Certain fuzzy hyperstructures from a fuzzy set

Abstract

Fuzzy set theory and also the hypergroups in the sense of Marty are both generalizations of some existing mathematical concepts which are used for modeling many real life situations.

The main purpose of this paper is the study of the link between fuzzy sets and fuzzy hypergroups and fuzzy semihypergroups. As a matter of fact, some commutative fuzzy hypergroups and fuzzy semihypergroups have been constructed from fuzzy set and some of their properties were investigated.

Keywords

Hypergroup fuzzy sets fuzzy hypergroup

1 Introduction

In 1965, L. Zadeh [1] introduced fuzzy set theory. Also, the concept of hypergroups was introduced in 1934 by F. Marty [2]. Both concepts are generalizations of some existing mathematical concepts which are used for modeling many real life situations. These have caught the attention of many researchers who are now working on them in different directions. Some interaction between them are also occurring in real life. As a matter of fact, in 1994, Dib [3] introduced the concept of fuzzy space, which corresponds to the concept of the universal set in the ordinary case and used them to define the fuzzy group.

Following the idea of Dib, Davvaz [4] studied the concept of fuzzy hypergroup, which depends on the concept of a fuzzy space and constructed fuzzy hypergroup from the fuzzy space. The hypergroup so constructed give a fuzzy hypergroup which is not from a fuzzy membership function of a set. In 2014 Corsini [5] introduced the notion of developing hyperstructures from fuzzy multiset and showed that it is a join space. This approach was supported by the work of [6]. More studies on fuzzy hypergroups, occuring by defining a hyperoperation on a set or on a hypergroup, can be found in [7 –10]. Connection between a hypergroup and a fuzzy set can be found in [10 –13]. The approach in [14] is to define a fuzzy subhypergroup on a hypergroup as though a fuzzy subgroup is defined on a group.

The hyperstructure in this paper is different from that constructed by [4 , 7– 10] because it neither requires an underlying hypergroupoid nor a fuzzy space but can easily be constructed from any given fuzzy subset. The main purpose of this paper therefore is to construct hyperstructures which are different from those ever constructed and to study some of their properties.

2 Preliminaries

In this section, the preliminary notions that are required in the sequel shall be formulated. Let H be a non-empty set and $P^{*} (H)$ be the set of all non-empty subsets of H. Then, the map $★ : H \times H ⟶ P^{*} (H)$ , where (x, y) ↦ x ★ y ⊆ H is called a hyperoperation and the couple (H, ★) is called a hypergroupoid.

For any two non-empty subsets A and B of H and x ∈ H, we define

A ★ B = ⋃ _a∈A,b∈Ba ★ b, A ★ x = A ★ {x} and x ★ B = {x} ★ B.

A hypergroupoid (H, ★) is called a semihypergroup if for all elements a, b, c of H we have (a ★ b) ★ c = a ★ (b ★ c).

A hypergroupoid (H, ★) is called a quasihypergroup if for all a ∈ H we have a ★ H = H ★ a = H.

A hypergroupoid (H, ★) which is both a semihypergroup and a quasihypergroup is called a hypergroup.

Definition 2.1. [2] Let (G, ∘) be a hypergroup. An element e ∈ G is called an identity if for all a ∈ G, a ∈ a ∘ e ∩ e ∘ a.

Definition 2.2. [15] Let (H, ★) be a hypergroup. Then it is canonical hypergroup if

it is commutative,

it has a scalar identity (also called scalar unit), which means that ∃e ∈ H, ∀x ∈ H, x ★ e = e ★ x = {x},

every element has a unique inverse, which means that for all x ∈ H, there exists a unique x^-1 ∈ H, such that e ∈ x ★ x^-1 ∩ x^-1 ★ x,

it is reversible, which means that if x ∈ y ★ z, then there exist the inverses y^-1 of y and z^-1 of z, such that z ∈ y^-1 ★ x and y ∈ x ★ z^-1.

Note that the identity of a canonical hypergroup is unique.

Definition 2.3. [1] Let X be a nonempty set. A Fuzzy subset A of a nonempty set X is characterized by the membership function μ_A: X →[0, 1] as A = {(x, μ_A (x)) : x ∈ X}.

Definition 2.4. [1] Let μ and ν be fuzzy sets on X. Then μ is said to be a subset of ν and is denoted μ ⊆ ν if μ (x) ≤ ν (x) for all x ∈ X.

Definition 2.5. [1] Let μ be a fuzzy subset of X. Then μ is said to be a an empty fuzzy set if μ (x) =0 ∀ x ∈ X.

Definition 2.6. [16] A multiset M drawn from the set X is represented by a count function C_M : X → N, where N represents the set of non-negative integers. C_M (x) is the number of occurrence of the element x in the multiset M. The multiset M drawn from X = {x₁, x₂, . . . , x_n} is represented by M = {x₁/m₁, x₂/m₂, ⋯ , x_n/m_n} where m_i is the number of occurrence of the element x_i, (i = 1, 2, ⋯ , n) in the multiset M.

Definition 2.7. [5] Given a fuzzy multiset $(H, A)$ , where $A$ is a function $A : H \to I \times I N$ such that for any $x \in H$ there is the pair (μ (x) , m (x)), where μ (x) ∈ [0, 1] and m (x) ∈ I N. For any $x, y, z \in H$ , define hyperoperations:

x ⨂ _μy = {z : min {μ (x) , μ (y)} ≤ μ (z) ≤ max {μ (x) , μ (y)}};

x ø _my = {z : min {m (x) , m (y)} ≤ m (z) ≤ max {m (x) , m (y)}};

x_m ⊠ _μy = x ⨂ _μy ∩ x ø _my.

Then,

(H, ⨂_{μ})

(H, ø_{m})

and

(H,_{m} ⊠_{μ})

are hypergroupoids.

3 Fuzzy hypergroups

In this section, let X be a finite nonempty set and μ : X → [0, 1] be a fuzzy membership function on X. $P (X \times [0, 1])$ is the family {(x, μ (x)) |x ∈ X and μ (x) ∈ [0, 1]} such that for all $α \in P^{*} (X \times [0, 1])$ , α (x) ≠0 and α (x) ≤ μ (x), for all x ∈ X. This is no doubt but all the nonempty fuzzy subsets of μ. Also, we can write X × I_μ for X × [0, 1] since the membership function from X to [0, 1] is μ.

Definition 3.1. Let X be a finite nonempty set and μ : X → [0, 1] be a fuzzy membership function on X. The composition $\circ : (X \times I_{μ}) \times (X \times I_{μ}) \to P^{*} (X \times I_{μ})$ is a fuzzy hyperoperation.

Definition 3.2. Let x, y ∈ X. With the notations in Definition 3.1, the following are fuzzy hyperoperations:

(x, μ (x)) ∘ (y, μ (y)) = {(z, μ (z)) |z ∈ X and μ (z) ≥ μ (x) ∧ μ (y)}

(x, μ (x)) ∘ (y, μ (y)) = {(z, μ (z)) |z ∈ X and μ (z) ≥ μ (x) ∨ μ (y)}

(x, μ (x)) ∘ (y, μ (y)) = {(z, μ (z)) |z ∈ X and μ (z) ≤ μ (x) ∧ μ (y)}

(x, μ (x)) ∘ (y, μ (y)) = {(z, μ (z)) |z ∈ X and μ (z) ≤ μ (x) ∨ μ (y)} .

Example 1. Let X = {a, b, c} and the membership function μ : X → [0, 1], such that μ (a) =0.2, μ (b) =0.6 and μ (c) =0.3. Applying the various types of hyperoperations, the following tables are obtained:

type I:

∘

(a, 0.2)

(b, 0.6)

(c, 0.3)

(a, 0.2)

{(a, 0.2) , (b, 0.6) , (c, 0.3)}

(b, 0.6)

{(a, 0.2) , (b, 0.6) , (c, 0.3)}

{(b, 0.6)}

{(b, 0.6) , (c, 0.3)}

(c, 0.3)

{(a, 0.2) , (b, 0.6) , (c, 0.3)}

{(b, 0.6) , (c, 0.3)}

type II:

∘

(a, 0.2)

(b, 0.6)

(c, 0.3)

(a, 0.2)

{(a, 0.2) , (b, 0.6) , (c, 0.3)}

{(b, 0.6)}

{(b, 0.6) , (c, 0.3)}

(b, 0.6)

{(b, 0.6)}

(c, 0.3)

{(b, 0.6) , (c, 0.3)}

{(b, 0.6)}

{(b, 0.6) , (c, 0.3)}

type III:

∘

(a, 0.2)

(b, 0.6)

(c, 0.3)

(a, 0.2)

{(a, 0.2)}

(b, 0.6)

{(a, 0.2)}

{(a, 0.2) , (b, 0.6) , (c, 0.3)}

{(a, 0.2) , (c, 0.3)}

(c, 0.3)

{(a, 0.2)}

{(a, 0.2) , (c, 0.3)}

type IV:

∘

(a, 0.2)

(b, 0.6)

(c, 0.3)

(a, 0.2)

{(a, 0.2)}

{(a, 0.2) , (b, 0.6) , (c, 0.3)}

{(a, 0.2) , (c, 0.3)}

(b, 0.6)

{(a, 0.2) , (b, 0.6) , (c, 0.3)}

(c, 0.3)

{(a, 0.2) , (c, 0.3)}

{(a, 0.2) , (b, 0.6) , (c, 0.3)}

{(a, 0.2) , (c, 0.3)}

Example 2. Let $X = ℤ \ {0}$ , where $ℤ$ is a set of integers. Let α, β ∈ [0, 1] be such that α < β and μ a fuzzy subset of X defined as

$μ (x) = {\begin{matrix} α, if x is even, \\ β, if x is odd . \end{matrix}$ (1)

Type I hyperoperation yeilds:

$\begin{matrix} (x, μ (x)) \circ (y, μ (y)) \\ = {\begin{matrix} X, if x or y (or both) is (are) even, \\ Odd integers, if both x and y are odd . \end{matrix} \end{matrix}$ (2)

Type II hyperoperation yeilds:

$\begin{matrix} (x, μ (x)) \circ (y, μ (y)) \\ = {\begin{matrix} Odd integers, if x or y (or both) is (are) odd, \\ X, if both x and y are even . \end{matrix} \end{matrix}$ (3)

Type III hyperoperation yeilds:

$\begin{matrix} (x, μ (x)) \circ (y, μ (y)) \\ = {\begin{matrix} Even integers, if x or y (or both) is (are) even, \\ X, if both x and y are odd . \end{matrix} \end{matrix}$ (4)

Type IV hyperoperation yeilds:

$\begin{matrix} (x, μ (x)) \circ (y, μ (y)) \\ = {\begin{matrix} X, if x or y (or both) is (are) odd, \\ Even integers, if both x and y are even . \end{matrix} \end{matrix}$ (5)

Theorem 3.3. Let μ : X → [0, 1] be a fuzzy subset of a nonempty set X. Then (X × [0, 1] , ∘) with respect to type I is a fuzzy hypergroup of type I.

Proof. For type I. Let x, y, z ∈ X. Without loss generality, let μ (x) ≤ μ (y). Three cases are possible: μ (x) ≤ μ (y) ≤ μ (z), μ (x) ≤ μ (z) ≤ μ (y) and μ (z) ≤ μ (x) ≤ μ (y). We prove the first case and the others can be done similarly.

Then, (x, μ (x)) ∘ (y, μ (y)) = {(u, μ (u)) |u ∈ X and μ (u) ≥ μ (x) ∧ μ (y)} (x, μ (x)) ∘ (y, μ (y)) = {(u, μ (u)) |u ∈ X and μ (u) ≥ μ (x)}.

Also, we have that (y, μ (y)) ∘ (z, μ (z)) = {(a, μ (a)) |a ∈ X and μ (a) ≥ μ (y) ∧ μ (z)}.

(y, μ (y)) ∘ (z, μ (z)) = {(a, μ (a)) |a ∈ X and μ (a) ≥ μ (y)}.

$\begin{matrix} (x, μ (x)) \circ (a, μ (a)) \\ = {(f, μ (f)) | f \in X and μ (f) \\ \geq μ (x) \land μ (a)} = {(f, μ (f)) | μ (f) \geq μ (x)} . \end{matrix}$ (6)

(u, μ (u)) ∘ (z, μ (z)) = {(v, μ (v)) |v ∈ X and μ (v) ≥ μ (u) ∧ μ (z)} = {(v, μ (v)) |μ (v) ≥ μ (x)}, where (u, μ (u)) is as in early part of showing (1). (2)

Using (1) and (2) we obtain that ((x, μ (x)) ∘ (y, μ (y))) ∘ (z, μ (z)) = (x, μ (x)) ∘ ((y, μ (y)) ∘ (z, μ (z))). Thus the associativity holds.

It is obvious that (x, μ (x)) ∘ (X × [0, 1]) = (X × [0, 1]) ∘ (x, μ (x)) = (X × [0, 1]).

Therefore, (X × [0, 1]) with the hyperoperation defined on μ by type I, is a fuzzy hypergroup of type I.

Theorem 3.4. Let μ : X → [0, 1] be a fuzzy subset of a nonempty set X. Then (X × [0, 1] , ∘) with respect to IV is a fuzzy hypergroup of IV.

Proof. The proof is similar to that of I.□

Remark 1. In Example 1, it can be observed that if, in Definition 2.7, a fuzzy multiset (which is a generalization of a fuzzy set) is replaced by a fuzzy set, then the hypergroupoids of type (i) of Corsini is not the same as the hypergroup of type I in this work. Hence, Corsini’s hyperstructures are different from those constructed and do not generalize the ones constructed in this paper. More precisely, applying type (i) of Corsini to that example, the following table is obtained.

type (i):

∘

(a, 0.2)

(b, 0.6)

(c, 0.3)

(a, 0.2)

{(a, 0.2) , (b, 0.6) , (c, 0.3)}

{(a, 0.2) , (c, 0.3)}

(b, 0.6)

{(a, 0.2) , (b, 0.6) , (c, 0.3)}

(b, 0.6)

{(b, 0.6) , (c, 0.3)}

(c, 0.3)

{(a, 0.2) , (c, 0.3)}

{(b, 0.6) , (c, 0.3)}

(c, 0.3)

Theorem 3.5. Let μ : X → [0, 1] be a fuzzy subset of a nonempty set X. Then (X × [0, 1] , ∘) with respect to type II is a fuzzy semihypergroup of type II.

Proof. For type II. Let x, y, z ∈ X. Without loss generality, let μ (x) ≤ μ (y). Three cases are possible: μ (x) ≤ μ (y) ≤ μ (z), μ (x) ≤ μ (z) ≤ μ (y) and μ (z) ≤ μ (x) ≤ μ (y). We prove the first case and the others can be done similarly.

Then, (x, μ (x)) ∘ (y, μ (y)) = {(u, μ (u)) |u ∈ X and μ (u) ≥ μ (x) ∨ μ (y)}

(x, μ (x)) ∘ (y, μ (y)) = {(u, μ (u)) |u ∈ X and μ (u) ≥ μ (y)}.

(y, μ (y)) ∘ (z, μ (z)) = {(a, μ (a)) |a ∈ X and μ (a) ≥ μ (y) ∨ μ (z)}.

(y, μ (y)) ∘ (z, μ (z)) = {(a, μ (a)) |a ∈ X and μ (a) ≥ μ (z)}.

(x, μ (x)) ∘ (a, μ (a)) = {(f, μ (f)) |f ∈ X and μ (f) ≥ μ (x) ∨ μ (a)} = {(f, μ (f)) |μ (f) ≥ μ (a) ≥ μ (z)}. (1)

(u, μ (u)) ∘ (z, μ (z)) = {(v, μ (v)) |v ∈ X and μ (v) ≥ μ (u) ∨ μ (z)} = {(v, μ (v)) |μ (v) ≥ μ (z)}. (2)

Using (1) and (2) we obtain that ((x, μ (x)) ∘ (y, μ (y))) ∘ (z, μ (z)) = (x, μ (x)) ∘ ((y, μ (y)) ∘ (z, μ (z))). Thus the associativity holds.

Therefore, (X × [0, 1]) in this case, with the hyperoperation defined on μ by type II, is a fuzzy semihypergroup of type II.□

Theorem 3.6. Let μ : X → [0, 1] be a fuzzy subset of a nonempty set X. Then (X × [0, 1] , ∘) with respect to III is a fuzzy semihypergroup of III.

Proof. The proof is similar to that of II.□

Remark 2. For II and III, (x, μ (x)) ∘ (X × [0, 1]) ≠ (X × [0, 1]) ∘ (x, μ (x)) ≠ (X × [0, 1]) as can be seen in Example 1.

The fuzzy hyperstructures defined in Definition 3.2 are commutative, since, if μ (x) ≤ μ (y) ≤ μ (z),

(x, μ (x)) ∘ (y, μ (y)) = {(a, μ (a)) |μ (a) ≥ μ (x) ∧ (y)} = {(a, μ (a)) |μ (a) ≥ μ (y) ∧ (x)} = (y, μ (y)) ∘ (x, μ (x)).

Theorem 3.7. If for the four types of fuzzy hyperoperations we associate the following hyperoperations:

x ∘ y = {z ∈ X|μ (z) ≥ μ (x) ∧ μ (y)}

x ∘ y = {z ∈ X|μ (z) ≥ μ (x) ∨ μ (y)}

x ∘ y = {z ∈ X|μ (z) ≤ μ (x) ∧ μ (y)}

x ∘ y = {z ∈ X|μ (z) ≤ μ (x) ∨ μ (y)},

which are not fuzzy, to each fuzzy hypergroup (fuzzy semihypergroup) (X × I_μ, ∘) of types I and IV (II and III) there is an associated hypergroup (semihypergroup) (X, ∘), such that the corresponding types are isomorphic.

Proof. The results of these non-fuzzy hyperoperations are hypergroups and semihypergroups as the case may be. This is further illustrated in Example 3.

□

Example 3. Refer to Example 1 and the following tables are obtained:

type I:

∘

{a, b, c}

{b, c}

{a, b, c}

{b, c}

type II:

∘

{a, b, c}

{b, c}

type III:

∘

{a, b, c}

{a, c}

type IV:

∘

{a, b, c}

{a, c}

{a, b, c}

{a, c}

{a, b, c}

{a, c}

Proposition 3.8. Define a relation on $P (X \times I_{μ})$ by λ ∼ ν ⇔ λ (x) ≤ ν (x) for $λ, ν \in P (X \times I_{μ})$ . Then, $(P (X \times I_{μ}), \sim)$ is a partially ordered set. Furthermore, for (x, μ (x)) , (y, μ (y)) ∈ (X × I_μ) such that (x, μ (x)) ∼ (y, μ (y)) ⇔ μ (x) ≤ μ (y), ((X × I_μ) , ∼) is also a partially ordered set.

Proof. This is left to the reader.□

Definition 3.9. Let ((X × I_μ) , ∼) be as described in Proposition 3.8. An element e is called the smallest (biggest) element in (X × I_μ) if μ (e) ≤ μ (x) (μ (e) ≥ μ (x)) ∀ x ∈ (X × I_μ) .

The following proposition gives a characterization of the identity of each fuzzy hyperstructure.

Proposition 3.10. Let (X × I_μ, ∘) be a fuzzy semihypergroup.

If (X × I_μ, ∘) is of type I or type IV, then all the elements are identity.

If (X × I_μ, ∘) is of type II, then the smallest element is the identity.

If (X × I_μ, ∘) is of type III, then the biggest element is the identity.

Proof.

Assume that (X × I_μ, ∘) is of type I.

Let (e, μ (e)) be any element of (X × I_μ, ∘). For x ∈ X, either μ (e) ≤ μ (x) or μ (x) ≤ μ (e) assume that μ (e) ≤ μ (x).

(x, μ (x)) ∘ (e, μ (e)) = {(z, μ (z)) |μ (z) ≥ μ (x) ∧ μ (e)} = {(z, μ (z)) |μ (z) ≥ μ (e)}. Thus (x, μ (x)) ∈ (x, μ (x)) ∘ (e, μ (e)).

Assume that μ (x) ≤ μ (e)

(x, μ (x)) ∘ (e, μ (e)) = {(z, μ (z)) |μ (z) ≥ μ (x) ∧ μ (e)} = {(z, μ (z)) |μ (z) ≥ μ (x)}. Thus (x, μ (x)) ∈ (x, μ (x)) ∘ (e, μ (e)).

Therefore (e, μ (e)) is an identity element of (X × I_μ, ∘).

For the type IV the proof is similar.

Assume that (X × I_μ, ∘) is of type II.

Let e ∈ X, such that (e, μ (e)) is the smallest element in (X × I_μ, ∘) and (x, μ (x)) ∈ X × I_μ. (x, μ (x)) ∘ (e, μ (e)) = {(z, μ (z)) |μ (z) ≥ μ (x) ∨ μ (e)} = {(z, μ (z)) |μ (z) ≥ μ (x)}. Thus (x, μ (x)) ∈ (x, μ (x)) ∘ (e, μ (e)).

Assume that (X × I_μ, ∘) is of type III.

Let e ∈ X, such that (e, μ (e)) is the biggest element in (X × I_μ, ∘) and (x, μ (x)) ∈ X × I_μ. (x, μ (x)) ∘ (e, μ (e)) = {(z, μ (z)) |μ (z) ≤ μ (x) ∧ μ (e)} = {(z, μ (z)) |μ (z) ≤ μ (x)}. Thus (x, μ (x)) ∈ (x, μ (x)) ∘ (e, μ (e)).□

Example 4. (a, 0.2) is the identity element for the type II and (b, 0.6) is the identity element for the type III in the Example 1. Both of them have one identity element each.

4 Morphisms on fuzzy hyperstructures

As in the case of most studies of hyperstructures, it is of interest to see when the algebraic properties of one fuzzy hyperstructure are preserved under some transformations. Let X, Y be two nonempty sets and μ, ν be two membership functions on X, Y respectively.

Definition 4.1. A function $F$ from (X × I_μ) into (Y × I_ν) is defined as an ordered pair $F = (f; \bar{f})$ , where f is a function from X into Y, and $\bar{f} : I_{μ} \to I_{ν}$ , called co-membership function, is such that $μ (x) \mapsto \bar{f} (μ (x)) = ν (f (x))$ .

Remark 3. It is important to note that if f is one-to-one (onto) so are $\bar{f}$ and $F$ .

Definition 4.2. Let μ be a fuzzy subset of a nonempty set X. A morphism $\bar{f}$ is said to be increasing (decreasing) if, for μ (x) ≤ μ (y), $\bar{f} (μ (x)) \leq \bar{f} (μ (y))$ ( $\bar{f} (μ (x)) \geq \bar{f} (μ (y))$ ).

Definition 4.3. [2] Let (G, ∘ ₁) and (H, ∘ ₂) be two hypergroups. A morphism from (G, ∘ ₁) into (H, ∘ ₂) is a map morph : (G, ∘ ₁) → (H, ∘ ₂) such that morph (a ∘ ₁b) ⊆ morph (a) ∘ ₂morph (b).

The next theorem gives some conditions on a function to become a morphism on fuzzy hypergroups.

Theorem 4.4. Let $F : (X \times I_{μ}, \circ_{μ}) \to (Y \times I_{ν}, \circ_{ν})$ be a function on fuzzy semihypergroups, where $F = (f; \bar{f})$ .

If $\bar{f} : I_{μ} \to I_{ν}$ is increasing, then $F$ is a morphism of the fuzzy hypergroups (semihypergroups) of the same type such as type I or IV (type II or III).

If $\bar{f} : I_{μ} \to I_{ν}$ is decreasing, then $F$ is a morphism of the fuzzy hypergroups (semihypergroups) from type I into type IV (type II into type III).

Proof. Let $F : (X \times I_{μ}, \circ_{μ}) \to (Y \times I_{ν}, \circ_{ν})$ be a function on the fuzzy semihypergroups.

Assume that $\bar{f} : I_{μ} \to I_{ν}$ is increasing and without any loss of generality assume also that (X × I_μ, ∘ _μ) and (Y × I_ν, ∘ _ν) are two fuzzy hypergroups of the type I.

Let (x, μ (x)) , (y, μ (y)) ∈ X × I_μ.

$\begin{matrix} F ((x, μ (x)) \circ_{μ} (y, μ (y))) \\ = F ({(z, μ (z)) | μ (z) \geq μ (x) \land μ (y)}) \\ = {(f (z), \bar{f} (μ (z))) | \bar{f} (μ (z)) \geq \bar{f} (μ (x) \land μ (y))} \\ = {(f (z), \bar{f} (μ (z))) | \bar{f} (μ (z)) \geq \bar{f} (μ (x)) \land \bar{f} (μ (y))} \\ = {(f (z), \bar{f} (μ (z))) | ν (f (z)) \geq ν (f (x)) \land ν (f (y))} (1) \end{matrix}$

$\begin{matrix} F ((x, μ (x)) \circ_{ν} F ((y, μ (y)) \\ = (f (x), \bar{f} (μ (x))) \circ_{ν} (f (y), \bar{f} (μ (y))) \\ = {(t, ν (t)) | ν (t) \geq \bar{f} (μ (x)) \land \bar{f} (μ (y))} \\ = {(t, ν (t)) | ν (t) \geq ν (f (x)) \land ν (f (y))} (2) \end{matrix}$ By using (1) and (2) it is clear that $F ((x, μ (x)) \circ_{μ} (y, μ (y))) \subseteq F ((x, μ (x)) \circ_{ν} F ((y, μ (y))$ . Thus $F$ is a morphism.

For the types II, III and IV the proofs are similar to the type I.

Assume that $\bar{f} : I_{μ} \to I_{ν}$ is decreasing.

Assume that (X × I_μ, ∘ _μ) is the type I and (Y × I_ν, ∘ _ν) is the type IV.

Let (x, μ (x)) , (y, μ (y)) ∈ X × I_μ.

$\begin{matrix} F ((x, μ (x)) \circ_{μ} (y, μ (y))) \\ = F ({(z, μ (z)) | μ (z) \geq μ (x) \land μ (y)}) \\ = {(f (z), \bar{f} (μ (z))) | \bar{f} (μ (z)) \leq \bar{f} (μ (x) \land μ (y))} \\ = {(f (z), \bar{f} (μ (z))) | \bar{f} (μ (z)) \leq \bar{f} (μ (x)) \lor \bar{f} (μ (y))} \\ = {(f (z), \bar{f} (μ (z))) | ν (f (z)) \leq ν (f (x)) \lor ν (f (y))} (3) \end{matrix}$

$\begin{matrix} F ((x, μ (x)) \circ_{ν} F ((y, μ (y)) \\ = (f (x), \bar{f} (μ (x))) \circ_{ν} (f (y), \bar{f} (μ (y))) \\ = {(t, ν (t)) | ν (t) \leq \bar{f} (μ (x)) \lor \bar{f} (μ (y))} \\ = {(t, ν (t)) | ν (t) \leq ν (f (x)) \lor ν (f (y))} (4) \end{matrix}$ Using (3) and (4) it is clear that $F ((x, μ (x)) \circ_{μ} (y, μ (y))) \subseteq F ((x, μ (x)) \circ_{ν} F ((y, μ (y))$ . Thus $F$ is a morphism from type I to type IV.

Furthermore, assume that (X × I_μ, ∘ _μ) is the type II and (Y × I_ν, ∘ _ν) is the type III.

Let (x, μ (x)) , (y, μ (y)) ∈ X × I_μ.

$\begin{matrix} F ((x, μ (x)) \circ_{μ} (y, μ (y))) \\ = F ({(z, μ (z)) | μ (z) \geq μ (x) \lor μ (y)}) \\ = {(f (z), \bar{f} (μ (z))) | \bar{f} (μ (z)) \leq \bar{f} (μ (x) \lor μ (y))} \\ = {(f (z), \bar{f} (μ (z))) | \bar{f} (μ (z)) \leq \bar{f} (μ (x)) \land \bar{f} (μ (y))} \\ = {(f (z), \bar{f} (μ (z))) | ν (f (z)) \leq ν (f (x)) \land ν (f (y))} (5) \end{matrix}$

$\begin{matrix} F ((x, μ (x)) \circ_{ν} F ((y, μ (y)) \\ = (f (x), \bar{f} (μ (x))) \circ_{ν} (f (y), \bar{f} (μ (y))) \\ = {(t, ν (t)) | ν (t) \leq \bar{f} (μ (x)) \land \bar{f} (μ (y))} \\ = {(t, ν (t)) | ν (t) \leq ν (f (x)) \land ν (f (y))} (6) \end{matrix}$ Using (5) and (6) it is clear that $F ((x, μ (x)) \circ_{μ} (y, μ (y))) \subseteq F ((x, μ (x)) \circ_{ν} F ((y, μ (y))$ . Thus $F$ is a morphism from type II to type III.□

Corollary 4.5. Let $F : (X \times I_{μ}, \circ_{μ}) \to (Y \times I_{ν}, \circ_{ν})$ be a function on hyperfuzzy hypergroups, where $F = (f; \bar{f})$ . If f : X → Y bijection, then

If $\bar{f} : I_{μ} \to I_{ν}$ is increasing, then $F$ is a isomorphism of the fuzzy hypergroups (semihypergoups) of the same type as type I or IV (type II or III).

If $\bar{f} : I_{μ} \to I_{ν}$ is decreasing, then $F$ is a isomorphism of the fuzzy hypergroups (semihypergroups) from type I into type IV (type II into type III).

Proof.

By Theorem 4.4(1), $F$ is a morphism of the same type of fuzzy hypergroups or fuzzy semihypergroups. Also, by Remark 3, $F$ is a bijection. Hence, it is an isomorphism.

The proof is similar to that of (1).□

5 Conclusion

In this paper, it has been shown from any fuzzy subset of a set, four different fuzzy hyperstructures (two fuzzy hypergroups and two fuzzy semihypergroups) can be constructed. It has also been shown that to each of these fuzzy hyperstructures is associated a classical hyperstructure. It was established that the hyperoperations yielding these hyperstructures have not been anywhere defined in the literature.

Furthermore, the conditions for two fuzzy hyperstructures to be homomorphic or isomorphic were established. One could investigate further if hyperstructures of I and IV are join spaces.

Footnotes

Acknowledgments

The first author is supported by Foundation of Chongqing Municipal Key Laboratory of Institutions of Higher Education ([2017]3), Foundation of Chongqing Development and Reform Commission (2017[1007]), and Foundation of Chongqing Three Gorges University. The third author is supported by the Ministry of Defence of the Czech Republic, project code: DZRO 217. The also wishes to thank the reviewers for their useful comments to improve on the quality of this work.

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