On Fuzzy Corsini's Hyperoperations

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.

(ProQuest: ... denotes non-US-ASCII text omitted.)

Yuming Feng 1, 2 and P. Corsini 2

Recommended by Said Abbasbandy

1, School of Mathematics and Statistics, Chongqing Three Gorges University, Wanzhou, Chongqing 404100, China
2, Dipartimento di Ingegneria Civile e Architettura, Università degli Studi di Udine, Via delle Scienze, 206, 33100 Udine, Italy

Received 22 February 2012; Accepted 7 May 2012

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 Y9;H,*YA; 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 [8]).

Let Y9;H,RYA; 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: [figure omitted; refer to PDF] 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[arrow right][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 [=, single dot above]{x∈H|"A(x)...5;p} .

A fuzzy binary relation R on a nonempty set H is a function R:H×H[arrow right][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: [figure omitted; refer to PDF]

A partial fuzzy hypergroupoid Y9;H,*YA; is a nonempty set endowed with a fuzzy hyperoperation *:H×H[arrow right]F(H) . Moreover, Y9;H,*YA; is called a fuzzy hypergroupoid if for all x,y∈H , there exists at least one z∈H , such that (x*y)(z)...0;0 holds.

Given a fuzzy hyperoperation *:H×H[arrow right]F(H) , for all a∈H , B∈F(H) , the fuzzy subset a*B of H is defined by [figure omitted; refer to PDF]

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 Y9;H,RYA; 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[arrow right]F(H) , for any x,y,z∈H , as follows: [figure omitted; refer to PDF] _*R is called a fuzzy Corsini's hyperoperation (briefly, F-C-hyperoperation ) associated with R . The fuzzy hyperstructure Y9;H,_*R YA; 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.

Table 1

R	a	b
a	0.1	0.2
b	0.3	0.4

From the previous definition, by calculating, for example, (a _*R a)(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/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[composite function]S)(x,y)[=, single dot above] _...z∈H (R(x,z)⋀S(z,y)) , for all x,y∈H .

Proposition 2.5.

A partial F-C-hypergroupoid Y9;H,_*R YA; is a F-C-hypergroupoid if and only if supp (R[composite function]R)=H×H , where supp (R[composite function]R)={(x,y)|"(R[composite function]R)(x,y)...0;0} .

Proof.

Suppose that Y9;H,_*R YA; is a hypergroupoid. For any x,y∈H , there exists at least one z∈H , such that (x _*R y)(z)...0;0 holds.

So (R[composite function]R)(x,y)=_...z∈H (R(x,z)⋀R(z,y))...0;0 . Thus (x,y)∈supp (R[composite function]R) . And we conclude that H×H⊆supp (R[composite function]R) .

supp (R[composite function]R)⊆H×H is obvious. And so supp (R[composite function]R)=H×H .

Conversely, if supp (R[composite function]R)=H×H , then for any x,y∈H , (x,y)∈H×H=supp (R[composite function]R) . So (R[composite function]R)(x,y)=_...z∈H (R(x,z)⋀R(z,y))...0;0 . That is, there exists at least one z∈H such that (x _*R y)(z)...0;0 holds. And so Y9;H,_*R YA; 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 _*p y=_(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: [figure omitted; refer to PDF] _Rp is called the p-cut of the fuzzy relation R .

Proposition 2.8.

Let Y9;H,_*R YA; be a partial F-C-hypergroupoid. Then _{(*R )p} is a C-hyperoperation associated with _Rp , for all 0<p...4;1 .

Proof.

For any 0<p...4;1 and for any x,y∈H , we have [figure omitted; refer to PDF]

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...4;1 .

3. Basic Properties of F-C-Hyperoperations

In this section, we list some basic properties of F-C-hyperoperations.

Proposition 3.1.

Let Y9;H,_*R YA; be a partial or nonpartial F-C-hypergroupoid defined on H...0;∅ . Then, for all x,y,a,b∈H , we have [figure omitted; refer to PDF]

Proof.

For any x,y,a,b,z∈H , we have that (x _*R y∩a _*R b)(z)=(x _*R y)(z)⋀(a _*R b)(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 _*R b∩a _*R y)(z) .

So [figure omitted; refer to PDF] for all x,y,a,b∈H .

Proposition 3.2.

Let Y9;H,_*R YA; be a partial F-C-hypergroupoid and x,y∈H , x _*R y=∅ . Then,

(1) x _*R H∩H _*R y=∅ ;

(2) If H=x _*R H then H _*R y=∅ ;

(3) If H=H _*R x then y _*R H=∅ .

Proof.

(1) Supposing x _*R H∩H _*R y...0;∅ , then there exist a,b∈H , such that x _*R a∩b _*R y...0;∅ . So from the previous proposition, we have x_*R y∩b_*R a...0;∅ . This is a contradiction.

(2) From H=x _*R H and x _*R H∩H _*R y=∅ , we have that H∩H _*R y=∅ , and so, H _*R y=∅ .

(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 some a∈H , _{(a *R a)p} =H ;

(2) for all x,y∈H , a∈_{(x *R y)p} .

Proof.

Let _{(a *R a)p} =H . Then, for all x,y∈H , we have that (a _*R a)(x)...5;p,(a _*R a)(y)...5;p , that is R(a,x)...5;p,R(x,a)...5;p,R(a,y)...5;p,R(y,a)...5;p and so R(x,a)⋀R(a,y)...5;p . Thus a∈_{(x *R y)p} , for all x,y∈H .

Conversely, let a∈(x _*R y_)p , for all x,y∈H . Specially, we have a∈(a _*R x_)p and a∈(x _*R a_)p . Thus, R(a,x)...5;p and R(x,a)...5;p . And so x∈(a_*R a_)p .

Proposition 3.4.

Let Y9;H,_*R YA; be a partial or nonpartial F-C-hypergroupoid defined on H...0;∅ . Then, for all a,b∈H , p∈(0,1] , we have [figure omitted; refer to PDF]

Proof.

For any a,b∈H , we have that [figure omitted; refer to PDF]

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 , [figure omitted; refer to PDF]

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...4;p...4;0.1 .

Proposition 4.3.

Letting Y9;H,_*R YA; be a partial F-C-hypergroupoid defined on H...0;∅ , R is p-fuzzy reflexive. Then, for all a,b∈H , p∈(0,1] , the following are equivalent:

(1) R(a,b)...5;p ;

(2) a∈_{(a *R b)p} ;

(3) b∈_{(a *R b)p} .

Proof.

"(1)[implies] (2)"

From R(a,a)...5;p and R(a,b)...5;p we have that R(a,a)⋀R(a,b)...5;p which shows that a∈_{(a *R b)p} .

"(2)[implies] (3)"

From a∈_{(a *R b)p} we have that R(a,b)...5;p . Since R(b,b)...5;p , so R(a,b)⋀R(b,b)...5;p which implies that b∈_{(a *R b)p} .

"(3)[implies] (1)"

It is obvious.

Proposition 4.4.

Letting Y9;H,_*R YA; be a partial F-C-hypergroupoid defined on H...0;∅ , R is p-fuzzy reflexive. Then, for any a∈H , we have that [figure omitted; refer to PDF]

Proof.

From R(a,a)...5;p we have R(a,a)⋀R(a,a)...5;p . That is a∈_{(a *R a)p} .

Proposition 4.5.

Letting Y9;H,_*R YA; be a partial F-C-hypergroupoid defined on H...0;∅ , R is p-fuzzy reflexive. Then, for any a,b∈H , p∈(0,1] , we have that [figure omitted; refer to PDF]

Proof.

From b∈_{(a*R a)p} we have that R(a,b)⋀R(b,a)...5;p . So R(a,b)...5;p and R(b,a)...5;p . Thus R(a,a)⋀R(a,b)...5;p and R(b,a)⋀R(a,a)...5;p . That is (a _*R b)(a)...5;p and (b _*R a)(a)...5;p . So (a _*R b∩b _*R a )(a)...5;p . Thus a∈_{(a *R b∩b *R a)p} .

Conversely, suppose that a∈_{(a *R b∩b *R a)p} . Then (a _*R b)(a)⋀(b _*R a)(a)...5;p . Thus R(a,a)⋀R(a,b)⋀R(b,a)⋀R(a,a)...5;p . So R(a,b)⋀R(b,a)...5;p . That is b∈_{(a *R a)p} .

Corollary 4.6.

Letting Y9;H,_*R YA; be a partial F-C-hypergroupoid defined on H...0;∅ , R is p-fuzzy reflexive. Then, for any a,b∈H , p∈(0,1] , we have that [figure omitted; refer to PDF]

Proposition 4.7.

Letting Y9;H,_*R YA; be a partial F-C-hypergroupoid defined on H...0;∅ , R is p-fuzzy reflexive. Then, for any a,b∈H , we have that [figure omitted; refer to PDF]

Proof.

If c∈_{(a *R b)p} , then R(a,c)...5;p and R(c,b)...5;p . Thus c∈_{(a *R c)p} and c∈_{(c *R b)p} . So c∈_{(a *R c∩c *R b)p} .

Conversely, if c∈_{(a *R c∩c *R b)p} , then (a _*R c)(c)⋀(c _*R b)(c)...5;p . Thus R(a,c)⋀R(c,c)⋀R(c,c)⋀R(c,b)...5;p . And so R(a,c)⋀R(c,b)...5;p . Thus c∈_{(a *R b)p} .

Proposition 4.8.

Letting Y9;H,_*R YA; be a partial F-C-hypergroupoid defined on H...0;∅ , R is p-fuzzy reflexive. Then, for any a,b,c∈H , p∈(0,1] , the following are equivalent:

(1) c∈_{(a *R b)p} ;

(2) a∈_{(a *R c)p} and b∈_{(c *R b)p} ;

(3) a∈_{(a *R c)p} and c∈_{(c *R b)p} .

Proof.

"(1)[implies] (2)"

Suppose that c∈_{(a *R b)p} . Then R(a,c)...5;p and R(c,b)...5;p . So R(a,a)⋀R(a,c)...5;p and R(c,b)⋀R(b,b)...5;p . Thus a∈_{(a *R c)p} and b∈_{(c *R b)p} .

"(2)[implies] (3)"

Suppose that b∈_{(c *R b)p} . Then R(c,b)...5;p . Thus R(c,c)⋀R(c,b)...5;p . And so c∈_{(c *R b)p} .

"(3)[implies] (1)"

From a∈_{(a *R c)p} and c∈_{(c *R b)p} , we have that R(a,c)...5;p and R(c,b)...5;p . Thus R(a,c)⋀R(c,b)...5;p . So c∈_{(a *R b)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 , [figure omitted; refer to PDF]

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...4;p...4;0.2 .

Proposition 5.3.

Letting Y9;H,_*R YA; be a partial F-C-hypergroupoid defined on H...0;∅ , R is p-fuzzy symmetric relation. Then, for all a,b∈H , we have that [figure omitted; refer to PDF]

Proof.

For all a,b∈H , two cases are possible.

(1) If _{(a *R b)p} =∅ , then _{(a *R b)p} ⊆_{(b *R a)p} .

(2) If _{(a *R b)p} ...0;∅ , let x∈_{(a *R b)p} . Then R(a,x)...5;p and R(x,b)...5;p .

Since R is p-fuzzy symmetric, so R(x,a)...5;p and R(b,x)...5;p . Thus (b _*R a)(x)=R(b,x)⋀R(x,a)...5;p . So x∈_{(b *R a)p} . And in this case, we also have that _{(a*R b)p} ⊆_{(b*R a)p} .

The remaining part can be proved by exchanging a and b .

Proposition 5.4.

Let Y9;H,_*R YA; be a partial F-C-hypergroupoid defined on H...0;∅ , p∈(0,1] , if

(1) for all a,b∈H , _{(a *R b)p} =_{(b *R a)p} ,

(2) for any x∈H , there exists a y∈H , such that R(x,y)...5;p .

Then R is a p-fuzzy symmetric binary relation on H .

Proof.

For all a,b∈H , suppose that R(a,b)...5;p . We need to show that R(b,a)...5;p .

Since for b∈H , there exists a x∈H , such that R(b,x)...5;p . So R(a,b)⋀R(b,x)...5;p . That is, b∈_{(a *R x)p} =_{(x *R a)p} . And so R(x,b)⋀R(b,a)...5;p . And finally we have that R(b,a)...5;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 , [figure omitted; refer to PDF]

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...4;p...4;0.1 .

Proposition 6.3.

Letting Y9;H,_*R YA; be a partial F-C-hypergroupoid defined on H...0;∅ , R is a p-fuzzy transitive relation on H , p∈(0,1] . Then for all x,y∈H , we have that [figure omitted; refer to PDF]

Proof.

(1) If _{(x *R x)p} =∅ , then obviously _{(x *R x)p} ⊆_{(x *R y)p} .

Supposing that _{(x *R x)p} ...0;∅ , then for any w∈_{(x *R x)p} , we have that R(x,w)⋀R(w,x)...5;p , that is, R(x,w)...5;p and R(w,x)...5;p . From R(w,x)...5;p and R(x,y)...5;p we have that R(w,y)...5;p . From R(x,w)...5;p and R(w,y)...5;p we conclude that w∈_{(x*R y)p} .

So _{(x *R x)p} ⊆_{(x *R y)p} .

(2) If _{(y*R y)p} =∅ , then obviously _{(y *R y)p} ⊆_{(x *R y)p} .

Supposing that _{(y *R y)p} ...0;∅ , then for any w∈_{(y *R y)p} , we have that R(y,w)⋀R(w,y)...5;p , that is, R(y,w)...5;p and R(w,y)...5;p . From R(y,w)...5;p and R(x,y)...5;p we have that R(x,w)...5;p . From R(x,w)...5;p and R(w,y)...5;p we conclude that w∈_{(x*R y)p} .

So _{(y*R y)p} ⊆_{(x *R y)p} .

Proposition 6.4.

Letting Y9;H,_*R YA; be a partial F-C-hypergroupoid defined on H...0;∅ , R is a p-fuzzy transitive binary relation. For any a,b,c∈H , we have that

(1) _{((a *R b)p *R c)p} ⊆_{(a *R c)p} ;

(2) _{(a *R (b *R c)p )p} ⊆_{(a *R c)p} .

Proof.

(1) If _{((a *R b) p *R c)p} =∅ , then it is obvious that _{((a *R b)p *R c)p} ⊆_{(a *R c)p} .

Suppose that _{((a *R b)p *R c)p} ...0;∅ . Then for any w∈_{((a *R b)p *R c)p} , there exists a _w1 ∈_{(a *R b)p} such that w∈_{(w1 *R c)p} . That is R(a,_w1 )...5;p , R(_w1 ,b)...5;p , R(_w1 ,w)...5;p and R(w,c)...5;p . From R(a,_w1 )...5;p and R(_w1 ,w)...5;p , we have that R(a,w)...5;p . Thus R(a,w)⋀R(w,c)...5;p⋀p=p . That is, w∈_{(a *R c)p} . So _{((a *R b)p *R c)p} ⊆_{(a *R c)p} .

(2) Can be proved similarly.

Acknowledgment

The paper is partially supported by CSC.

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