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.

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.

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.