`Journal of Applied MathematicsVolume 2012, Article ID 685681, 9 pageshttp://dx.doi.org/10.1155/2012/685681`
Research Article

## On Fuzzy Corsini's Hyperoperations

1School of Mathematics and Statistics, Chongqing Three Gorges University, Wanzhou, Chongqing 404100, China
2Dipartimento 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

Copyright © 2012 Yuming Feng and P. Corsini. 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.

#### 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.

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.

#### References

1. J. Chvalina, Functional Graphs, Quasi-Ordered Sets and Commutative Hyper-Graph, Vydavatelstvi Masarykovy Univerzity, Brno, Czech Republic, 1995.
2. J. Chvalina, “Commutative hypergroups in the sense of Marty and ordered sets,” in Proceedings of the Proceedings of the Summer School on General Algebra and Ordered Sets, pp. 19–30, Olomouc, Czech Republic, 1994.
3. P. Corsini and V. Leoreanu, “Hypergroups and binary relations,” Algebra Universalis, vol. 43, no. 4, pp. 321–330, 2000.
4. Y. Feng, “Algebraic hyperstructures obtained from algebraic structures with fuzzy binary relations,” Italian Journal of Pure and Applied Mathematics, no. 25, pp. 157–164, 2009.
5. D. Hort, “A construction of hypergroups from ordered structures and their morphisms,” Journal of Discrete Mathematical Sciences & Cryptography, vol. 6, no. 2-3, pp. 139–150, 2003.
6. I. G. Rosenberg, “Hypergroups and join spaces determined by relations,” Italian Journal of Pure and Applied Mathematics, no. 4, pp. 93–101, 1998.
7. S. I. Spartalis, “Hypergroupoids obtained from groupoids with binary relations,” Italian Journal of Pure and Applied Mathematics, no. 16, pp. 201–210, 2004.
8. P. Corsini, “Binary relations and hypergroupoids,” Italian Journal of Pure and Applied Mathematics, no. 7, pp. 11–18, 2000.
9. P. Corsini and V. Leoreanu, Applications of Hyperstructure Theory, vol. 5 of Advances in Mathematics, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2003.
10. P. Corsini and V. Leoreanu, “Survey on new topics of Hyperstructure Theory and its applications,” in Proceedings of the 8th International Congress on Algebraic Hyperstructures and Applications (AHA '03), pp. 1–37, 2003.
11. V. Leoreanu and L. Leoreanu, “Hypergroups associated with hypergraphs,” Italian Journal of Pure and Applied Mathematics, no. 4, pp. 119–126, 1998.
12. V. Leoreanu, “Weak mutually associative hyperstructures. II,” in Proceedings of the 8th International Congress on (AHA '03), pp. 183–189, 2003.
13. S. I. Spartalis and C. Mamaloukas, “Hyperstructures associated with binary relations,” Computers & Mathematics with Applications, vol. 51, no. 1, pp. 41–50, 2006, Elsevier.
14. S. I. Spartalis, “The hyperoperation relation and the Corsini's partial or not-partial hypergroupoids (A classification),” Italian Journal of Pure and Applied Mathematics, no. 24, pp. 97–112, 2008.
15. S. I. Spartalis, M. Konstantinidou-Serafimidou, and A. Taouktsoglou, “$C$-hypergroupoids obtained by special binary relations,” Computers & Mathematics with Applications, vol. 59, no. 8, pp. 2628–2635, 2010.