Research Article  Open Access
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
Abstract
We generalize the concept of Chyperoperation and introduce the concept of FChyperoperation. We list some basic properties of FChyperoperation and the relationship between the concept of Chyperoperation and the concept of FChyperoperation. We also research FChyperoperations 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, Chyperoperation) 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 pcut 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 nonempty set and is a fuzzy relation on . We define a fuzzy hyperoperation , for any , as follows: is called a fuzzy Corsini's hyperoperation (briefly, FChyperoperation) associated with . The fuzzy hyperstructure is called a partial FChypergroupoid.
Remark 2.2. It is obvious that the concept of FChyperoperation is a generalization of the concept of Chyperoperation.
Example 2.3. Letting be a nonempty 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 FChypergroupoid.
Definition 2.4. Supposing , are two fuzzy relations on a nonempty set , the composition of and is a fuzzy relation on and is defined by , for all .
Proposition 2.5. A partial FChypergroupoid is a FChypergroupoid 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 nonempty set, is a fuzzy hyperoperation of , the hyperoperation is defined by , for all , . is called the pcut of .
Definition 2.7. Letting be a fuzzy relation on a nonempty set , we define a binary relation on , for all , as follows: is called the pcut of the fuzzy relation .
Proposition 2.8. Let be a partial FChypergroupoid. Then is a Chyperoperation associated with , for all .
Proof. For any and for any , we have
From the definition of Chyperoperation, we conclude that is a Chyperoperation associated with .
Thus we complete the proof.
From the previous proposition and the construction of the FChyperoperation, we can easily conclude that a fuzzy hyperoperation is a FChyperoperation if and only if every pcut of the FChyperoperation is a Chyperoperation. That is, consider the following.
Proposition 2.9. Let be a nonempty set and let be a fuzzy hyperoperation of , then the fuzzy hyperoperation is an FChyperoperation associated with a fuzzy relation on if and only if is a Chyperoperation associated with , for any .
3. Basic Properties of FCHyperoperations
In this section, we list some basic properties of FChyperoperations.
Proposition 3.1. Let be a partial or nonpartial FChypergroupoid defined on . Then, for all , we have
Proof. For any , we have that .
So
for all .
Proposition 3.2. Let be a partial FChypergroupoid 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 FChyperoperation defined on the nonempty 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 FChypergroupoid defined on . Then, for all , , we have
Proof. For any , we have that
The remaining part can be proved similarly.
4. FCHyperoperations Associated with pFuzzy Reflexive Relations
In this section, we will assume that is a pfuzzy reflexive relation on a nonempty set.
Definition 4.1. A fuzzy relation on a nonempty set is called pfuzzy reflexive if for any ,
Example 4.2. The fuzzy relation introduced in Example 2.3 is 0.1fuzzy reflexive. Of course, it is pfuzzy reflexive, where .
Proposition 4.3. Letting be a partial FChypergroupoid defined on , is pfuzzy 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 FChypergroupoid defined on , is pfuzzy reflexive. Then, for any , we have that
Proof. From we have . That is .
Proposition 4.5. Letting be a partial FChypergroupoid defined on , is pfuzzy 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 FChypergroupoid defined on , is pfuzzy reflexive. Then, for any , , we have that
Proposition 4.7. Letting be a partial FChypergroupoid defined on , is pfuzzy 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 FChypergroupoid defined on , is pfuzzy 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. FCHyperoperations Associated with pFuzzy Symmetric Relations
In this section, we will assume that is a pfuzzy symmetric relation on a nonempty set.
Definition 5.1. A fuzzy binary relation on a nonempty set is called pfuzzy symmetric if for any ,
Example 5.2. The fuzzy relation introduced in Example 2.3 is 0.2fuzzy symmetric. Of course, it is pfuzzy reflexive, where .
Proposition 5.3. Letting be a partial FChypergroupoid defined on , is pfuzzy 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 pfuzzy 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 FChypergroupoid defined on , , if(1) for all , ,(2) for any , there exists a , such that . Then is a pfuzzy 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. FCHyperoperations Associated with pFuzzy Transitive Relations
In this section, we will assume that is a pfuzzy transitive relation on a nonempty set.
Definition 6.1. A fuzzy binary relation on a nonempty set is called pfuzzy transitive if for any ,
Example 6.2. The fuzzy relation introduced in Example 2.3 is 0.1fuzzy transitive. Of course, it is pfuzzy transitive, where .
Proposition 6.3. Letting be a partial FChypergroupoid defined on , is a pfuzzy 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 FChypergroupoid defined on , is a pfuzzy 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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Copyright
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.