Fuzzy soft hyperalgebras

Abstract

We introduce the notion of fuzzy soft hypealgebras as an extension of the notion of soft hyperalgebras as well as soft algebras. Also, some basic properties of fuzzy soft sets and homomorphisms between fuzzy soft hypealgebras are presented and discussed. Finally, we study the image and inverse image of a fuzzy soft hypealgebra under a fuzzy soft hyperalgebra homomorphism.

Keywords

Soft set soft hyperalgebra fuzzy soft hyperalgebra homomorphism

1 Introduction

The theory of algebraic hyperstructures is a branch of classical algebraic theory. This theory was first initiated by Marty in 1934 [19] when he defined the hypergroups and began to investigate their properties with applications to groups and algebraic functions. Later on, researchers have observed that this theory also have many applications in both pure and applied siences [8]. Soft set theory [20] is a general mathematical tool which was desined to deal with uncertainties. There are some applications of soft set theory in [16 , 34–36]. Before the introduction of soft set theory, mathematical theories such as the theory of fuzzy sets, rough sets and probability theory were used as tools to deal with uncertainties and vagueness. Now soft set theory has been applied in various areas of mathematics such as hyperalgebra, fuzzy algebra and fuzzy hyperalgebra to develop algebraic structures. The study of soft hyperstuctures was initiated by Yamark et al. [31] as an extension to the theory of soft sets and the theory of hyperstructures. They defined the notion of soft hypergroupoid and softsubhypergroupoids and proceeded to study some of the basic properties of these concepts.

Several aspects of homomorphisms, subalgebras and subdirect decompositions of hyperalgebras (also called multialgebras) are studied by Picket in [23, 24] and by Hansoul in [12]. In [26] Schweigert studied the congruences of multialgebras and in [28] the exponentiation of universal hyperalgebras introduced. In [5] Ameri and Zahedi was introduced and studied notion of hyperalgebraic systems. Also some more basic properties of multialgebras such as, identities, term function and fundamental relation, and direct limit of multialgebras has been studied by Pelea and et. al. in [22]. Ameri and Rosenberg in [2] studied congruence and strong congruences of multialgebras as a generalization of congruences in ordinary algebras and in [1] studied L-multialgebra(also called L- hyperalgebras, where L is a lattice) and fuzzy congruences of multialgebras. Some other applications of fuzzy sets and hypersteuctures are studied by Hoskova, Maturo and Cristea in [9 , 13–15]. The authors in [3] introduced and studied fuzzy hyperalgebras. Now in this paper we define and study the fuzzy soft hyperalgebra notion in connections with soft hyperalgebras. In this regard, in Section 2, we recall some basic concepts of hyperstructures and soft set theory. In Section 3, we define soft hyperalgebra and fuzzy soft hyperalgebra and we study some properties of them. We establish a connection between fuzzy soft hyperalgebras and soft hyperalgebras. We prove that how we can associate to every fuzzy soft hyperalgebra a soft hyperalgebra. In Section 4, we study the homomorphism of fuzzy soft hyperalgebras and the image and inverse image of a fuzzy soft hyperalgebra under a fuzzy soft hyperalgebra homomorphism.

2 Preliminaries

In this section we present some basic definitions and properties of hyperalgebras which will be used in the next section. We use H for a fixed nonvoid set and P^* (H) for the family of all nonvoid subsets of H. Also for a positive integer n we denote for Hⁿ the set of n-tuples over H (for more see [7]).

[3] For a positive integer n, an n-ary hyperoperation β on H is a function β : Hⁿ ⟶ P^* (H). We say that n is the arity of β. A subset S of H is closed under the n-ary hyperoperation β if (x₁, …, x_n) ∈ Sⁿ implies that β (x₁, …, x_n) ⊆ S. A nullary hyperoperation on H is just an element of P^* (H); i.e. a nonvoid subset of H. A hyperalgebra $ℍ = 〈 H, (β_{i} ∣ i \in I) 〉$ is the set H with together a collection (β_i ∣ i ∈ I) of hyperoperations on H.

A subset S of a hyperalgebra $ℍ$ =〈H, (β_i ∣ i ∈ I) 〉 is a subhyperalgebra of $ℍ$ if S is closed under each hyperoperation β_i, for all i ∈ I, that is β_i (a₁, . . . , a_{n
_i}) ⊆ S, whenever (a₁, . . . , a_{n
_i}) ∈ S^{n
_i}. The type of $ℍ$ is the map from I into the set $ℕ^{*}$ of nonnegative integers assigning to each i ∈ I the arity of β_i. Two hyperalgebras of the same type are called similar hyperalgebras.

For n > 0 we extend an n-ary hyperoperation β on H to an n-ary operation $\bar{β}$ on P^* (H) by setting for all A₁, . . . , A_n ∈ P^* (H) $\begin{matrix} \bar{β} (A_{1}, . . ., A_{n}) = ⋃ {β (a_{1}, . . ., a_{n}) | a_{i} \in \\ A_{i} (i = 1, . . ., n)} \end{matrix}$

It is easy to see that $〈 P^{*} (H), ({\bar{β}}_{i} : i \in I) 〉$ is an algebra of the same type of $ℍ$ .

[3] Let $ℍ$ =〈H, (β_i : i ∈ I) 〉 and $\bar{ℍ}$ = $〈 \bar{H}, ({\bar{β}}_{i} : i \in I) 〉$ be two similar hyperalgebras. A map h from $ℍ$ into $\bar{ℍ}$ is called

a homomorphism if for every i ∈ I and all (a₁, . . . , a_{n
_i}) ∈ H^n
_i $h (β_{i} ((a_{1}, . . ., a_{n_{i}})) \subseteq {\bar{β}}_{i} (h (a_{1}), . . ., h (a_{n_{i}}));$

a good homomorphism if for every i ∈ I and all (a₁, . . . , a_{n
_i}) ∈ H^n
_i $h (β_{i} ((a_{1}, . . ., a_{n_{i}})) = {\bar{β}}_{i} (h (a_{1}), . . ., h (a_{n_{i}})) .$

The following definitions are overtaken from [6 , 27].

Suppose that X is an initial universe set and K be a set of parameters. A pair (F, K) is called a soft set over X if and only if F is a mapping from K into the set of all subsets of the set X, i.e., $F : K ⟶ P (X)$ where $P (X)$ is the power set of X [20].

In other words, a soft set is a parameterized family of subsets of the set X. Every set F (e), for every e ∈ K, from this family may be considered as the set of e-elements of the soft set (F, K), or considered as the set of e-approximate elements of the soft set. According to this manner, we can view a soft set (F, K) as consisting of collection of approximations: $(F, K) = {F (e) | e \in K} .$

For a soft set (F, K), the set Supp(F, K) = {x ∈ K : F (x) ≠ φ} is called the support of the soft set (F, K). Thus a null soft set is a soft set with an empty support and a soft set (F, K) is said to be non-null if Supp(F, K) ≠ φ.

Example 2.1. [6] (1) A soft set (F, K) describes the attractiveness of the houses which Mr. A is going to buy.

X = the set of houses under consideration.

K = the set of parameters. Each parameter is a word or sentence.

K = {expensive: beautiful; wooden; cheap; in the green surrounding; modern; in good repair; in bad repair}.

In this case, to define a soft set means to point out expensive houses, beautiful houses, and so on. It is worth noting that the sets F (ϵ) may be empty for some ϵ ∈ K.

(2) [20] Suppose A is a fuzzy set of the universe X. Take the parameter set K = [0, 1], and define the mapping $F : K ⟶ P (X)$ as follows: $F (α) = {x \in X | A (x) \geq α}, α \in [0, 1] .$

In other words, F (α) is α-level set of A.

According to this manner and by using the decomposition theorem of fuzzy sets, we see that a fuzzy set can be uniquely represented as a soft set.

For two soft sets (F, K) and (G, B) over a common universe X, we say that (F, A) is a soft subset of (G, B) and write (F, A) ⊑ (G, B) if

(i) A ⊂ B, and

(ii) For each a ∈ A, F (a) ⊆ G (a).

Union of two soft sets (F, A) and (G, B) over a common universe X is the soft set (H, C), where C = A ∪ B and for all c ∈ C

$H (c) = {\begin{matrix} F (c), & if c \in A - B \\ G (c), & if c \in B - A \\ F (c) \cup G (c), & if c \in A \cap B . \end{matrix}$ We write (F, A) ⊔ (G, B) = (H, C).

Also, intersection of two soft sets (F, A) and (G, B) over a common universe X is the soft set (H, C), where C = A ∩ B and H (c) = F (c) ∩ G (c), for all c ∈ C.

We write (F, A) ⊓ (G, B) = (H, C).

If (F, A) and (G, B) are two soft sets, then (F, A) AND (G, B) is denoted by (F, A) ∧ (G, B) and it is defined as (H, A × B) where H (a, b) = F (a) ∩ G (b), for all (a, b) ∈ A × B.

[21] Let I denote the unite real inerval [0, 1]. Denote the set of all fuzzy sets on X by I^X, that is the set of all functions from X into I. For μ, ν ∈ I^X we say that μ is contained in ν and we write μ ⊆ ν if μ (x) ≤ ν (x), for all x ∈ X. For μ, ν ∈ I^X, the intersection and union, μ ∪ ν, μ ∩ ν ∈ I^X are defined by $\begin{matrix} (μ \cup ν) (x) & = μ (x) \lor ν (x) and \\ (μ \cap ν) (x) = μ (x) \land ν (x), for all x \in X . \end{matrix}$

Also for μ ∈ L^X, a ∈ I, μ_a is defined by

$μ_{a} = {x \in M | μ (x) \geq a},$

μ_a is called a-cut or (a-level subset) of μ.

Let K be a set of parameters and A ⊂ K. A pair (μ, A) is called a fuzzy soft set over X, where μ is a mapping from A into I^X. That is, for each a ∈ A, μ (a) = μ_a : X ⟶ I, is a fuzzy set on X.

Definition 2.2. [18] For two fuzzy soft sets (μ, A) and (ϑ, B) over a common universe X, we say that (μ, A) is a fuzzy soft subset of (ϑ, B) and write (μ, A) ⊑ (ϑ, B) if

(i) A ⊂ B, and

(ii) For each a ∈ A, μ_a ≤ ϑ_a i.e μ_a is a fuzzy subset of ϑ_a.

Definition 2.3. [18] Union of two fuzzy soft sets (μ, A) and (ϑ, B) over a common universe X is the fuzzy soft set (γ, C), where C = A ∪ B and for all c ∈ C $γ (c) = {\begin{matrix} μ_{c}, & if c \in A - B \\ ϑ_{c}, & if c \in B - A \\ μ_{c} \lor ϑ_{c}, & if c \in A \cap B . \end{matrix}$ We write (μ, A) ⊔ (ϑ, B) = (γ, C).

Definition 2.4. [18] Intersection of two fuzzy soft sets (μ, A) and (ϑ, B) over a common universe X is the fuzzy soft set (γ, C), where C = A ∩ B and γ_c = μ_c ∧ ϑ_c, for all c ∈ C.

We write (μ, A) ⊓ (ϑ, B) = (γ, C).

Definition 2.5. [18] If (μ, A) and (ϑ, B) are two fuzzy soft sets, then (μ, A) AND (ϑ, B) is denoted by (μ, A) ∧ (ϑ, B) = (γ, A × B) where γ (a, b) = γ_a,b = μ_a ∧ ϑ_b, for all (a, b) ∈ A × B.

Remark 2.6. [18] Obviously, a classical soft set (F, K) over a universe X can be seen as a fuzzy soft set (μ, K) according to this manner, for e ∈ K, the image of e under μ is defined as the characteristic function of the set F (e), i.e., $μ_{e} (a) = χ_{F (e)} (a) = {\begin{matrix} 1, & if a \in F (e) \\ 0, & otherwise . \end{matrix}$

Definition 2.7. [18] Let φ be a function from X into Y, and let μ and ϑ be fuzzy subsets of X, Y, respectively. The fuzzy subsets φ (μ) of Y and φ^-1 (ϑ) of X are defined by ∀y ∈ Y, $φ (μ) (y) = {\begin{matrix} ⋁_{φ (x) = y} μ (x), & if φ^{- 1} (y) \neq \emptyset \\ 0, & otherwise \end{matrix}$

and ∀x ∈ X $φ^{- 1} (ϑ) (x) = ϑ (φ (x)) .$ Then φ (μ) is called the image of μ under φ, and φ^-1 (ϑ) is called the preimage (or inverse image) of ϑ under φ.

Definition 2.8. [18] Let φ : X ⟶ Y and ψ : A ⟶ B be two functions, where A and B are parameter sets for the crisp sets X and Y, respectively. Then the pair (φ, ψ) is called a fuzzy soft function from X to Y.

Definition 2.9. [18] Let (μ, A) and (ϑ, B) be two fuzzy soft sets over X and Y, respectively, and let (φ, ψ) be a fuzzy soft function from X to Y.

(1) The image of (μ, A) under the soft function (φ, ψ), denoted by (φ, ψ) (μ, A), is the fuzzy soft

set over Y defined by (φ, ψ) (μ, A) = (φ (μ) , ψ (A)), where $φ (μ)_{k} (y) = {\begin{matrix} ⋁_{φ (x) = y} ⋁_{ψ (a) = k} μ_{a} (x); & x \in φ^{- 1} (y) \\ 0, & otherwise, \end{matrix}$

for all k ∈ ψ (A) and y ∈ Y.

(2) The pre-image of (ϑ, B) under the fuzzy soft function (φ, ψ), denoted by (φ, ψ) ^-1 (ϑ, B), is the fuzzy soft set over X defined by (φ, ψ) ^-1 (ϑ, B) = (φ^-1 (ϑ) , ψ^-1 (B)), where $φ^{- 1} (ϑ)_{a} (x) = ϑ_{ψ (a)} (φ (x)),$

for all a ∈ ψ^-1 (B) and x ∈ X.

If φ and ψ are injective (surjective), then (φ, ψ) is said to be injective (surjective).

3 Connection between fuzzy soft hyperalgebras and soft hyperalgebras

In this section, first we introduce the fuzzy soft hyperalgebra notion and we present connection between this notion and soft hyperalgebra notion in Theorem 3.7.

Definition 3.1. Let 〈H, (β_i) _i∈I〉 be a hyperalgebra and (F, A) be a soft set over H. Then (F, A) is a soft hyperalgebra if ∀a∈ Supp(F, A), F (a) be a sub-hyperalgebra of H.

Example 3.2. Soft hypergroups, soft hypermodules and soft hyperlattices are examples of soft hyperalgebras (for more see [17 , 30]).

Definition 3.3. Let 〈H, (β_i) _i∈I〉 be a hyperalgebra and (μ, A) be a fuzzy soft set over H. Then (μ, A) is a fuzzy soft hyperalgebra over H, if for all a ∈ H, ∀i ∈ I and ∀ x₁, …, x_{n
_i} ∈ H, ∀ z ∈ β_i (x₁, …, x_{n
_i}) $⋀_{z \in β_{i} (x_{1}, \dots, x_{n_{i}})} μ_{a} (z) \geq min {μ_{a} (x_{1}), \dots, μ_{a} (x_{n_{i}})} .$

and if 〈H, (β_i) _i∈I〉 has null hyperoperation such as β, then ∀a ∈ H, ∀ z ∈ β, μ_a (z) is constant and it is greatest.

Example 3.4. Fuzzy soft hypergroups, fuzzy soft hypermodules and fuzzy soft hyperlattices are examples of fuzzy soft hyperalgebras (for more see [10, 18]).

Theorem 3.5. Let 〈H, (β_i) _i∈I〉 be a hyperalgebra and (μ, A) and (ϑ, B) be two fuzzy soft hyperalgebras over H. Then

(μ, A) ⊓ (ϑ, B) is a fuzzy soft hyperalgebra over H.

If A∩ B = ∅, then (μ, A) ⊔ (ϑ, B) is a fuzzy soft hyperalgebra over H.

(μ, A) ∧ (ϑ, B) is a fuzzy soft hyperalgebra over H.

Proof. (i) Let (μ, A) ⊓ (ϑ, B) = (γ, C) where C = A ∩ B and γ_c = μ_c (x) ∧ ϑ_c (x), ∀ c ∈ C, x ∈ H. For all c ∈ C, i ∈ I and x₁, …, x_{n
_i} ∈ H and z ∈ β_i (x₁, …, x_n), we have $⋀_{z \in β_{i} (x_{1}, \dots, x_{n_{i}})} μ_{c} (z) \geq min (μ_{c} (x_{1}), \dots, μ_{c} (x_{n_{i}}))$ and $⋀_{z \in β_{i} (x_{1}, \dots, x_{n_{i}})} ϑ_{c} (z) \geq min (ϑ_{c} (x_{1}), \dots, ϑ_{c} (x_{n_{i}})) .$ Therefore $\begin{matrix} ⋀_{z \in β_{i} (x_{1}, \dots, x_{n_{i}})} (μ_{c} \land ϑ_{c}) (z) = ⋀_{z \in β_{i} (x_{1}, \dots, x_{n_{i}})} (μ_{c} (z) \land ϑ_{c} (z)) \\ = ⋀_{z \in β_{i} (x_{1}, \dots, x_{n_{i}})} μ_{c} (z) \land ⋀_{z \in β_{i} (x_{1}, \dots, x_{n_{i}})} ϑ_{c} (z) \\ \geq min (μ_{c} (x_{1}), \dots, μ_{c} (x_{n_{i}})) \land min (ϑ_{c} (x_{1}), \dots, ϑ_{c} (x_{n_{i}})) \\ = min {μ_{c} (x_{1}) \land ϑ_{c} (x_{1}), \dots, μ_{c} (x_{n_{i}}) \land ϑ_{c} (x_{n})} . \end{matrix}$ Also, if β is a null hyperoperation over H, z ∈ β, we have $\begin{matrix} (μ_{c} \land ϑ_{c}) (z) = μ_{c} (z) \land ϑ_{c} (z) \\ \geq μ_{c} (x) \land ϑ_{c} (x) \forall x \in H \\ = (μ_{c} \land ϑ_{c}) (x) . \end{matrix}$

(ii) Let (μ, A) ⊔ (ϑ, B) = (γ, C). Since A∩ B = ∅, we have c ∈ A - B or c ∈ B - A for all c ∈ C. If c ∈ A - B, then γ_c = μ_c is a fuzzy sub-hyperalgebra of H and if c ∈ B - A, then γ_c = ϑ_c is a fuzzy sub-hyperalgebra of H. Therefore (μ, A) ⊔ (ϑ, B) is a fuzzy soft hyperalgebra over H.

(iii) Since, for all a ∈ A and for all b ∈ B, μ_a and ϑ_b are fuzzy sub hyperalgebras of H, and so is γ (a, b) = γ_a,b = μ_a ∧ ϑ_b for all (a, b) ∈ A × B. Since, intersection of two fuzzy sub-hyperalgebras is also a fuzzy sub-hyperalgebra, (γ, A × B) = (μ, A) ∧ (ϑ, B) is a fuzzy soft hyperalgebra over H.□

Definition 3.6. Let (μ, A) be a fuzzy soft set over hyperalgebra 〈H, (β_i) _i∈I〉. The soft set $(μ, A)_{α} = {(μ_{a})_{α} | a \in A}$ for all α ∈ (0, 1], is called an α-level soft set of the fuzzy soft set (μ, A), where (μ_a) _α is an α-level subset of the fuzzy set μ_a.

Theorem 3.7. Let (μ, A) be a fuzzy soft set over a hyperalgebra 〈H, (β_i) _i∈I〉. Then (μ, A) is a fuzzy soft hyperalgebra over H if and only if for all a ∈ A and for arbitrary α ∈ (0, 1] with (μ_a) _α ≠ φ, the α-level soft set (μ, A) _α is a soft hyperalgebra over H.

Proof. If 〈H, (β_i) _i∈I〉 is a hyperalgebra and (μ, A) is a fuzzy soft hyperalgebra over H, then for all a ∈ A, μ_a is a fuzzy sub-hyperalgebra of H. For every α ∈ (0, 1] where (μ_a) _α≠ ∅, and ∀i ∈ I, we consider x₁, x₂, …, x_{n
_i} ∈ (μ_a) _α, thus $μ_{a} (x_{1}) \geq α, \dots, μ_{a} (x_{n_{i}}) \geq α .$ Therefore, for all z ∈ β_i (x₁, x₂, …, x_{n
_i}) we have $\begin{matrix} ⋀_{z \in β_{i} (x_{1}, \dots, x_{n_{i}})} μ_{a} (z) \geq min {μ_{a} (x_{1}), \dots, μ_{a} (x_{n_{i}})} \\ \geq min {α, \dots, α} = α . \end{matrix}$ Hence β_i (x₁, x₂, …, x_{n
_i}) ⊆ (μ_a) _α.

Now, let β be a null hyperoperation in 〈H, (β_i) _i∈I〉. For all a ∈ A, z ∈ β, since μ_a is a fuzzy sub-hyperalgebra over H, we have $\begin{matrix} μ_{a} (z) = m \geq μ_{a} (x) \forall x \in H \\ μ_{a} (z) \geq α . \end{matrix}$ Hence β ⊆ (μ_a) _α. We obtain (μ_a) _α is a sub-hyperalgebra of H for all a ∈ A. This means that (μ, A) _α is a soft hyperalgebra over H.

⟸ If (μ, A) is not a fuzzy soft hyperalgebra over H, then there exists a ∈ A such that μ_a is not a fuzzy soft hyperalgebra of H. Hence, there is i ∈ I and x₁, …, x_{n
_i} ∈ H and z ∈ β_i (x₁, …, x_{n
_i}) such that $⋀_{z \in β_{i} (x_{1}, \dots, x_{n_{i}})} μ_{a} (z) < min {μ_{a} (x_{1}), \dots, μ_{a} (x_{n_{i}})} .$ Let ⋀_{z∈β_i(x₁,…,x_{n
_i})}μ_a (z) = λ and μ_a (x₁) = δ₁,…, μ_a (x_{n
_i}) = δ_{n
_i}. We obtain λ < min {δ₁, …, δ_{n
_i}}. Suppose that $α = \frac{λ + min {δ_{1}, \dots, δ_{n_{i}}}}{2},$ then λ < α < min {δ₁, …, δ_{n
_i}}. Since ⋀_{z∈β_i(x₁,…,x_{n
_i})}μ_a (z) = λ < α we obtain, β_i (x₁, …, x_{n
_i}) ⊈ (μ_a) _α.

On the other hand, min {δ₁, …, δ_{n
_i}} > α implies that μ_a (x₁) ≥ α, … , μ_a (x_{n
_i}) ≥ α, that is; x₁, …, x_{n
_i} ∈ (μ_a) _α. This contradicts with the fact (μ, A) _α is a soft hyperalgebra over H. Therefore, for all i ∈ I and x₁, …, x_{n
_i} ∈ H, z ∈ β_i (x₁, …, x_{n
_i}) we obtain ⋀_{z∈β_i(x₁,…,x_{n
_i})}μ_a (z) ≥ min {μ_a (x₁) , …, μ_a (x_{n
_i})}.

Moreover, suppose that β is null hyperoperation in 〈H, (β_i) _i∈I〉 and z ∈ β such that μ_a (z) < μ_a (x), for some x ∈ H, then let α = μ_a (x). Since (μ_a) _α is a sub-hyperalgebra of H, we have β ⊆ (μ_a) _α. This is contradiction. Hence for all x ∈ H, μ_a (x) ≤ μ_a (z).

Finally, if z₁, z₂ ∈ β and μ_a (z₁) ≠ μ_a (z₂), we can’t conclude that for all x ∈ H, z ∈ β: μ_a (x) ≤ μ_a (z). This completes the proof.□

Theorem 3.8. Let 〈H, (β_i) _i∈I〉 be a hyperalgebra and (μ, A) be a fuzzy soft hyperalgebra over H. If β is a null hyperoperation in H, then for all a ∈ A, consider the set $(μ, A) |_{β} = {(μ_{a}) |_{β} = {x \in H | μ_{a} (x) = μ_{a} (β)}} .$ Then the soft set (μ, A) |_β is a soft hyperalgebra over H.

Proof. Let a ∈ A, i ∈ I and x₁, …, x_{n
_i} ∈ (μ_a) |_β and z ∈ β_i (x₁, …, x_{n
_i}), we have $\begin{matrix} ⋀_{z \in β_{i} (x_{1}, \dots, x_{n_{i}})} μ_{a} (z) \geq min {μ_{a} (x_{1}), \dots, μ_{a} (x_{n_{i}})} \\ = min {μ_{a} (β), \dots, μ_{a} (β)} = μ_{a} (β) . \end{matrix}$ On the other hand, we always have ⋀_{z∈β_i(x₁,…,x_{n
_i})}μ_a (z) ⩽ μ_a (β). Hence ⋀_{z∈β_i(x₁,…,x_{n
_i})}μ_a (z) = μ_a (β), then β_i (x₁, …, x_{n
_i}) ⊆ (μ_a) |_β. It is obvious that ∀z ∈ β′, where β′ is a null hyperoperation, z ∈ (μ_a) _β, therefore (μ_a) _β is a sub-hyperalgebra of H and (μ, A) |_β is a soft hyperalgebra over H.□

4 Homomorphisms of soft hyperalgebras

Let us define

Definition 4.1. Let 〈H, (β_i) _i∈I〉 be a hyperalgebra and (μ, A) be a fuzzy soft hyperalgebra over H. If β is a null hyperoperation in H and δ ∈ (0, 1], then

(μ, A) is said to be a δ_β-identity fuzzy soft hyperalgebra over H, if for every x ∈ H and a ∈ A $μ_{a} (x) = {\begin{matrix} δ & if x \in β \\ 0 & otherwise \end{matrix}$

(μ, a) is said to be a δ-absolute fuzzy soft hyperalgebra over H if μ_a (x) = δ, for all x ∈ H and a ∈ A.

Theorem 4.2. Let 〈H, (β_i) _i∈I〉 and $〈 H^{'}, (β_{i}^{'})_{i \in I} 〉$ be a two hyperalgebras with the same type and f : H ⟶ H′ be a homomorphism. If β′ is a null hyperoperation in H′, then we define (ker f) _β′ = {x ∈ H| f (x) ∈ β′}.

(1) If (μ, A) is a fuzzy soft hyperalgebra over H, then (f (μ) , A) is a δ_β′-identity fuzzy soft hyperalgebra over H′,if for x ∈ H and a ∈ A $μ_{a} (x) = {\begin{matrix} δ & ifx \in (ker f)_{β^{'}} \\ 0 & otherwise \end{matrix}$

(2) If (μ, A) is a δ-absolute fuzzy soft hyperalgebra over H, then (f (μ) , A) is a δ-absolute fuzzy soft hyperalgebra over H′, where f (μ) _a = f (μ_a).

Proof. (1) ∀z ∈ β′ $\begin{matrix} f (μ)_{a} (z) = ⋁_{x \in f^{- 1} (z)} μ_{a} (x) \\ = ⋁_{x \in (ker f)_{β^{'}}} μ_{a} (x) = δ \end{matrix}$

If y ∈ H′, y ∉ β′, then f (μ_a) (y) =0. Thus (f (μ) , A) is a δ_β′-identity fuzzy soft hyperalgebra over H′.

(2) For all y ∈ H′, f (μ_a) (y) = ⋁ _x∈f^-1(y)μ_a (x) = δ. Hence, (f (μ) , A) is a δ-absolute fuzzy soft hyperalgebra over H′ .□

Definition 4.3. Let (μ, A) and (ϑ, B) be two fuzzy soft hyperalgebras over H and H′, respectively and (φ, ψ) be a fuzzy soft function from H to H′. If φ is a homomorphism from H to H′, then (φ, ψ) is said to be fuzzy soft hyperalgebra homomorphism. If φ is a isomorphism from H to H′ and ψ is a one-to-one mapping from A onto B, then (φ, ψ) is said to be fuzzy soft isomorphism.

Theorem 4.4. Let 〈H, (β_i) _i∈I〉 and $〈 H^{'}, (β_{i}^{'})_{i \in I} 〉$ be two hyperalgebras with the same type. If (μ, A) is a fuzzy soft hyperalgebra over H and (φ, ψ) be a fuzzy soft homomorphism from H to H′, then (φ, ψ) (μ, A) is a fuzzy soft hyperalgebra over H′.

Proof. Let k ∈ ψ (A), i ∈ I and y₁, y₂, …, y_{n
_i} ∈ H′. If φ^-1 (y₁) =∅ or φ^-1 (y₂) =∅ or … or φ^-1 (y_{n
_i}) =∅, the proof is straightforward. Suppose that there are x₁, x₂, …, x_{n
_i} ∈ H such that φ (x₁) = y₁, … , φ (x_{n
_i}) = y_{n
_i}. Hence for all z ∈ β_i (x₁, …, x_{n
_i}) and $z^{'} \in β_{i}^{'} (y_{1}, \dots, y_{n_{i}})$ such that φ (z) = z′, $\begin{matrix} ⋀_{z^{'} \in β_{i}^{'} (y_{1}, \dots, y_{n_{i}})} φ (μ)_{k} (z^{'}) = ⋀_{z^{'} \in β_{i}^{'} (y_{1}, \dots, y_{n_{i}})} (⋁_{φ (z) = z^{'}} ⋁_{φ (a) = k} μ_{a} (z)) \\ \geq ⋁_{φ (z) = z^{'}} ⋁_{ψ (a) = k} (⋀_{z \in β_{i} (x_{1}, \dots, x_{n_{i}})} μ_{a} (z)) \\ \geq ⋁_{ψ (a) = k} ⋀_{z \in β_{i} (x_{1}, \dots, x_{n_{i}})} μ_{a} (z) \\ \geq ⋁_{ψ (a) = k} min {μ_{a} (x_{1}), μ_{a} (x_{2}), \dots, μ_{a} (x_{n_{i}})} \\ = min {⋁_{ψ (a) = k} μ_{a} (x_{1}), \dots, ⋁_{ψ (a) = k} μ_{a} (x_{n_{i}})} \end{matrix}$ Since this inequality is satisfied for each x₁, x₂, …, x_{n
_i} ∈ H, such that φ (x₁) = y₁, …, φ (x_{n
_i}) = y_{n
_i}, we set δ₁ = ⋁ _ψ(a)=kμ_a (x₁) , … , δ_{n
_i} = ⋁ _ψ(a)=kμ_a (x_{n
_i}) and since min is a continues T-norm we have $\begin{matrix} ⋀_{z^{'} \in β_{i}^{'} (y_{1}, \dots, y_{n_{i}})} φ (μ)_{k} (z^{'}) \geq ⋁_{φ (x_{n_{i}}) = y_{n_{i}}} (\dots ⋁_{φ (x_{1}) = y_{1}} min {δ_{1}, \dots, δ_{n_{i}}}) \\ = min {⋁_{φ (x_{1}) = y_{1}} δ_{1}, \dots, ⋁_{φ (x_{n_{i}}) = y_{n_{i}}} δ_{n_{i}}} \\ = min {φ (μ)_{k} (y_{1}), \dots, φ (μ)_{k} (y_{n_{i}})} . \end{matrix}$ Moreover, if β′ is a null hyperoperation in H′ and z ∈ β′, we have $\begin{matrix} φ (μ)_{k} (z) = ⋁_{φ (x) = z} ⋁_{ψ (a) = k} μ_{a} (x) . (★) \end{matrix}$ We know φ^-1 (β′) in H is a null hyperoperation. Thus μ_a (x) such that φ (x) = z is constant therefore φ (μ) _k (z) is constant.

On the other hand in (★), since μ_a (x) ≥ μ_a (t) for all t ∈ H, we conclude that φ (μ) _k (z) ≥ φ (μ) _k (s), for all s ∈ H′. This completes the proof.□

Theorem 4.5. Let 〈H, (β_i) _i∈I〉 and $〈 H^{'}, (β_{i}^{'})_{i \in I} 〉$ be two hyperalgebras with the same type, and (ϑ, B) be a fuzzy soft hyperalgebra over H′ and (φ, ψ) be a fuzzy soft homomorphism from H to H′. Then (φ, ψ) ^-1 (ϑ, B) is a fuzzy soft hyperalgebraover H.

Proof. Suppose that a ∈ ψ^-1 (B), i ∈ I and x₁, …, x_{n
_i} ∈ H. For all z ∈ β_i (x_i, …, x_{n
_i}) we have $\begin{matrix} ⋀_{z \in β_{i} (x_{1}, \dots, x_{n_{i}})} φ^{- 1} (ϑ)_{a} (z) = ⋀_{z \in β_{i} (x_{1}, \dots, x_{n_{i}})} ϑ_{ψ (a)} (φ (z)) \\ \geq min {ϑ_{ψ (a)} (φ (x_{1})), \dots, ϑ_{ψ (a)} (φ (x_{n_{i}}))} \\ = min {φ^{- 1} (ϑ)_{a} (x_{1}), \dots, φ^{- 1} (ϑ)_{a} (x_{n_{i}})} . \end{matrix}$ Moreover, if β is a null hyperoperation in H and z ∈ β, we have $φ^{- 1} (ϑ)_{a} (z) = ϑ_{ψ (a)} (φ (z)) .$ Since φ (β) is a null hyperoperation in H and φ (z) ∈ φ (β), ϑ_ψ(a) (φ (z)) is constant, for all z ∈ β. Also, since ϑ_ψ(a) (φ (z)) has greatest degree, φ^-1 (ϑ) _a (z) is too.□

5 Conclusion

We applied soft sets and fuzzy soft sets to algebraic hyperalgebras, as a generalization of classical algebraic structures and introduced and studied notions of soft hyperalgebras and fuzzy soft hyperalgebras. Also, we established a connection between soft fuzzy hyperalgebras and sof hyperalgebras. Finally, we introduced and studied homomorphisms of soft hyperalgebras.

Footnotes

Acknowledgment

The second author is partially supported by Center of Excellence of Algebraic Hyperstructures and its Applications of Tarbiat Modares University (CEAHA), Tehran, Iran.

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