On Fuzzy Corsini's Hyperoperations

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=∪{𝑎∗𝑏∣𝑎∈𝐴,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 𝐴_𝑝, for any 𝑝∈(0,1], and defined by 𝐴_𝑝≐{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 𝑎,𝑏∈[0,1], we use 𝑎∧𝑏 to stand for the minimum of 𝑎 and 𝑏 and 𝑎∨𝑏 to denote the maximum of 𝑎 and 𝑏.

Given 𝐴,𝐵∈𝐹(𝐻), we will use the following definitions: ((𝐴𝐴𝐴𝐴∪∩⊆=𝐵𝐵𝐵𝐵))((𝑥𝑥≐≐))𝐴𝐴≐≐((𝐴𝐴𝑥𝑥))((𝑥𝑥≤=))𝐵𝐵∨∧𝐵𝐵((𝑥𝑥(())𝑥𝑥,,)),,∀∀𝑥𝑥∀∀∈∈𝑥𝑥∈∈𝐻𝐻𝐻𝐻,,,.(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 𝑎∈𝐻, 𝐵∈𝐹(𝐻), the fuzzy subset 𝑎∗𝐵 of 𝐻 is defined by (𝑎∗𝐵)(𝑥)≐∨_𝐵(𝑏)>0(𝑎∗𝑏)(𝑥).(1.3)

𝐵∗𝑎, 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={𝑎,𝑏} be a non-empty set, 𝑅 is a fuzzy relation on 𝐻 as described in Table 1.

Table 1 

From the previous definition, by calculating, for example, (𝑎∗_𝑅𝑎)(𝑎)=𝑅(𝑎,𝑎)∧𝑅(𝑎,𝑎)=0.1∧0.1=0.1, 𝑅(𝑎∗𝑏)(𝑎)=𝑅(𝑎,𝑎)∧𝑅(𝑎,𝑏)=0.1∧0.2=0.1, we can obtain Table 2 which is a partial F-C-hypergroupoid.

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 (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 𝑅_𝑝 on 𝐻, for all 𝑝∈(0,1], as follows: 𝑥𝑅_𝑝𝑦≐𝑅(𝑥,𝑦)≥𝑝.(2.2)𝑅_𝑝 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 𝑅_𝑝, for all 0<𝑝≤1.

Proof. For any 0<𝑝≤1 and for any 𝑥,𝑦∈𝐻, we have 𝑥∗_𝑅_𝑝𝑦=={𝑥𝑧∗∈_𝑅𝐻𝑦∣_𝑝𝑅=(𝑥𝑧,𝑧∈)≥𝐻𝑝∣,𝑅𝑥(𝑧∗,_𝑅𝑦𝑦)≥(𝑧𝑝)}≥=𝑝𝑧=∈{𝐻𝑧∈∣𝑥𝐻𝑅_𝑝∣𝑅(𝑥,𝑧)∧𝑅(𝑧,𝑦)≥𝑝}𝑧,𝑧𝑅_𝑝𝑦.(2.3)
From the definition of C-hyperoperation, we conclude that (∗_𝑅)_𝑝 is a C-hyperoperation associated with 𝑅_𝑝.
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 𝑅_𝑝, 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,𝑎,𝑏∈𝐻, we have 𝑥∗_𝑅𝑦∩𝑎∗_𝑅𝑏=𝑥∗_𝑅𝑏∩𝑎∗_𝑅𝑦.(3.1)

Proof. For any 𝑥,𝑦,𝑎,𝑏,𝑧∈𝐻, we have that (𝑥∗_𝑅𝑦∩𝑎∗_𝑅𝑏)(𝑧)=(𝑥∗_𝑅𝑦)(𝑧)∧(𝑎∗_𝑅𝑏)(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)=(𝑥∗_𝑅𝑏∩𝑎∗_𝑅𝑦)(𝑧).
So 𝑥∗_𝑅𝑦∩𝑎∗_𝑅𝑏=𝑥∗_𝑅𝑏∩𝑎∗_𝑅𝑦,(3.2) for all 𝑥,𝑦,𝑎,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 𝑎,𝑏∈𝐻, such that 𝑥∗_𝑅𝑎∩𝑏∗_𝑅𝑦≠∅. So from the previous proposition, we have 𝑥∗_𝑅𝑦∩𝑏∗_𝑅𝑎≠∅. 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 𝑎∈𝐻, (𝑎∗_𝑅𝑎)_𝑝=𝐻;(2) for all 𝑥,𝑦∈𝐻, 𝑎∈(𝑥∗_𝑅𝑦)_𝑝.

Proof. Let (𝑎∗_𝑅𝑎)_𝑝=𝐻. Then, for all 𝑥,𝑦∈𝐻, we have that (𝑎∗_𝑅𝑎)(𝑥)≥𝑝,(𝑎∗_𝑅𝑎)(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∈(𝑥∗_𝑅𝑦)_𝑝, for all 𝑥,𝑦∈𝐻.
Conversely, let 𝑎∈(𝑥∗_𝑅𝑦)_𝑝, for all 𝑥,𝑦∈𝐻. Specially, we have 𝑎∈(𝑎∗_𝑅𝑥)_𝑝 and 𝑎∈(𝑥∗_𝑅𝑎)_𝑝. Thus, 𝑅(𝑎,𝑥)≥𝑝 and 𝑅(𝑥,𝑎)≥𝑝. And so 𝑥∈(𝑎∗_𝑅𝑎)_𝑝.

Proposition 3.4. Let ⟨𝐻,∗_𝑅⟩ be a partial or nonpartial F-C-hypergroupoid defined on 𝐻≠∅. Then, for all 𝑎,𝑏∈𝐻, 𝑝∈(0,1], we have 𝑎∈𝑏∗_𝑅𝑏_𝑝⟺𝑏∈𝑎∗_𝑅𝑎_𝑝.(3.3)

Proof. For any 𝑎,𝑏∈𝐻, we have that 𝑎∈𝑏∗_𝑅𝑏_𝑝⟹⟹𝑅𝑏(𝑎∗,_𝑅𝑏𝑏)∧(𝑎𝑅)(≥𝑏,𝑝𝑎)⟹≥𝑝𝑅⟹(𝑏,𝑎)𝑎∧∗_𝑅𝑅(𝑎,𝑏)≥𝑝𝑎(𝑏)≥𝑝⟹𝑏∈𝑎∗_𝑅𝑎_𝑝.(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 𝑎,𝑏∈𝐻, 𝑝∈(0,1], the following are equivalent:(1)𝑅(𝑎,𝑏)≥𝑝;(2)𝑎∈(𝑎∗_𝑅𝑏)_𝑝;(3)𝑏∈(𝑎∗_𝑅𝑏)_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∈(𝑎∗_𝑅𝑏)_𝑝.
“(2)⇒(3)”
From 𝑎∈(𝑎∗_𝑅𝑏)_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∈(𝑎∗_𝑅𝑏)_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 𝑎∈𝐻, we have that 𝑎∈𝑎∗_𝑅𝑎_𝑝.(4.2)

Proof. From 𝑅(𝑎,𝑎)≥𝑝 we have 𝑅(𝑎,𝑎)∧𝑅(𝑎,𝑎)≥𝑝. That is 𝑎∈(𝑎∗_𝑅𝑎)_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,𝑏∈𝐻, 𝑝∈(0,1], we have that 𝑏∈𝑎∗_𝑅𝑎_𝑝⟺𝑎∈𝑎∗_𝑅𝑏∩𝑏∗_𝑅𝑎_𝑝.(4.3)

Proof. From 𝑏∈(𝑎∗_𝑅𝑎)_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 (𝑎∗_𝑅𝑏)(𝑎)≥𝑝 and (𝑏∗_𝑅𝑎)(𝑎)≥𝑝. So (𝑎∗_𝑅𝑏∩𝑏∗_𝑅𝑎)(𝑎)≥𝑝. Thus 𝑎∈(𝑎∗_𝑅𝑏∩𝑏∗_𝑅𝑎)_𝑝.
Conversely, suppose that 𝑎∈(𝑎∗_𝑅𝑏∩𝑏∗_𝑅𝑎)_𝑝. Then (𝑎∗_𝑅𝑏)(𝑎)∧(𝑏∗_𝑅𝑎)(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∈(𝑎∗_𝑅𝑎)_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,𝑏∈𝐻, 𝑝∈(0,1], we have that 𝑏∈𝑎∗_𝑅𝑎_𝑝⟺𝑎∈𝑏∗_𝑅𝑏_𝑝⟺𝑎∈𝑎∗_𝑅𝑏∩𝑏∗_𝑅𝑎_𝑝⟺𝑏∈𝑎∗_𝑅𝑏∩𝑏∗_𝑅𝑎_𝑝.(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,𝑏∈𝐻, we have that 𝑐∈𝑎∗_𝑅𝑏_𝑝⟺𝑐∈𝑎∗_𝑅𝑐∩𝑐∗_𝑅𝑏_𝑝.(4.5)

Proof. If 𝑐∈(𝑎∗_𝑅𝑏)_𝑝, then 𝑅(𝑎,𝑐)≥𝑝 and 𝑅(𝑐,𝑏)≥𝑝. Thus 𝑐∈(𝑎∗_𝑅𝑐)_𝑝 and 𝑐∈(𝑐∗_𝑅𝑏)_𝑝. So 𝑐∈(𝑎∗_𝑅𝑐∩𝑐∗_𝑅𝑏)_𝑝.
Conversely, if 𝑐∈(𝑎∗_𝑅𝑐∩𝑐∗_𝑅𝑏)_𝑝, then (𝑎∗_𝑅𝑐)(𝑐)∧(𝑐∗_𝑅𝑏)(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∈(𝑎∗_𝑅𝑏)_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,𝑐∈𝐻, 𝑝∈(0,1], the following are equivalent:(1)𝑐∈(𝑎∗_𝑅𝑏)_𝑝;(2)𝑎∈(𝑎∗_𝑅𝑐)_𝑝 and 𝑏∈(𝑐∗_𝑅𝑏)_𝑝;(3)𝑎∈(𝑎∗_𝑅𝑐)_𝑝 and 𝑐∈(𝑐∗_𝑅𝑏)_𝑝.

Proof. “(1)⇒(2)”
Suppose that 𝑐∈(𝑎∗_𝑅𝑏)_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∈(𝑎∗_𝑅𝑐)_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 𝑎∈(𝑎∗_𝑅𝑐)_𝑝 and 𝑐∈(𝑐∗_𝑅𝑏)_𝑝, we have that 𝑅(𝑎,𝑐)≥𝑝 and 𝑅(𝑐,𝑏)≥𝑝. Thus 𝑅(𝑎,𝑐)∧𝑅(𝑐,𝑏)≥𝑝. So 𝑐∈(𝑎∗_𝑅𝑏)_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 𝑎,𝑏∈𝐻, we have that 𝑎∗_𝑅𝑏_𝑝=𝑏∗_𝑅𝑎_𝑝.(5.2)

Proof. For all 𝑎,𝑏∈𝐻, two cases are possible.(1) If (𝑎∗_𝑅𝑏)_𝑝=∅, then (𝑎∗_𝑅𝑏)_𝑝⊆(𝑏∗_𝑅𝑎)_𝑝.(2) If (𝑎∗_𝑅𝑏)_𝑝≠∅, let 𝑥∈(𝑎∗_𝑅𝑏)_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 (𝑏∗_𝑅𝑎)(𝑥)=𝑅(𝑏,𝑥)∧𝑅(𝑥,𝑎)≥𝑝. So 𝑥∈(𝑏∗_𝑅𝑎)_𝑝. And in this case, we also have that (𝑎∗_𝑅𝑏)_𝑝⊆(𝑏∗_𝑅𝑎)_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,𝑏∈𝐻, (𝑎∗_𝑅𝑏)_𝑝=(𝑏∗_𝑅𝑎)_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 𝑎,𝑏∈𝐻, suppose that 𝑅(𝑎,𝑏)≥𝑝. We need to show that 𝑅(𝑏,𝑎)≥𝑝.
Since for 𝑏∈𝐻, there exists a 𝑥∈𝐻, such that 𝑅(𝑏,𝑥)≥𝑝. So 𝑅(𝑎,𝑏)∧𝑅(𝑏,𝑥)≥𝑝. That is, 𝑏∈(𝑎∗_𝑅𝑥)_𝑝=(𝑥∗_𝑅𝑎)_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,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 𝑎,𝑏,𝑐∈𝐻, we have that(1)((𝑎∗_𝑅𝑏)_𝑝∗_𝑅𝑐)_𝑝⊆(𝑎∗_𝑅𝑐)_𝑝;(2)(𝑎∗_𝑅(𝑏∗_𝑅𝑐)_𝑝)_𝑝⊆(𝑎∗_𝑅𝑐)_𝑝.

Proof. (1) If ((𝑎∗_𝑅𝑏)_𝑝∗_𝑅𝑐)_𝑝=∅, then it is obvious that ((𝑎∗_𝑅𝑏)_𝑝∗_𝑅𝑐)_𝑝⊆(𝑎∗_𝑅𝑐)_𝑝.
Suppose that ((𝑎∗_𝑅𝑏)_𝑝∗_𝑅𝑐)_𝑝≠∅. Then for any 𝑤∈((𝑎∗_𝑅𝑏)_𝑝∗_𝑅𝑐)_𝑝, there exists a 𝑤₁∈(𝑎∗_𝑅𝑏)_𝑝 such that 𝑤∈(𝑤₁∗_𝑅𝑐)_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)≥𝑝∧𝑝=𝑝. That is, 𝑤∈(𝑎∗_𝑅𝑐)_𝑝. So ((𝑎∗_𝑅𝑏)_𝑝∗_𝑅𝑐)_𝑝⊆(𝑎∗_𝑅𝑐)_𝑝.
(2) Can be proved similarly.

Acknowledgment

The paper is partially supported by CSC.