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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 685681, 9 pages
http://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

Academic Editor: Said Abbasbandy

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 [3], Feng [4], Hort [5], Rosenberg [6], Spartalis [7], and so on.

A partial hypergroupoid 𝐻, 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,𝐵𝐏(𝐻){}, then we define 𝐴𝐵={𝑎𝑏𝑎𝐴,𝑏𝐵}, 𝑥𝐵={𝑥}𝐵 and 𝐴𝑦=𝐴{𝑦}.

A Corsini's hyperoperation was first introduced by Corsini [8] and studied by many researchers; for example, see [3, 815].

Definition 1.1 (see [8]). Let 𝐻,𝑅 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: 𝑅𝐻×𝐻𝑃(𝐻)𝑥𝑅𝑦={𝑧𝐻𝑥𝑅𝑧,𝑧𝑅𝑦},(1.1) where 𝑃(𝐻) denotes the family of all the subsets of 𝐻.

A fuzzy subset 𝐴 of a nonempty set 𝐻 is a function 𝐴𝐻[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 𝑥𝐻.

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{𝑥𝐻𝐴(𝑥)𝑝}.

A fuzzy binary relation 𝑅 on a nonempty set 𝐻 is a function 𝑅𝐻×𝐻[0,1]. In the following, sometimes we use fuzzy relation to refer to fuzzy binary relation.

For any 𝑎,𝑏[0,1], we use 𝑎𝑏 to stand for the minimum of 𝑎 and 𝑏 and 𝑎𝑏 to denote the maximum of 𝑎 and 𝑏.

Given 𝐴,𝐵𝐹(𝐻), we will use the following definitions: 𝐴𝐵𝐴(𝑥)𝐵(𝑥),𝑥𝐻,𝐴=𝐵𝐴(𝑥)=𝐵(𝑥),𝑥𝐻,(𝐴𝐵)(𝑥)𝐴(𝑥)𝐵(𝑥),𝑥𝐻,(𝐴𝐵)(𝑥)𝐴(𝑥)𝐵(𝑥),𝑥𝐻.(1.2)

A partial fuzzy hypergroupoid 𝐻, is a nonempty set endowed with a fuzzy hyperoperation 𝐻×𝐻𝐹(𝐻). Moreover, 𝐻, is called a fuzzy hypergroupoid if for all 𝑥,𝑦𝐻, there exists at least one 𝑧𝐻, such that (𝑥𝑦)(𝑧)0 holds.

Given a fuzzy hyperoperation 𝐻×𝐻𝐹(𝐻), for all 𝑎𝐻, 𝐵𝐹(𝐻), the fuzzy subset 𝑎𝐵 of 𝐻 is defined by (𝑎𝐵)(𝑥)𝐵(𝑏)>0(𝑎𝑏)(𝑥).(1.3)

𝐵𝑎, 𝐴𝐵 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 𝑥𝐵 and 𝐵(𝑥)=0, for all 𝑥𝐻𝐵.

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 𝐻,𝑅 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 𝑅𝐻×𝐻𝐹(𝐻), for any 𝑥,𝑦,𝑧𝐻, as follows: 𝑥𝑅𝑦(𝑧)𝑅(𝑥,𝑧)𝑅(𝑧,𝑦).(2.1)𝑅 is called a fuzzy Corsini's hyperoperation (briefly, F-C-hyperoperation) associated with 𝑅. The fuzzy hyperstructure 𝐻,𝑅 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.

tab1
Table 1

From the previous definition, by calculating, for example, (𝑎𝑅𝑎)(𝑎)=𝑅(𝑎,𝑎)𝑅(𝑎,𝑎)=0.10.1=0.1, 𝑅(𝑎𝑏)(𝑎)=𝑅(𝑎,𝑎)𝑅(𝑎,𝑏)=0.10.2=0.1, we can obtain Table 2 which is a partial F-C-hypergroupoid.

tab2
Table 2

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 (𝑅𝑆)(𝑥,𝑦)𝑧𝐻(𝑅(𝑥,𝑧)𝑆(𝑧,𝑦)), for all 𝑥,𝑦𝐻.

Proposition 2.5. A partial F-C-hypergroupoid 𝐻,𝑅 is a F-C-hypergroupoid if and only if supp(𝑅𝑅)=𝐻×𝐻, where supp(𝑅𝑅)={(𝑥,𝑦)(𝑅𝑅)(𝑥,𝑦)0}.

Proof. Suppose that 𝐻,𝑅 is a hypergroupoid. For any 𝑥,𝑦𝐻, there exists at least one 𝑧𝐻, such that (𝑥𝑅𝑦)(𝑧)0 holds.
So (𝑅𝑅)(𝑥,𝑦)=𝑧𝐻(𝑅(𝑥,𝑧)𝑅(𝑧,𝑦))0. Thus (𝑥,𝑦)supp(𝑅𝑅). And we conclude that 𝐻×Hsupp(𝑅𝑅).
supp(𝑅𝑅)𝐻×𝐻 is obvious. And so supp(𝑅𝑅)=𝐻×𝐻.
Conversely, if supp(𝑅𝑅)=𝐻×𝐻, then for any 𝑥,𝑦𝐻, (𝑥,𝑦)𝐻×𝐻=supp(𝑅𝑅). So (𝑅𝑅)(𝑥,𝑦)=𝑧𝐻(𝑅(𝑥,𝑧)𝑅(𝑧,𝑦))0. That is, there exists at least one 𝑧𝐻 such that (𝑥𝑅𝑦)(𝑧)0 holds. And so 𝐻,𝑅 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 𝑥𝑝𝑦=(𝑥𝑦)𝑝, for all 𝑥,𝑦𝐻, 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: 𝑥𝑅𝑝𝑦𝑅(𝑥,𝑦)𝑝.(2.2)𝑅𝑝 is called the p-cut of the fuzzy relation 𝑅.

Proposition 2.8. Let 𝐻,𝑅 be a partial F-C-hypergroupoid. Then (𝑅)𝑝 is a C-hyperoperation associated with 𝑅𝑝, for all 0<𝑝1.

Proof. For any 0<𝑝1 and for any 𝑥,𝑦𝐻, we have 𝑥𝑅𝑝𝑦=𝑥𝑅𝑦𝑝=𝑧𝐻𝑥𝑅𝑦=(𝑧)𝑝{𝑧𝐻𝑅(𝑥,𝑧)𝑅(𝑧,𝑦)𝑝}={𝑧𝐻𝑅(𝑥,𝑧)𝑝,𝑅(𝑧,𝑦)𝑝}=𝑧𝐻𝑥𝑅𝑝𝑧,𝑧𝑅𝑝𝑦.(2.3)
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<𝑝1.

3. Basic Properties of F-C-Hyperoperations

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

Proposition 3.1. Let 𝐻,𝑅 be a partial or nonpartial F-C-hypergroupoid defined on 𝐻. Then, for all 𝑥,𝑦,𝑎,𝑏𝐻, we have 𝑥𝑅𝑦𝑎𝑅𝑏=𝑥𝑅𝑏𝑎𝑅𝑦.(3.1)

Proof. For any 𝑥,𝑦,𝑎,𝑏,𝑧𝐻, we have that (𝑥𝑅𝑦𝑎𝑅𝑏)(𝑧)=(𝑥𝑅𝑦)(𝑧)(𝑎𝑅𝑏)(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)=(𝑥𝑅𝑏𝑎𝑅𝑦)(𝑧).
So 𝑥𝑅𝑦𝑎𝑅𝑏=𝑥𝑅𝑏𝑎𝑅𝑦,(3.2) for all 𝑥,𝑦,𝑎,𝑏𝐻.

Proposition 3.2. Let 𝐻,𝑅 be a partial F-C-hypergroupoid and 𝑥,𝑦𝐻, 𝑥𝑅𝑦=. Then,(1)𝑥𝑅𝐻𝐻𝑅𝑦=;(2) If 𝐻=𝑥𝑅𝐻 then 𝐻𝑅𝑦=;(3) If 𝐻=𝐻𝑅𝑥 then 𝑦𝑅𝐻=.

Proof. (1) Supposing 𝑥𝑅𝐻𝐻𝑅𝑦, then there exist 𝑎,𝑏𝐻, such that 𝑥𝑅𝑎𝑏𝑅𝑦. So from the previous proposition, we have 𝑥𝑅𝑦𝑏𝑅𝑎. This is a contradiction.
(2) From 𝐻=𝑥𝑅𝐻 and 𝑥𝑅𝐻𝐻𝑅𝑦=, we have that 𝐻𝐻𝑅𝑦=, and so, 𝐻𝑅𝑦=.
(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 𝑎𝐻, (𝑎𝑅𝑎)𝑝=𝐻;(2) for all 𝑥,𝑦𝐻, 𝑎(𝑥𝑅𝑦)𝑝.

Proof. Let (𝑎𝑅𝑎)𝑝=𝐻. Then, for all 𝑥,𝑦𝐻, we have that (𝑎𝑅𝑎)(𝑥)𝑝,(𝑎𝑅𝑎)(𝑦)𝑝, that is 𝑅(𝑎,𝑥)𝑝,𝑅(𝑥,𝑎)𝑝,𝑅(𝑎,𝑦)𝑝,𝑅(𝑦,𝑎)𝑝 and so 𝑅(𝑥,𝑎)𝑅(𝑎,𝑦)𝑝. Thus 𝑎(𝑥𝑅𝑦)𝑝, for all 𝑥,𝑦𝐻.
Conversely, let 𝑎(𝑥𝑅𝑦)𝑝, for all 𝑥,𝑦𝐻. Specially, we have 𝑎(𝑎𝑅𝑥)𝑝 and 𝑎(𝑥𝑅𝑎)𝑝. Thus, 𝑅(𝑎,𝑥)𝑝 and 𝑅(𝑥,𝑎)𝑝. And so 𝑥(𝑎𝑅𝑎)𝑝.

Proposition 3.4. Let 𝐻,𝑅 be a partial or nonpartial F-C-hypergroupoid defined on 𝐻. Then, for all 𝑎,𝑏𝐻, 𝑝(0,1], we have 𝑎𝑏𝑅𝑏𝑝𝑏𝑎𝑅𝑎𝑝.(3.3)

Proof. For any 𝑎,𝑏𝐻, we have that 𝑎𝑏𝑅𝑏𝑝𝑏𝑅𝑏(𝑎)𝑝𝑅(𝑏,𝑎)𝑅(𝑎,𝑏)𝑝𝑅(𝑎,𝑏)𝑅(𝑏,𝑎)𝑝𝑎𝑅𝑎(𝑏)𝑝𝑏𝑎𝑅𝑎𝑝.(3.4)
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 𝑥𝐻, 𝑅(𝑥,𝑥)𝑝.(4.1)

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 0𝑝0.1.

Proposition 4.3. Letting 𝐻,𝑅 be a partial F-C-hypergroupoid defined on 𝐻, 𝑅 is p-fuzzy reflexive. Then, for all 𝑎,𝑏𝐻, 𝑝(0,1], the following are equivalent:(1)𝑅(𝑎,𝑏)𝑝;(2)𝑎(𝑎𝑅𝑏)𝑝;(3)𝑏(𝑎𝑅𝑏)𝑝.

Proof. “(1)(2)”
From 𝑅(𝑎,𝑎)𝑝 and 𝑅(𝑎,𝑏)𝑝 we have that 𝑅(𝑎,𝑎)𝑅(𝑎,𝑏)𝑝 which shows that 𝑎(𝑎𝑅𝑏)𝑝.
“(2)(3)”
From 𝑎(𝑎𝑅𝑏)𝑝 we have that 𝑅(𝑎,𝑏)𝑝. Since 𝑅(𝑏,𝑏)𝑝, so 𝑅(𝑎,𝑏)𝑅(𝑏,𝑏)𝑝 which implies that 𝑏(𝑎𝑅𝑏)𝑝.
“(3)(1)”
It is obvious.

Proposition 4.4. Letting 𝐻,𝑅 be a partial F-C-hypergroupoid defined on 𝐻, 𝑅 is p-fuzzy reflexive. Then, for any 𝑎𝐻, we have that 𝑎𝑎𝑅𝑎𝑝.(4.2)

Proof. From 𝑅(𝑎,𝑎)𝑝 we have 𝑅(𝑎,𝑎)𝑅(𝑎,𝑎)𝑝. That is 𝑎(𝑎𝑅𝑎)𝑝.

Proposition 4.5. Letting 𝐻,𝑅 be a partial F-C-hypergroupoid defined on 𝐻, 𝑅 is p-fuzzy reflexive. Then, for any 𝑎,𝑏𝐻, 𝑝(0,1], we have that 𝑏𝑎𝑅𝑎𝑝𝑎𝑎𝑅𝑏𝑏𝑅𝑎𝑝.(4.3)

Proof. From 𝑏(𝑎𝑅𝑎)𝑝 we have that 𝑅(𝑎,𝑏)𝑅(𝑏,𝑎)𝑝. So 𝑅(𝑎,𝑏)𝑝 and 𝑅(𝑏,𝑎)𝑝. Thus 𝑅(𝑎,𝑎)𝑅(𝑎,𝑏)𝑝 and 𝑅(𝑏,𝑎)𝑅(𝑎,𝑎)𝑝. That is (𝑎𝑅𝑏)(𝑎)𝑝 and (𝑏𝑅𝑎)(𝑎)𝑝. So (𝑎𝑅𝑏𝑏𝑅𝑎)(𝑎)𝑝. Thus 𝑎(𝑎𝑅𝑏𝑏𝑅𝑎)𝑝.
Conversely, suppose that 𝑎(𝑎𝑅𝑏𝑏𝑅𝑎)𝑝. Then (𝑎𝑅𝑏)(𝑎)(𝑏𝑅𝑎)(𝑎)𝑝. Thus 𝑅(𝑎,𝑎)𝑅(𝑎,𝑏)𝑅(𝑏,𝑎)𝑅(𝑎,𝑎)𝑝. So 𝑅(𝑎,𝑏)𝑅(𝑏,𝑎)𝑝. That is 𝑏(𝑎𝑅𝑎)𝑝.

Corollary 4.6. Letting 𝐻,𝑅 be a partial F-C-hypergroupoid defined on 𝐻, 𝑅 is p-fuzzy reflexive. Then, for any 𝑎,𝑏𝐻, 𝑝(0,1], we have that 𝑏𝑎𝑅𝑎𝑝𝑎𝑏𝑅𝑏𝑝𝑎𝑎𝑅𝑏𝑏𝑅𝑎𝑝𝑏𝑎𝑅𝑏𝑏𝑅𝑎𝑝.(4.4)

Proposition 4.7. Letting 𝐻,𝑅 be a partial F-C-hypergroupoid defined on 𝐻, 𝑅 is p-fuzzy reflexive. Then, for any 𝑎,𝑏𝐻, we have that 𝑐𝑎𝑅𝑏𝑝𝑐𝑎𝑅𝑐𝑐𝑅𝑏𝑝.(4.5)

Proof. If 𝑐(𝑎𝑅𝑏)𝑝, then 𝑅(𝑎,𝑐)𝑝 and 𝑅(𝑐,𝑏)𝑝. Thus 𝑐(𝑎𝑅𝑐)𝑝 and 𝑐(𝑐𝑅𝑏)𝑝. So 𝑐(𝑎𝑅𝑐𝑐𝑅𝑏)𝑝.
Conversely, if 𝑐(𝑎𝑅𝑐𝑐𝑅𝑏)𝑝, then (𝑎𝑅𝑐)(𝑐)(𝑐𝑅𝑏)(𝑐)𝑝. Thus 𝑅(𝑎,𝑐)𝑅(𝑐,𝑐)𝑅(𝑐,𝑐)𝑅(𝑐,𝑏)𝑝. And so 𝑅(𝑎,𝑐)𝑅(𝑐,𝑏)𝑝. Thus 𝑐(𝑎𝑅𝑏)𝑝.

Proposition 4.8. Letting 𝐻,𝑅 be a partial F-C-hypergroupoid defined on 𝐻, 𝑅 is p-fuzzy reflexive. Then, for any 𝑎,𝑏,𝑐𝐻, 𝑝(0,1], the following are equivalent:(1)𝑐(𝑎𝑅𝑏)𝑝;(2)𝑎(𝑎𝑅𝑐)𝑝 and 𝑏(𝑐𝑅𝑏)𝑝;(3)𝑎(𝑎𝑅𝑐)𝑝 and 𝑐(𝑐𝑅𝑏)𝑝.

Proof. “(1)(2)”
Suppose that 𝑐(𝑎𝑅𝑏)𝑝. Then 𝑅(𝑎,𝑐)𝑝 and 𝑅(𝑐,b)𝑝. So 𝑅(𝑎,𝑎)𝑅(𝑎,𝑐)𝑝 and 𝑅(𝑐,𝑏)𝑅(𝑏,𝑏)𝑝. Thus 𝑎(𝑎𝑅𝑐)𝑝 and 𝑏(𝑐𝑅𝑏)𝑝.
“(2)(3)”
Suppose that 𝑏(𝑐𝑅𝑏)𝑝. Then 𝑅(𝑐,𝑏)𝑝. Thus 𝑅(𝑐,𝑐)𝑅(𝑐,𝑏)𝑝. And so 𝑐(𝑐𝑅𝑏)𝑝.
“(3)(1)”
From 𝑎(𝑎𝑅𝑐)𝑝 and 𝑐(𝑐𝑅𝑏)𝑝, we have that 𝑅(𝑎,𝑐)𝑝 and 𝑅(𝑐,𝑏)𝑝. Thus 𝑅(𝑎,𝑐)𝑅(𝑐,𝑏)𝑝. So 𝑐(𝑎𝑅𝑏)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,𝑦𝐻, 𝑅(𝑥,𝑦)𝑝𝑅(𝑦,𝑥)𝑝.(5.1)

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 0𝑝0.2.

Proposition 5.3. Letting 𝐻,𝑅 be a partial F-C-hypergroupoid defined on 𝐻, 𝑅 is p-fuzzy symmetric relation. Then, for all 𝑎,𝑏𝐻, we have that 𝑎𝑅𝑏𝑝=𝑏𝑅𝑎𝑝.(5.2)

Proof. For all 𝑎,𝑏𝐻, two cases are possible.(1) If (𝑎𝑅𝑏)𝑝=, then (𝑎𝑅𝑏)𝑝(𝑏𝑅𝑎)𝑝.(2) If (𝑎𝑅𝑏)𝑝, let 𝑥(𝑎𝑅𝑏)𝑝. Then 𝑅(𝑎,𝑥)𝑝 and 𝑅(𝑥,𝑏)𝑝. Since 𝑅 is p-fuzzy symmetric, so 𝑅(𝑥,𝑎)𝑝 and 𝑅(𝑏,𝑥)𝑝. Thus (𝑏𝑅𝑎)(𝑥)=𝑅(𝑏,𝑥)𝑅(𝑥,𝑎)𝑝. So 𝑥(𝑏𝑅𝑎)𝑝. And in this case, we also have that (𝑎𝑅𝑏)𝑝(𝑏𝑅𝑎)𝑝.
The remaining part can be proved by exchanging 𝑎 and 𝑏.

Proposition 5.4. Let 𝐻,𝑅 be a partial F-C-hypergroupoid defined on 𝐻, 𝑝(0,1], if(1) for all 𝑎,𝑏𝐻, (𝑎𝑅𝑏)𝑝=(𝑏𝑅𝑎)𝑝,(2) for any 𝑥H, there exists a 𝑦𝐻, such that 𝑅(𝑥,𝑦)𝑝. Then 𝑅 is a p-fuzzy symmetric binary relation on 𝐻.

Proof. For all 𝑎,𝑏𝐻, suppose that 𝑅(𝑎,𝑏)𝑝. We need to show that 𝑅(𝑏,𝑎)𝑝.
Since for 𝑏𝐻, there exists a 𝑥𝐻, such that 𝑅(𝑏,𝑥)𝑝. So 𝑅(𝑎,𝑏)𝑅(𝑏,𝑥)𝑝. That is, 𝑏(𝑎𝑅𝑥)𝑝=(𝑥𝑅𝑎)𝑝. And so 𝑅(𝑥,𝑏)𝑅(𝑏,𝑎)𝑝. And finally we have that 𝑅(𝑏,𝑎)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,𝑧𝐻, 𝑅(𝑥,𝑦)𝑝,𝑅(𝑦,𝑧)𝑝𝑅(𝑥,𝑧)𝑝.(6.1)

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 0𝑝0.1.

Proposition 6.3. Letting 𝐻,𝑅 be a partial F-C-hypergroupoid defined on 𝐻, 𝑅 is a p-fuzzy transitive relation on 𝐻, 𝑝(0,1]. Then for all 𝑥,𝑦𝐻, we have that 𝑅(𝑥,𝑦)𝑝𝑥𝑅𝑥𝑦𝑅𝑦𝑝𝑥𝑅𝑦𝑝.(6.2)

Proof. (1) If (𝑥𝑅𝑥)𝑝=, then obviously (𝑥𝑅𝑥)𝑝(𝑥𝑅𝑦)𝑝.
Supposing that (𝑥𝑅𝑥)𝑝, then for any 𝑤(𝑥𝑅𝑥)𝑝, we have that 𝑅(𝑥,𝑤)𝑅(𝑤,𝑥)𝑝, that is, 𝑅(𝑥,𝑤)𝑝 and 𝑅(𝑤,𝑥)𝑝. From 𝑅(𝑤,𝑥)𝑝 and 𝑅(𝑥,𝑦)𝑝 we have that 𝑅(𝑤,𝑦)𝑝. From 𝑅(𝑥,𝑤)𝑝 and 𝑅(𝑤,𝑦)𝑝 we conclude that 𝑤(𝑥𝑅𝑦)𝑝.
So (𝑥𝑅𝑥)𝑝(𝑥𝑅𝑦)𝑝.
(2) If (𝑦𝑅𝑦)𝑝=, then obviously (𝑦𝑅𝑦)𝑝(𝑥𝑅𝑦)𝑝.
Supposing that (𝑦𝑅𝑦)𝑝, then for any 𝑤(𝑦𝑅𝑦)𝑝, we have that 𝑅(𝑦,𝑤)𝑅(𝑤,𝑦)𝑝, that is, 𝑅(𝑦,𝑤)𝑝 and 𝑅(𝑤,𝑦)𝑝. From 𝑅(𝑦,𝑤)𝑝 and 𝑅(𝑥,𝑦)𝑝 we have that 𝑅(𝑥,𝑤)𝑝. From 𝑅(𝑥,𝑤)𝑝 and 𝑅(𝑤,𝑦)𝑝 we conclude that 𝑤(𝑥𝑅𝑦)𝑝.
So (𝑦𝑅𝑦)𝑝(𝑥𝑅𝑦)𝑝.

Proposition 6.4. Letting 𝐻,𝑅 be a partial F-C-hypergroupoid defined on 𝐻, 𝑅 is a p-fuzzy transitive binary relation. For any 𝑎,𝑏,𝑐𝐻, we have that(1)((𝑎𝑅𝑏)𝑝𝑅𝑐)𝑝(𝑎𝑅𝑐)𝑝;(2)(𝑎𝑅(𝑏𝑅𝑐)𝑝)𝑝(𝑎𝑅𝑐)𝑝.

Proof. (1) If ((𝑎𝑅𝑏)𝑝𝑅𝑐)𝑝=, then it is obvious that ((𝑎𝑅𝑏)𝑝𝑅𝑐)𝑝(𝑎𝑅𝑐)𝑝.
Suppose that ((𝑎𝑅𝑏)𝑝𝑅𝑐)𝑝. Then for any 𝑤((𝑎𝑅𝑏)𝑝𝑅𝑐)𝑝, there exists a 𝑤1(𝑎𝑅𝑏)𝑝 such that 𝑤(𝑤1𝑅𝑐)𝑝. That is 𝑅(𝑎,𝑤1)𝑝, 𝑅(𝑤1,𝑏)𝑝, 𝑅(𝑤1,𝑤)𝑝 and 𝑅(𝑤,𝑐)𝑝. From 𝑅(𝑎,𝑤1)𝑝 and 𝑅(𝑤1,𝑤)𝑝, we have that 𝑅(𝑎,𝑤)𝑝. Thus 𝑅(𝑎,𝑤)𝑅(𝑤,𝑐)𝑝𝑝=𝑝. That is, 𝑤(𝑎𝑅𝑐)𝑝. So ((𝑎𝑅𝑏)𝑝𝑅𝑐)𝑝(𝑎𝑅𝑐)𝑝.
(2) Can be proved similarly.

Acknowledgment

The paper is partially supported by CSC.

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