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Research Article | Open Access

Volume 2012 |Article ID 685681 | https://doi.org/10.1155/2012/685681

Yuming Feng, P. Corsini, "On Fuzzy Corsini's Hyperoperations", Journal of Applied Mathematics, vol. 2012, Article ID 685681, 9 pages, 2012. https://doi.org/10.1155/2012/685681

# On Fuzzy Corsini's Hyperoperations

Academic Editor: Said Abbasbandy
Received22 Feb 2012
Accepted07 May 2012
Published03 Jul 2012

#### 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 , Feng , Hort , Rosenberg , Spartalis , 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  and studied by many researchers; for example, see [3, 815].

Definition 1.1 (see ). 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.

 𝑅 𝑎 𝑏 𝑎 0.1 0.2 𝑏 0.3 0.4

From the previous definition, by calculating, for example, , , 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 , 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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