Journal of Applied Mathematics

Journal of Applied Mathematics / 2012 / Article

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 [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,𝐵∈𝐏(𝐻)−{∅}, then we define 𝐴∗𝐵=∪{ğ‘Žâˆ—ğ‘âˆ£ğ‘Žâˆˆğ´,𝑏∈𝐵}, 𝑥∗𝐵={𝑥}∗𝐵 and 𝐴∗𝑦=𝐴∗{𝑦}.

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 ⟨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: ∗𝑅∶𝐻×𝐻⟶𝑃(𝐻)∶𝑥∗𝑅𝑦={𝑧∈𝐻∣𝑥𝑅𝑧,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 𝐴∶𝐻→[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 𝑥∈𝐻.

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≐{𝑥∈𝐻∣𝐴(𝑥)≥𝑝}.

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

For any ğ‘Ž,𝑏∈[0,1], we use ğ‘Žâˆ§ğ‘ to stand for the minimum of ğ‘Ž and 𝑏 and ğ‘Žâˆ¨ğ‘ to denote the maximum of ğ‘Ž and 𝑏.

Given 𝐴,𝐵∈𝐹(𝐻), we will use the following definitions: 𝐴⊆𝐵≐𝐴(𝑥)≤𝐵(𝑥),∀𝑥∈𝐻,𝐴=𝐵≐𝐴(𝑥)=𝐵(𝑥),∀𝑥∈𝐻,(𝐴∪𝐵)(𝑥)≐𝐴(𝑥)∨𝐵(𝑥),∀𝑥∈𝐻,(𝐴∩𝐵)(𝑥)≐𝐴(𝑥)∧𝐵(𝑥),∀𝑥∈𝐻.(1.2)

A partial fuzzy hypergroupoid ⟨𝐻,∗⟩ is a nonempty set endowed with a fuzzy hyperoperation ∗∶𝐻×𝐻→𝐹(𝐻). Moreover, ⟨𝐻,∗⟩ is called a fuzzy hypergroupoid if for all 𝑥,𝑦∈𝐻, there exists at least one 𝑧∈𝐻, such that (𝑥∗𝑦)(𝑧)≠0 holds.

Given a fuzzy hyperoperation ∗∶𝐻×𝐻→𝐹(𝐻), for all ğ‘Žâˆˆğ», 𝐵∈𝐹(𝐻), the fuzzy subset ğ‘Žâˆ—ğµ of 𝐻 is defined by (ğ‘Žâˆ—ğµ)(𝑥)≐∨𝐵(𝑏)>0(ğ‘Žâˆ—ğ‘)(𝑥).(1.3)

ğµâˆ—ğ‘Ž, 𝐴∗𝐵 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 𝑥∈𝐵 and 𝐵(𝑥)=0, for all 𝑥∈𝐻−𝐵.

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 ∗𝑅∶𝐻×𝐻→𝐹(𝐻), for any 𝑥,𝑦,𝑧∈𝐻, as follows: 𝑥∗𝑅𝑦(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={ğ‘Ž,𝑏} be a non-empty set, 𝑅 is a fuzzy relation on 𝐻 as described in Table 1.


𝑅 ğ‘Ž 𝑏

ğ‘Ž 0.10.2
𝑏 0.30.4

From the previous definition, by calculating, for example, (ğ‘Žâˆ—ğ‘…ğ‘Ž)(ğ‘Ž)=𝑅(ğ‘Ž,ğ‘Ž)∧𝑅(ğ‘Ž,ğ‘Ž)=0.1∧0.1=0.1, 𝑅(ğ‘Žâˆ—ğ‘)(ğ‘Ž)=𝑅(ğ‘Ž,ğ‘Ž)∧𝑅(ğ‘Ž,𝑏)=0.1∧0.2=0.1, we can obtain Table 2 which is a partial F-C-hypergroupoid.


∗ 𝑅 ğ‘Ž 𝑏

ğ‘Ž 0 . 1 / ğ‘Ž + 0 . 2 / 𝑏 0 . 1 / ğ‘Ž + 0 . 2 / 𝑏
𝑏 0 . 1 / ğ‘Ž + 0 . 3 / 𝑏 0 . 2 / ğ‘Ž + 0 . 4 / 𝑏

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 ⋁(𝑅∘𝑆)(𝑥,𝑦)≐𝑧∈𝐻(𝑅(𝑥,𝑧)∧𝑆(𝑧,𝑦)), for all 𝑥,𝑦∈𝐻.

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

Proof. Suppose that ⟨𝐻,∗𝑅⟩ is a hypergroupoid. For any 𝑥,𝑦∈𝐻, there exists at least one 𝑧∈𝐻, such that (𝑥∗𝑅𝑦)(𝑧)≠0 holds.
So ⋁(𝑅∘𝑅)(𝑥,𝑦)=𝑧∈𝐻(𝑅(𝑥,𝑧)∧𝑅(𝑧,𝑦))≠0. Thus (𝑥,𝑦)∈supp(𝑅∘𝑅). And we conclude that 𝐻×H⊆supp(𝑅∘𝑅).
supp(𝑅∘𝑅)⊆𝐻×𝐻 is obvious. And so supp(𝑅∘𝑅)=𝐻×𝐻.
Conversely, if supp(𝑅∘𝑅)=𝐻×𝐻, then for any 𝑥,𝑦∈𝐻, (𝑥,𝑦)∈𝐻×𝐻=supp(𝑅∘𝑅). So ⋁(𝑅∘𝑅)(𝑥,𝑦)=𝑧∈𝐻(𝑅(𝑥,𝑧)∧𝑅(𝑧,𝑦))≠0. That is, there exists at least one 𝑧∈𝐻 such that (𝑥∗𝑅𝑦)(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 𝑥∗𝑝𝑦=(𝑥∗𝑦)𝑝, for all 𝑥,𝑦∈𝐻, 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: 𝑥𝑅𝑝𝑦≐𝑅(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<𝑝≤1.

Proof. For any 0<𝑝≤1 and for any 𝑥,𝑦∈𝐻, we have 𝑥∗𝑅𝑝𝑦=𝑥∗𝑅𝑦𝑝=𝑧∈𝐻∣𝑥∗𝑅𝑦=(𝑧)≥𝑝{𝑧∈𝐻∣𝑅(𝑥,𝑧)∧𝑅(𝑧,𝑦)≥𝑝}={𝑧∈𝐻∣𝑅(𝑥,𝑧)≥𝑝,𝑅(𝑧,𝑦)≥𝑝}=𝑧∈𝐻∣𝑥𝑅𝑝𝑧,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<𝑝≤1.

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,ğ‘Ž,𝑏∈𝐻, we have ğ‘¥âˆ—ğ‘…ğ‘¦âˆ©ğ‘Žâˆ—ğ‘…ğ‘=ğ‘¥âˆ—ğ‘…ğ‘âˆ©ğ‘Žâˆ—ğ‘…ğ‘¦.(3.1)

Proof. For any 𝑥,𝑦,ğ‘Ž,𝑏,𝑧∈𝐻, we have that (ğ‘¥âˆ—ğ‘…ğ‘¦âˆ©ğ‘Žâˆ—ğ‘…ğ‘)(𝑧)=(𝑥∗𝑅𝑦)(𝑧)∧(ğ‘Žâˆ—ğ‘…ğ‘)(𝑧)=𝑅(𝑥,𝑧)∧𝑅(𝑧,𝑦)∧𝑅(ğ‘Ž,𝑧)∧𝑅(𝑧,𝑏)=𝑅(𝑥,𝑧)∧𝑅(𝑧,𝑏)∧𝑅(ğ‘Ž,𝑧)∧𝑅(𝑧,𝑦)=(ğ‘¥âˆ—ğ‘…ğ‘âˆ©ğ‘Žâˆ—ğ‘…ğ‘¦)(𝑧).
So ğ‘¥âˆ—ğ‘…ğ‘¦âˆ©ğ‘Žâˆ—ğ‘…ğ‘=ğ‘¥âˆ—ğ‘…ğ‘âˆ©ğ‘Žâˆ—ğ‘…ğ‘¦,(3.2) for all 𝑥,𝑦,ğ‘Ž,𝑏∈𝐻.

Proposition 3.2. Let ⟨𝐻,∗𝑅⟩ be a partial F-C-hypergroupoid and 𝑥,𝑦∈𝐻, 𝑥∗𝑅𝑦=∅. Then,(1)𝑥∗𝑅𝐻∩𝐻∗𝑅𝑦=∅;(2) If 𝐻=𝑥∗𝑅𝐻 then 𝐻∗𝑅𝑦=∅;(3) If 𝐻=𝐻∗𝑅𝑥 then 𝑦∗𝑅𝐻=∅.

Proof. (1) Supposing 𝑥∗𝑅𝐻∩𝐻∗𝑅𝑦≠∅, then there exist ğ‘Ž,𝑏∈𝐻, such that ğ‘¥âˆ—ğ‘…ğ‘Žâˆ©ğ‘âˆ—ğ‘…ğ‘¦â‰ âˆ…. So from the previous proposition, we have ğ‘¥âˆ—ğ‘…ğ‘¦âˆ©ğ‘âˆ—ğ‘…ğ‘Žâ‰ âˆ…. This is a contradiction.
(2) From 𝐻=𝑥∗𝑅𝐻 and 𝑥∗𝑅𝐻∩𝐻∗𝑅𝑦=∅, we have that 𝐻∩𝐻∗𝑅𝑦=∅, and so, 𝐻∗𝑅𝑦=∅.
(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 ğ‘Žâˆˆğ», (ğ‘Žâˆ—ğ‘…ğ‘Ž)𝑝=𝐻;(2) for all 𝑥,𝑦∈𝐻, ğ‘Žâˆˆ(𝑥∗𝑅𝑦)𝑝.

Proof. Let (ğ‘Žâˆ—ğ‘…ğ‘Ž)𝑝=𝐻. Then, for all 𝑥,𝑦∈𝐻, we have that (ğ‘Žâˆ—ğ‘…ğ‘Ž)(𝑥)≥𝑝,(ğ‘Žâˆ—ğ‘…ğ‘Ž)(𝑦)≥𝑝, that is 𝑅(ğ‘Ž,𝑥)≥𝑝,𝑅(𝑥,ğ‘Ž)≥𝑝,𝑅(ğ‘Ž,𝑦)≥𝑝,𝑅(𝑦,ğ‘Ž)≥𝑝 and so 𝑅(𝑥,ğ‘Ž)∧𝑅(ğ‘Ž,𝑦)≥𝑝. Thus ğ‘Žâˆˆ(𝑥∗𝑅𝑦)𝑝, for all 𝑥,𝑦∈𝐻.
Conversely, let ğ‘Žâˆˆ(𝑥∗𝑅𝑦)𝑝, for all 𝑥,𝑦∈𝐻. Specially, we have ğ‘Žâˆˆ(ğ‘Žâˆ—ğ‘…ğ‘¥)𝑝 and ğ‘Žâˆˆ(ğ‘¥âˆ—ğ‘…ğ‘Ž)𝑝. Thus, 𝑅(ğ‘Ž,𝑥)≥𝑝 and 𝑅(𝑥,ğ‘Ž)≥𝑝. And so 𝑥∈(ğ‘Žâˆ—ğ‘…ğ‘Ž)𝑝.

Proposition 3.4. Let ⟨𝐻,∗𝑅⟩ be a partial or nonpartial F-C-hypergroupoid defined on 𝐻≠∅. Then, for all ğ‘Ž,𝑏∈𝐻, 𝑝∈(0,1], we have î€·ğ‘Žâˆˆğ‘âˆ—ğ‘…ğ‘î€¸ğ‘î€·âŸºğ‘âˆˆğ‘Žâˆ—ğ‘…ğ‘Žî€¸ğ‘.(3.3)

Proof. For any ğ‘Ž,𝑏∈𝐻, we have that î€·ğ‘Žâˆˆğ‘âˆ—ğ‘…ğ‘î€¸ğ‘âŸ¹î€·ğ‘âˆ—ğ‘…ğ‘î€¸î€·(ğ‘Ž)≥𝑝⟹𝑅(𝑏,ğ‘Ž)∧𝑅(ğ‘Ž,𝑏)≥𝑝⟹𝑅(ğ‘Ž,𝑏)∧𝑅(𝑏,ğ‘Ž)â‰¥ğ‘âŸ¹ğ‘Žâˆ—ğ‘…ğ‘Žî€¸î€·(𝑏)â‰¥ğ‘âŸ¹ğ‘âˆˆğ‘Žâˆ—ğ‘…ğ‘Žî€¸ğ‘.(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 𝑥∈𝐻, 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 0≤𝑝≤0.1.

Proposition 4.3. Letting ⟨𝐻,∗𝑅⟩ be a partial F-C-hypergroupoid defined on 𝐻≠∅, 𝑅 is p-fuzzy reflexive. Then, for all ğ‘Ž,𝑏∈𝐻, 𝑝∈(0,1], the following are equivalent:(1)𝑅(ğ‘Ž,𝑏)≥𝑝;(2)ğ‘Žâˆˆ(ğ‘Žâˆ—ğ‘…ğ‘)𝑝;(3)𝑏∈(ğ‘Žâˆ—ğ‘…ğ‘)𝑝.

Proof. “(1)⇒(2)”
From 𝑅(ğ‘Ž,ğ‘Ž)≥𝑝 and 𝑅(ğ‘Ž,𝑏)≥𝑝 we have that 𝑅(ğ‘Ž,ğ‘Ž)∧𝑅(ğ‘Ž,𝑏)≥𝑝 which shows that ğ‘Žâˆˆ(ğ‘Žâˆ—ğ‘…ğ‘)𝑝.
“(2)⇒(3)”
From ğ‘Žâˆˆ(ğ‘Žâˆ—ğ‘…ğ‘)𝑝 we have that 𝑅(ğ‘Ž,𝑏)≥𝑝. Since 𝑅(𝑏,𝑏)≥𝑝, so 𝑅(ğ‘Ž,𝑏)∧𝑅(𝑏,𝑏)≥𝑝 which implies that 𝑏∈(ğ‘Žâˆ—ğ‘…ğ‘)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 ğ‘Žâˆˆğ», we have that î€·ğ‘Žâˆˆğ‘Žâˆ—ğ‘…ğ‘Žî€¸ğ‘.(4.2)

Proof. From 𝑅(ğ‘Ž,ğ‘Ž)≥𝑝 we have 𝑅(ğ‘Ž,ğ‘Ž)∧𝑅(ğ‘Ž,ğ‘Ž)≥𝑝. That is ğ‘Žâˆˆ(ğ‘Žâˆ—ğ‘…ğ‘Ž)𝑝.

Proposition 4.5. Letting ⟨𝐻,∗𝑅⟩ be a partial F-C-hypergroupoid defined on 𝐻≠∅, 𝑅 is p-fuzzy reflexive. Then, for any ğ‘Ž,𝑏∈𝐻, 𝑝∈(0,1], we have that î€·ğ‘âˆˆğ‘Žâˆ—ğ‘…ğ‘Žî€¸ğ‘î€·âŸºğ‘Žâˆˆğ‘Žâˆ—ğ‘…ğ‘âˆ©ğ‘âˆ—ğ‘…ğ‘Žî€¸ğ‘.(4.3)

Proof. From 𝑏∈(ğ‘Žâˆ—ğ‘…ğ‘Ž)𝑝 we have that 𝑅(ğ‘Ž,𝑏)∧𝑅(𝑏,ğ‘Ž)≥𝑝. So 𝑅(ğ‘Ž,𝑏)≥𝑝 and 𝑅(𝑏,ğ‘Ž)≥𝑝. Thus 𝑅(ğ‘Ž,ğ‘Ž)∧𝑅(ğ‘Ž,𝑏)≥𝑝 and 𝑅(𝑏,ğ‘Ž)∧𝑅(ğ‘Ž,ğ‘Ž)≥𝑝. That is (ğ‘Žâˆ—ğ‘…ğ‘)(ğ‘Ž)≥𝑝 and (ğ‘âˆ—ğ‘…ğ‘Ž)(ğ‘Ž)≥𝑝. So (ğ‘Žâˆ—ğ‘…ğ‘âˆ©ğ‘âˆ—ğ‘…ğ‘Ž)(ğ‘Ž)≥𝑝. Thus ğ‘Žâˆˆ(ğ‘Žâˆ—ğ‘…ğ‘âˆ©ğ‘âˆ—ğ‘…ğ‘Ž)𝑝.
Conversely, suppose that ğ‘Žâˆˆ(ğ‘Žâˆ—ğ‘…ğ‘âˆ©ğ‘âˆ—ğ‘…ğ‘Ž)𝑝. Then (ğ‘Žâˆ—ğ‘…ğ‘)(ğ‘Ž)∧(ğ‘âˆ—ğ‘…ğ‘Ž)(ğ‘Ž)≥𝑝. Thus 𝑅(ğ‘Ž,ğ‘Ž)∧𝑅(ğ‘Ž,𝑏)∧𝑅(𝑏,ğ‘Ž)∧𝑅(ğ‘Ž,ğ‘Ž)≥𝑝. So 𝑅(ğ‘Ž,𝑏)∧𝑅(𝑏,ğ‘Ž)≥𝑝. That is 𝑏∈(ğ‘Žâˆ—ğ‘…ğ‘Ž)𝑝.

Corollary 4.6. Letting ⟨𝐻,∗𝑅⟩ be a partial F-C-hypergroupoid defined on 𝐻≠∅, 𝑅 is p-fuzzy reflexive. Then, for any ğ‘Ž,𝑏∈𝐻, 𝑝∈(0,1], we have that î€·ğ‘âˆˆğ‘Žâˆ—ğ‘…ğ‘Žî€¸ğ‘î€·âŸºğ‘Žâˆˆğ‘âˆ—ğ‘…ğ‘î€¸ğ‘î€·âŸºğ‘Žâˆˆğ‘Žâˆ—ğ‘…ğ‘âˆ©ğ‘âˆ—ğ‘…ğ‘Žî€¸ğ‘î€·âŸºğ‘âˆˆğ‘Žâˆ—ğ‘…ğ‘âˆ©ğ‘âˆ—ğ‘…ğ‘Žî€¸ğ‘.(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 ğ‘Ž,𝑏∈𝐻, we have that î€·ğ‘âˆˆğ‘Žâˆ—ğ‘…ğ‘î€¸ğ‘î€·âŸºğ‘âˆˆğ‘Žâˆ—ğ‘…ğ‘âˆ©ğ‘âˆ—ğ‘…ğ‘î€¸ğ‘.(4.5)

Proof. If 𝑐∈(ğ‘Žâˆ—ğ‘…ğ‘)𝑝, then 𝑅(ğ‘Ž,𝑐)≥𝑝 and 𝑅(𝑐,𝑏)≥𝑝. Thus 𝑐∈(ğ‘Žâˆ—ğ‘…ğ‘)𝑝 and 𝑐∈(𝑐∗𝑅𝑏)𝑝. So 𝑐∈(ğ‘Žâˆ—ğ‘…ğ‘âˆ©ğ‘âˆ—ğ‘…ğ‘)𝑝.
Conversely, if 𝑐∈(ğ‘Žâˆ—ğ‘…ğ‘âˆ©ğ‘âˆ—ğ‘…ğ‘)𝑝, then (ğ‘Žâˆ—ğ‘…ğ‘)(𝑐)∧(𝑐∗𝑅𝑏)(𝑐)≥𝑝. Thus 𝑅(ğ‘Ž,𝑐)∧𝑅(𝑐,𝑐)∧𝑅(𝑐,𝑐)∧𝑅(𝑐,𝑏)≥𝑝. And so 𝑅(ğ‘Ž,𝑐)∧𝑅(𝑐,𝑏)≥𝑝. Thus 𝑐∈(ğ‘Žâˆ—ğ‘…ğ‘)𝑝.

Proposition 4.8. Letting ⟨𝐻,∗𝑅⟩ be a partial F-C-hypergroupoid defined on 𝐻≠∅, 𝑅 is p-fuzzy reflexive. Then, for any ğ‘Ž,𝑏,𝑐∈𝐻, 𝑝∈(0,1], the following are equivalent:(1)𝑐∈(ğ‘Žâˆ—ğ‘…ğ‘)𝑝;(2)ğ‘Žâˆˆ(ğ‘Žâˆ—ğ‘…ğ‘)𝑝 and 𝑏∈(𝑐∗𝑅𝑏)𝑝;(3)ğ‘Žâˆˆ(ğ‘Žâˆ—ğ‘…ğ‘)𝑝 and 𝑐∈(𝑐∗𝑅𝑏)𝑝.

Proof. “(1)⇒(2)”
Suppose that 𝑐∈(ğ‘Žâˆ—ğ‘…ğ‘)𝑝. Then 𝑅(ğ‘Ž,𝑐)≥𝑝 and 𝑅(𝑐,b)≥𝑝. So 𝑅(ğ‘Ž,ğ‘Ž)∧𝑅(ğ‘Ž,𝑐)≥𝑝 and 𝑅(𝑐,𝑏)∧𝑅(𝑏,𝑏)≥𝑝. Thus ğ‘Žâˆˆ(ğ‘Žâˆ—ğ‘…ğ‘)𝑝 and 𝑏∈(𝑐∗𝑅𝑏)𝑝.
“(2)⇒(3)”
Suppose that 𝑏∈(𝑐∗𝑅𝑏)𝑝. Then 𝑅(𝑐,𝑏)≥𝑝. Thus 𝑅(𝑐,𝑐)∧𝑅(𝑐,𝑏)≥𝑝. And so 𝑐∈(𝑐∗𝑅𝑏)𝑝.
“(3)⇒(1)”
From ğ‘Žâˆˆ(ğ‘Žâˆ—ğ‘…ğ‘)𝑝 and 𝑐∈(𝑐∗𝑅𝑏)𝑝, we have that 𝑅(ğ‘Ž,𝑐)≥𝑝 and 𝑅(𝑐,𝑏)≥𝑝. Thus 𝑅(ğ‘Ž,𝑐)∧𝑅(𝑐,𝑏)≥𝑝. So 𝑐∈(ğ‘Žâˆ—ğ‘…ğ‘)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,𝑦∈𝐻, 𝑅(𝑥,𝑦)≥𝑝⟹𝑅(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 0≤𝑝≤0.2.

Proposition 5.3. Letting ⟨𝐻,∗𝑅⟩ be a partial F-C-hypergroupoid defined on 𝐻≠∅, 𝑅 is p-fuzzy symmetric relation. Then, for all ğ‘Ž,𝑏∈𝐻, we have that î€·ğ‘Žâˆ—ğ‘…ğ‘î€¸ğ‘=î€·ğ‘âˆ—ğ‘…ğ‘Žî€¸ğ‘.(5.2)

Proof. For all ğ‘Ž,𝑏∈𝐻, two cases are possible.(1) If (ğ‘Žâˆ—ğ‘…ğ‘)𝑝=∅, then (ğ‘Žâˆ—ğ‘…ğ‘)𝑝⊆(ğ‘âˆ—ğ‘…ğ‘Ž)𝑝.(2) If (ğ‘Žâˆ—ğ‘…ğ‘)𝑝≠∅, let 𝑥∈(ğ‘Žâˆ—ğ‘…ğ‘)𝑝. Then 𝑅(ğ‘Ž,𝑥)≥𝑝 and 𝑅(𝑥,𝑏)≥𝑝. Since 𝑅 is p-fuzzy symmetric, so 𝑅(𝑥,ğ‘Ž)≥𝑝 and 𝑅(𝑏,𝑥)≥𝑝. Thus (ğ‘âˆ—ğ‘…ğ‘Ž)(𝑥)=𝑅(𝑏,𝑥)∧𝑅(𝑥,ğ‘Ž)≥𝑝. So 𝑥∈(ğ‘âˆ—ğ‘…ğ‘Ž)𝑝. And in this case, we also have that (ğ‘Žâˆ—ğ‘…ğ‘)𝑝⊆(ğ‘âˆ—ğ‘…ğ‘Ž)𝑝.
The remaining part can be proved by exchanging ğ‘Ž and 𝑏.

Proposition 5.4. Let ⟨𝐻,∗𝑅⟩ be a partial F-C-hypergroupoid defined on 𝐻≠∅, 𝑝∈(0,1], if(1) for all ğ‘Ž,𝑏∈𝐻, (ğ‘Žâˆ—ğ‘…ğ‘)𝑝=(ğ‘âˆ—ğ‘…ğ‘Ž)𝑝,(2) for any 𝑥∈H, there exists a 𝑦∈𝐻, such that 𝑅(𝑥,𝑦)≥𝑝. Then 𝑅 is a p-fuzzy symmetric binary relation on 𝐻.

Proof. For all ğ‘Ž,𝑏∈𝐻, suppose that 𝑅(ğ‘Ž,𝑏)≥𝑝. We need to show that 𝑅(𝑏,ğ‘Ž)≥𝑝.
Since for 𝑏∈𝐻, there exists a 𝑥∈𝐻, such that 𝑅(𝑏,𝑥)≥𝑝. So 𝑅(ğ‘Ž,𝑏)∧𝑅(𝑏,𝑥)≥𝑝. That is, 𝑏∈(ğ‘Žâˆ—ğ‘…ğ‘¥)𝑝=(ğ‘¥âˆ—ğ‘…ğ‘Ž)𝑝. And so 𝑅(𝑥,𝑏)∧𝑅(𝑏,ğ‘Ž)≥𝑝. And finally we have that 𝑅(𝑏,ğ‘Ž)≥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,𝑧∈𝐻, 𝑅(𝑥,𝑦)≥𝑝,𝑅(𝑦,𝑧)≥𝑝⟹𝑅(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 0≤𝑝≤0.1.

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,𝑦∈𝐻, we have that 𝑅(𝑥,𝑦)≥𝑝⟹𝑥∗𝑅𝑥∪𝑦∗𝑅𝑦𝑝⊆𝑥∗𝑅𝑦𝑝.(6.2)

Proof. (1) If (𝑥∗𝑅𝑥)𝑝=∅, then obviously (𝑥∗𝑅𝑥)𝑝⊆(𝑥∗𝑅𝑦)𝑝.
Supposing that (𝑥∗𝑅𝑥)𝑝≠∅, then for any 𝑤∈(𝑥∗𝑅𝑥)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∈(𝑥∗𝑅𝑦)𝑝.
So (𝑥∗𝑅𝑥)𝑝⊆(𝑥∗𝑅𝑦)𝑝.
(2) If (𝑦∗𝑅𝑦)𝑝=∅, then obviously (𝑦∗𝑅𝑦)𝑝⊆(𝑥∗𝑅𝑦)𝑝.
Supposing that (𝑦∗𝑅𝑦)𝑝≠∅, then for any 𝑤∈(𝑦∗𝑅𝑦)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∈(𝑥∗𝑅𝑦)𝑝.
So (𝑦∗𝑅𝑦)𝑝⊆(𝑥∗𝑅𝑦)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 ğ‘Ž,𝑏,𝑐∈𝐻, we have that(1)((ğ‘Žâˆ—ğ‘…ğ‘)𝑝∗𝑅𝑐)𝑝⊆(ğ‘Žâˆ—ğ‘…ğ‘)𝑝;(2)(ğ‘Žâˆ—ğ‘…(𝑏∗𝑅𝑐)𝑝)𝑝⊆(ğ‘Žâˆ—ğ‘…ğ‘)𝑝.

Proof. (1) If ((ğ‘Žâˆ—ğ‘…ğ‘)𝑝∗𝑅𝑐)𝑝=∅, then it is obvious that ((ğ‘Žâˆ—ğ‘…ğ‘)𝑝∗𝑅𝑐)𝑝⊆(ğ‘Žâˆ—ğ‘…ğ‘)𝑝.
Suppose that ((ğ‘Žâˆ—ğ‘…ğ‘)𝑝∗𝑅𝑐)𝑝≠∅. Then for any 𝑤∈((ğ‘Žâˆ—ğ‘…ğ‘)𝑝∗𝑅𝑐)𝑝, there exists a 𝑤1∈(ğ‘Žâˆ—ğ‘…ğ‘)𝑝 such that 𝑤∈(𝑤1∗𝑅𝑐)𝑝. That is 𝑅(ğ‘Ž,𝑤1)≥𝑝, 𝑅(𝑤1,𝑏)≥𝑝, 𝑅(𝑤1,𝑤)≥𝑝 and 𝑅(𝑤,𝑐)≥𝑝. From 𝑅(ğ‘Ž,𝑤1)≥𝑝 and 𝑅(𝑤1,𝑤)≥𝑝, we have that 𝑅(ğ‘Ž,𝑤)≥𝑝. Thus 𝑅(ğ‘Ž,𝑤)∧𝑅(𝑤,𝑐)≥𝑝∧𝑝=𝑝. That is, 𝑤∈(ğ‘Žâˆ—ğ‘…ğ‘)𝑝. So ((ğ‘Žâˆ—ğ‘…ğ‘)𝑝∗𝑅𝑐)𝑝⊆(ğ‘Žâˆ—ğ‘…ğ‘)𝑝.
(2) Can be proved similarly.

Acknowledgment

The paper is partially supported by CSC.

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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.


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