)”
From a∈(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.
From R(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.
From b∈(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 that a∈(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.
If c∈(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, if c∈(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:
c∈(a*Rb)p;
a∈(a*Rc)p and b∈(c*Rb)p;
a∈(a*Rc)p and c∈(c*Rb)p.
Proof.
“(1)⇒(2)”
Suppose that c∈(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 that b∈(c*Rb)p. Then R(c,b)≥p. Thus R(c,c)∧R(c,b)≥p. And so c∈(c*Rb)p.
“(3)⇒(1)”
From a∈(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 that R is a p-fuzzy symmetric relation on a non-empty set.
Definition 5.1.
A fuzzy binary relation R 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 relation R 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 all a,b∈H, two cases are possible.
If (a*Rb)p=∅, then (a*Rb)p⊆(b*Ra)p.
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 exchanging a and b.
Proposition 5.4.
Let 〈H,*R〉 be a partial F-C-hypergroupoid defined on H≠∅, p∈(0,1], if
for all a,b∈H, (a*Rb)p=(b*Ra)p,
for any x∈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 all a,b∈H, suppose that R(a,b)≥p. We need to show that R(b,a)≥p.
Since for b∈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 that R is a p-fuzzy transitive relation on a non-empty set.
Definition 6.1.
A fuzzy binary relation R 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 relation R 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
((a*Rb)p*Rc)p⊆(a*Rc)p;
(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.
Acknowledgment
The paper is partially supported by CSC.
ChvalinaJ.ChvalinaJ.Commutative hypergroups in the sense of Marty and ordered setsProceedings of the Proceedings of the Summer School on General Algebra and Ordered Sets1994Olomouc, Czech Republic1930CorsiniP.LeoreanuV.Hypergroups and binary relationsFengY.Algebraic hyperstructures obtained from algebraic structures with fuzzy binary relationsHortD.A construction of hypergroups from ordered structures and their morphismsRosenbergI. G.Hypergroups and join spaces determined by relationsSpartalisS. I.Hypergroupoids obtained from groupoids with binary relationsCorsiniP.Binary relations and hypergroupoidsCorsiniP.LeoreanuV.CorsiniP.LeoreanuV.Survey on new topics of Hyperstructure Theory and its applicationsProceedings of the 8th International Congress on Algebraic Hyperstructures and Applications (AHA '03)2003137LeoreanuV.LeoreanuL.Hypergroups associated with hypergraphsLeoreanuV.Weak mutually associative hyperstructures. IIProceedings of the 8th International Congress on (AHA '03)2003183189SpartalisS. I.MamaloukasC.Hyperstructures associated with binary relationsSpartalisS. I.The hyperoperation relation and the Corsini's partial or not-partial hypergroupoids (A classification)SpartalisS. I.Konstantinidou-SerafimidouM.TaouktsoglouA.C-hypergroupoids obtained by special binary relations