Journal of Applied Mathematics

Volume 2012, Article ID 685681, 9 pages

http://dx.doi.org/10.1155/2012/685681

## On Fuzzy Corsini's Hyperoperations

^{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

Academic Editor: Said Abbasbandy

Copyright © 2012 Yuming Feng and P. Corsini.

#### 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.

#### 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 is a nonempty set with a function from to the set of subsets of .

A hypergroupoid is a nonempty set , endowed with a hyperoperation, that is, a function from to , the set of nonempty subsets of .

If , then we define , and .

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 be a a pair of sets where is a nonempty set and is a binary relation on . Corsini's hyperoperation (briefly, * C-hyperoperation*) associated with is defined in the following way:
where denotes the family of all the subsets of .

A fuzzy subset of a nonempty set is a function . The family of all the fuzzy subsets of is denoted by .

We use to denote a special fuzzy subset of which is defined by , for all .

For a fuzzy subset of a nonempty set , the * p-cut* of is denoted , for any , and defined by .

A fuzzy binary relation on a nonempty set is a function . In the following, sometimes we use fuzzy relation to refer to fuzzy binary relation.

For any , we use to stand for the minimum of and and to denote the maximum of and .

Given , we will use the following definitions:

A partial fuzzy hypergroupoid is a nonempty set endowed with a fuzzy hyperoperation . Moreover, is called a fuzzy hypergroupoid if for all , there exists at least one , such that holds.

Given a fuzzy hyperoperation , for all , , the fuzzy subset of is defined by

, can be defined similarly. When is a * crisp* subset of , we treat as a fuzzy subset by treating it as , for all and , for all .

#### 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 be a pair of sets where is a non-empty set and is a fuzzy relation on . We define a fuzzy hyperoperation , for any , as follows:
is called a * fuzzy Corsini's hyperoperation* (briefly, * F-C-hyperoperation*) associated with . The fuzzy hyperstructure 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 be a non-empty set, is a fuzzy relation on as described in Table 1.

From the previous definition, by calculating, for example, , , we can obtain Table 2 which is a partial F-C-hypergroupoid.

*Definition 2.4. *Supposing , are two fuzzy relations on a non-empty set , the composition of and is a fuzzy relation on and is defined by , for all .

Proposition 2.5. *A partial F-C-hypergroupoid is a F-C-hypergroupoid if and only if , where .*

*Proof. *Suppose that is a hypergroupoid. For any , there exists at least one , such that holds.

So . Thus . And we conclude that .

is obvious. And so .

Conversely, if , then for any , . So . That is, there exists at least one such that holds. And so is a hypergroupoid.

Thus we complete the proof.

*Definition 2.6. *Letting be a non-empty set, is a fuzzy hyperoperation of , the hyperoperation is defined by , for all , . is called the p-cut of .

*Definition 2.7. *Letting be a fuzzy relation on a non-empty set , we define a binary relation on , for all , as follows:
is called the p-cut of the fuzzy relation .

Proposition 2.8. *Let be a partial F-C-hypergroupoid. Then is a C-hyperoperation associated with , for all .*

*Proof. *For any and for any , we have

From the definition of C-hyperoperation, we conclude that is a C-hyperoperation associated with .

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 be a non-empty set and let be a fuzzy hyperoperation of , then the fuzzy hyperoperation is an F-C-hyperoperation associated with a fuzzy relation on if and only if is a C-hyperoperation associated with , for any .*

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

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

Proposition 3.1. *Let be a partial or nonpartial F-C-hypergroupoid defined on . Then, for all , we have
*

* Proof. *For any , we have that .

So
for all .

Proposition 3.2. *Let be a partial F-C-hypergroupoid and , . Then,*(1)*;*(2)* If then ;*(3)* If then .*

* Proof. *(1) Supposing , then there exist , such that . So from the previous proposition, we have . This is a contradiction.

(2) From and , we have that , and so, .

(3) is proved similar to (2).

Proposition 3.3. *Letting be the F-C-hyperoperation defined on the non-empty set , , then the following are equivalent:*(1)* for some , ;*(2)* for all , .*

* Proof. *Let . Then, for all , we have that , that is and so . Thus , for all .

Conversely, let , for all . Specially, we have and . Thus, and . And so .

Proposition 3.4. *Let be a partial or nonpartial F-C-hypergroupoid defined on . Then, for all , , we have
*

* Proof. *For any , we have that

The remaining part can be proved similarly.

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

In this section, we will assume that is a p-fuzzy reflexive relation on a non-empty set.

*Definition 4.1. *A fuzzy relation on a non-empty set is called *p-fuzzy reflexive* if for any ,

*Example 4.2. *The fuzzy relation introduced in Example 2.3 is 0.1-fuzzy reflexive. Of course, it is p-fuzzy reflexive, where .

Proposition 4.3. *Letting be a partial F-C-hypergroupoid defined on , is p-fuzzy reflexive. Then, for all , , the following are equivalent:*(1)*;*(2)*;*(3)*.*

* Proof. *“(1)*⇒*(2)”

From and we have that which shows that .

“(2)*⇒*(3)”

From we have that . Since , so which implies that .

“(3)*⇒*(1)”

It is obvious.

Proposition 4.4. *Letting be a partial F-C-hypergroupoid defined on , is p-fuzzy reflexive. Then, for any , we have that
*

* Proof. *From we have . That is .

Proposition 4.5. *Letting be a partial F-C-hypergroupoid defined on , is p-fuzzy reflexive. Then, for any , , we have that
*

* Proof. *From we have that . So and . Thus and . That is and . So . Thus .

Conversely, suppose that . Then . Thus . So . That is .

Corollary 4.6. *Letting be a partial F-C-hypergroupoid defined on , is p-fuzzy reflexive. Then, for any , , we have that
*

Proposition 4.7. *Letting be a partial F-C-hypergroupoid defined on , is p-fuzzy reflexive. Then, for any , we have that
*

* Proof. *If , then and . Thus and . So .

Conversely, if , then . Thus . And so . Thus .

Proposition 4.8. *Letting be a partial F-C-hypergroupoid defined on , is p-fuzzy reflexive. Then, for any , , the following are equivalent:*(1)*;*(2)* and ;*(3)* and .*

* Proof. *“(1)*⇒*(2)”

Suppose that . Then and . So and . Thus and .

“(2)*⇒*(3)”

Suppose that . Then . Thus . And so .

“(3)*⇒*(1)”

From and , we have that and . Thus . So .

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

In this section, we will assume that is a p-fuzzy symmetric relation on a non-empty set.

*Definition 5.1. *A fuzzy binary relation on a non-empty set is called *p-fuzzy symmetric* if for any ,

*Example 5.2. *The fuzzy relation introduced in Example 2.3 is 0.2-fuzzy symmetric. Of course, it is p-fuzzy reflexive, where .

Proposition 5.3. *Letting be a partial F-C-hypergroupoid defined on , is p-fuzzy symmetric relation. Then, for all , we have that
*

* Proof. *For all , two cases are possible.(1) If , then .(2) If , let . Then and . Since is p-fuzzy symmetric, so and . Thus . So . And in this case, we also have that .

The remaining part can be proved by exchanging and .

Proposition 5.4. *Let be a partial F-C-hypergroupoid defined on , , if*(1)* for all , ,*(2)* for any , there exists a , such that . **Then is a p-fuzzy symmetric binary relation on .*

* Proof. *For all , suppose that . We need to show that .

Since for , there exists a , such that . So . That is, . And so . And finally we have that .

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

In this section, we will assume that is a p-fuzzy transitive relation on a non-empty set.

*Definition 6.1. *A fuzzy binary relation on a non-empty set is called p-fuzzy transitive if for any ,

*Example 6.2. *The fuzzy relation introduced in Example 2.3 is 0.1-fuzzy transitive. Of course, it is p-fuzzy transitive, where .

Proposition 6.3. *Letting be a partial F-C-hypergroupoid defined on , is a p-fuzzy transitive relation on , . Then for all , we have that
*

* Proof. *(1) If , then obviously .

Supposing that , then for any , we have that , that is, and . From and we have that . From and we conclude that .

So .

(2) If , then obviously .

Supposing that , then for any , we have that , that is, and . From and we have that . From and we conclude that .

So .

Proposition 6.4. *Letting be a partial F-C-hypergroupoid defined on , is a p-fuzzy transitive binary relation. For any , we have that*(1)*;*(2)*.*

* Proof. *(1) If , then it is obvious that .

Suppose that . Then for any , there exists a such that . That is , , and . From and , we have that . Thus . That is, . So .

(2) Can be proved similarly.

#### Acknowledgment

The paper is partially supported by CSC.

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