Soft topological hyperstructure

Abstract

Molodtsov proposed the theory of soft sets which can be considered as a recent mathematical tool to deal with uncertainties. The main purpose of this paper is to give the definition of soft topological hypergrupoid by examining the concept of hypergrupoid which is one of the hyperystructures with soft set theory from the topological point of view. Also, the relation between soft topological hypergroupoids and soft hypergroupoids is examined and some theoretical results are obtained. By introducing the concept of soft good topological homomorphism, the category of soft topological hypergrupoids is constructed. At last, the definition of soft topological subhypergrupoid is presented and some related properties are studied.

Keywords

Soft set topological hypergroupoid soft hypergrupoid soft topological hypergroupoid

1 Introduction

Many complex problems in the real world contain some uncertainties. Existing methods to resolve these uncertainties are inadequate. At this point, different set theories such as fuzzy, rough, soft and near have been proposed. Of these, soft set theory, initiated by the Russian mathematician Molodtsov in 1999, is a powerful tool for modeling uncertainty [1]. He combined soft set theory with game theory, Riemann integration, Perron integration, theory of measurement. Subsequently, Maji et al. presented the first application of soft set theory in decision making problems [7]. In addition to the practical studies related to soft sets, many theoretical studies have been made especially in algebraic and topological aspects. Aktas and cagman introduced the concept of soft groups [3]. After them, Jun introduced the concept of soft ideals on BCK/ BCI-algebras [12]. Feng et al. presented the definitions of soft semiring, soft ideals on soft semiring and idealistic soft semiring [8]. By examining the actions of soft groups, Oguz et al. presented the relation between the soft action and soft symmetric group [13]. In the meantime, many different algebraic structures on soft sets have been studied by some authors [2 , 9]. Besides, topological studies on soft sets were initiated by Shabir and Naz [5]. By defining the notion of a soft topological space, they proposed the separation axioms in a soft topological space. Aygunoglu and Aygun studied soft product topologies and soft compactness [17]. Nazmul and Samanta introduced the concept of a soft topological group [11]. Oguz et al. defined soft topological categories and obtained some important properties [10].

Another focus of this study is hyperstructure theory. In 1934, this theory proposed by Marty at the eighth congress of Scandinavian Mathematicians [14]. Important hyperstructures are hypergroupoid, hypergroup, hyperring and hypermodule. Although many studies have been conducted in pure and applied sciences related to these structures, these researches have reached a very high potential in the last 30 years [16]. This theory, which is applied to different branches of mathematics such as geometry, fuzzy sets, rough set theory, optimization theory, dynamic systems and probability, has also been combined with topology and soft set theory. Topologically, Ho $\overset{ˇ}{s}$ ková-Mayerová introduced the notion of topological hypergroupoids [18]. After that, Heidari and et al. established several important characterizations by defining the notion of topological hypergroups as a generalization of topological groups [19]. Morever, the relationships between the soft sets and hyperstructures have been investigated by some researchers. The definitions of soft hypergroupoid and soft subhypergroupoid were presented by Yamak and et al. [20]. Later on, Selvachandran and Salleh obtained some structural results by studying on the soft hypergroups and soft subhypergroups [15]. Selvachandran described the concepts of soft hyperring and soft subhyperring [21].

In this paper, we aim to initiate the study of soft topological hypergrupoid which is a new concept by combining the concept of topological hypergrupoid with soft set theory. We introduce the concepts of soft topological subhypergrupoid and soft good topological homomorphism. Finally, we investigate several properties of union, intersection, ∨-union, ∧-intersection and product of the family of soft topological hypergroupoids and soft topological subhypergroupoids in detail.

2 Preliminaries

In this section, we recall some fundamental concepts and properties of soft sets and topological hypergroupoids in order to ensure integrity. For more details, see [1–4 , 18].

Molodtsov described the soft set in the following way [1]. Let X be an initial universe set and let E be a set of parameters. Let P (X) denotes the power set of X and A ⊂ E.

Definition 2.1. [1] A pair (F, A) is said to be a soft set over X, where F is a mapping defined by $F : A ⟶ P (X) .$ In general, a soft set over X can be considered as a parameterized family of subsets of the universe X.

Definition 2.2. [2] Let (F, A) and (G, B) be two soft sets over the common universe X. Then, (F, A) is called a soft subset of (G, B) if

i . A ⊆ B;

ii . F (a) and G (a) are identical approximations for all a ∈ A. We denote it as $(F, A) \tilde{\subset} (G, B)$ .

Definition 2.3. [2] A soft set (F, A) over X is called null soft set denoted by Φ, if F (a) =∅ for all a ∈ A.

Definition 2.4. [2] A soft set (F, A) over X is called absolute soft set denoted by $\tilde{A}$ , if F (a) = X for all a ∈ A.

Definition 2.5. [8] The support of a soft set (F, A) is defined as $Supp (F, A) = {a \in A | F (a) \neq \emptyset} .$ If Supp (F, A) is not equal to the empty set, then (F, A) is called non-null.

As a generalization, the following definitions are given for the nonempty family {(F_i, A_i) | i ∈ I} of soft sets over the common universe X.

Definition 2.6. [22] The restricted intersection of the family {(F_i, A_i) | i ∈ I} is a soft set $(F, A) = {⋂^{˜}}_{i \in I} (F_{i}, A_{i})$ such that A =⋂ _i∈IA_i ≠ ∅ and F (a) = ⋂ _i∈IF_i (a) for all a ∈ A_i.

Definition 2.7. [22] The restricted union of the family {(F_i, A_i) | i ∈ I} is a soft set $(F, A) = (⋃_{R})_{i \in I} (F_{i}, A_{i})$ such that A =⋂ _i∈IA_i ≠ ∅ and F (a) = ⋃ _i∈IF_i (a) for all a ∈ A_i.

Definition 2.8. [22] The extended union of the family {(F_i, A_i) | i ∈ I} is a soft set $(F, A) = {⋃^{˜}}_{i \in I} (F_{i}, A_{i})$ such that A = ⋃ _i∈IA_i and F (a) = ⋃ _i∈I(a)F_i (a), I (a) = {i ∈ I | a ∈ A_i} for all a ∈ A_i.

Definition 2.9. [22] The extended intersection of the family {(F_i, A_i) | i ∈ I} is a soft set $(F, A) = (⋂_{E})_{i \in I} (F_{i}, A_{i})$ such that A = ⋃ _i∈IA_i and F (a) = ⋂ _i∈I(a)F_i (a), I (a) = {i ∈ I | a ∈ A_i} for all a ∈ A_i.

Definition 2.10. [22] The ∧-intersection of the family {(F_i, A_i) | i ∈ I} is a soft set $(F, A) = {\tilde{⋀}}_{i \in I} (F_{i}, A_{i})$ such that A = Π_i∈IA_i and F ((a_i) _i∈I) = ⋂ _i∈IF_i (a_i) for all (a_i) _i∈I ∈ A_i.

Definition 2.11. [22] The ∨-intersection of the family {(F_i, A_i) | i ∈ I} is a soft set $(F, A) = {\tilde{⋁}}_{i \in I} (F_{i}, A_{i})$ such that A = Π_i∈IA_i and F ((a_i) _i∈I) = ⋃ _i∈IF_i (a_i) for all (a_i) _i∈I ∈ A_i.

Here, we review the notions of hypergroupoid and topological hypergroupoid.

Definition 2.12. [20] Let H be a non-empty set and P^* (H) denotes the family of non-empty subsets of H. Then, the mapping · : H × H ⟶ P^* (H) is said to be hyperoperation and the pair (H, ·) is called hypergroupoid.

Definition 2.13. [20] Let (H, ·) be a hypergroupoid and K be a non-empty subset of H. Then K is said to be a subhypergroupoid if (K, ·) is itself a hypergroupoid.

Definition 2.14. [20] For given two hypergroupoids (H, ·) and (H′, ★), the mapping f : H ⟶ H′ is called

•inclusion homomorphism if f (x · y) ⊆ f (x) ★ f (y) for all x, y ∈ H;

•good homomorphism if f (x · y) = f (x) ★ f (y) for all x, y ∈ H.

On the other hand, the definition of soft hypergrupoid is as follows:

Definition 2.15. [20] For a non-null soft set (F, A) over the hypergroupoid H, it is said that (F, A) is a soft hypergroupoid over H if F (a) is a subhypergroupoid of H for all a ∈ Supp (F, A).

Definition 2.16. [18] Let (H, ·) be a hypergroupoid with the topology τ and τ^∗ be a topology on P^∗ (H). Then the hyperoperation · is said to be a τ^∗-continuous if the mapping · : H × H ⟶ P^* (H) is continuous with respect to the topologies τ × τ and τ^∗.

Also, the triple (H, · , τ) is said to be a τ^∗-topological hypergroupoid if the hyperoperation ^′ · ′ is τ^∗-continuous.

Definition 2.17. [18] Let (H, ·) and (H′, ★) be two hypergroupoids with the topologies τ and τ′ on H and H′, respectively. A mapping f : H ⟶ H′ is called is said to be good topological homomorphism if the following conditions are satisfied for all x, y ∈ H:

i . f (x · y) = f (x) ★ f (y);

ii . f is continuous and open.

A good topological homomorphism is said to be a topological isomorphism if the mapping f is one to one and onto.

3 Soft topological hypergroupoids

In this section, we will describe the concept of soft topological hypergroupoid. Let (H, ·) be a hypergroupoid and P^∗ (H) denotes the non-empty subsets of H.

Definition 3.1. Let τ and τ^∗ be two topologies on the hypergroupoid H and P^∗ (H), respectively. Let F : A ⟶ P (H) be a mapping, where P (H) is the set of all subhypergroupoids of H, and A is the set of parameters. The pair (F, A) is called a soft topological hypergroupoid over H with the topology τ if the following conditions are satisfied:

i . F (a) is a subhypergroupoid of H for all a ∈ Supp (F, A);

ii . The hyperoperation · : F (a) × F (a) ⟶ P^∗ (F (a)) is continuous with respect to the topologies induced by τ × τ and τ^∗ for all a ∈ Supp (F, A).

Note that if H is a topological hypergroupoid, it is sufficient that only the first condition of the above definition is satisfied in order to the pair (F, A) to be defined as a soft topological hypergroupoid. In other words, the soft topological hypergroupoid (F, A) can be considered as a parameterized family of subhypergroupoids of the topological hypergroupoid H.

Theorem 3.1. Every soft hypergroupoid on a topological hypergroupoid is a soft topological hypergroupoid.

Proof. Let H be a topological hypergroupoid and let (F, A) be a soft hypergroupoid over H. Then F (a) is a subhypergroupoid of H for all a ∈ A. Thus, F (a) is a topological subhypergroupoid of H with recpect to the topologies induced by H and P^∗ (H) for all a ∈ A. Hence, (F, A) is also a soft topological hypergroupoid over H.

Example 3.1. Every soft topological groupoid is a soft topological hypergroupoid.

Example 3.2. Suppose that (H, τ) is a topological hypergroupoid. and ${H_{i}}_{i = 1}^{n}$ be all subhypergroupoid of H. Let A be a non-empty set and a₁, …, a_n be fixed elements of A. We define F (x) = H_i when x = a_i (i = 1, …, n) and F (x) =∅, otherwise. Then, (F, A, τ) is a soft topological hypergroupoid.

Remark 3.1. Each soft hypergroupoid H can be transformed into a soft topological hypergroupoid by equipping both H and P^∗ (H) with discrete or indiscrete topology. But every soft hypergroupoid over a hypergroupoid is not a soft topological hypergroupoid.

Example 3.3. Consider any soft hypergroupoid (F, A) over H. In this case, F (a) is a subhypergroupoid of H for all a ∈ A. If the topology τ on H (resp. τ^∗ on P^∗ (H)) is discrete (resp. indiscrete) topology, it is clear that the hyperoperation · : H × H ⟶ P^∗ (H) is continuous. Moreover, subspace topology τ_F(a) (resp. $τ_{F (a)}^{*}$ ) induced by discrete (resp. indiscrete) topology τ (resp.τ^∗) is discrete (resp. indiscrete) topology and each hyperoperation · : F (a) × F (a) ⟶ P^∗ (F (a)) is also continuous for all a ∈ A. Thus, since the condition ii of Definition 3.1 is satisfied, it can be said that (F, A) is a soft topological hypergroupoid over H with discrete (resp. indiscrete) topology.

Theorem 3.2. Let {(F_i, A_i) |i ∈ I} be a non-empty family of soft topological hypergroupoids over H with the topology τ.

i . The restricted intersection of the family {(Fi, Ai) |i ∈ I} with ⋂_i∈IA_i≠ ∅ is a soft topological hypergroupoid over H if ${⋂^{˜}}_{i \in I} (F_{i}, A_{i}) \neq \emptyset$ ;

ii . The extended intersection of the family {(F_i, A_i) |i ∈ I} is a soft topological hypergroupoid over H if $(⋂_{E})_{i \in I} (F_{i}, A_{i}) \neq \emptyset$ .

Proof. i . The restricted intersection of the family {(F_i, A_i) |i ∈ I} with ⋂_i∈IA_i≠ ∅ defined as the soft set ${⋂^{˜}}_{i \in I} (F_{i}, A_{i}) = (F, A)$ such that ⋂_i∈IF_i (a) for all a ∈ A. Take a ∈ Supp (F, A). Suppose ⋂_i∈IF_i (a)≠ ∅ so that F_i (a)≠ ∅ for all i ∈ I. Since {(F_i, A_i) |i ∈ I} is a non-empty family of soft topological hypergroupoids over H with the topology τ, F_i (a) is a topological subhypergroupoid of H for all i ∈ I. Then, ⋂_i∈IF_i (a) is a topological subhypergroupoid of H. Therefore, (F, A) is a soft topological hypergroupoid over H with the topology τ.

ii . The proof is similar to i .

Example 3.4. Assume that H = {α, β, γ} is a hypergroupoid with the hyperoperation $\cdot : H \times H ⟶ P^{*} (H), (x, y) ⟼ {x, y}$ for all $x, y \in ℕ$ . Choose τ = {∅ , H, {α}} and τ * = {∅ , H}. Then, it is clear that H is a topological hypergroupoid. Also, let the family {(F_i, A_i) |i ∈ I} on the topological hypergroupoid H be defined as follows: $\begin{matrix} F_{1} : A_{1} = {1} & ⟶ & P^{*} (H) \\ 1 & \mapsto & F_{1} (1) = {α} \\ F_{2} : A_{2} = {1, 2} & ⟶ & P^{*} (H) \\ x & \mapsto & F_{2} (x) = {\begin{matrix} {α} & , 2 ∤ x \\ H & , 2 ∣ x \end{matrix} \\ ⋮ \\ F_{i} : A_{i} = {1, 2, \dots, i} & ⟶ & P^{*} (H) \\ x & \mapsto & F_{i} (x) = {\begin{matrix} {α} & , 2 ∤ x \\ H & , 2 ∣ x \end{matrix} \end{matrix}$ Then, it is easy to verify that ⋂_i∈IA_i = {α} such that the restricted intersection of the family {(Fi, Ai) |i ∈ I} is the pair (F, A) where A = ⋂ _i∈IA_i = {1} and F (1) = ⋃ _i∈IF_i (1) = {α}. Thus, it induces a soft topological hypergroupoid over H with the topology τ.

In this way, it can be shown that the extended intersection of the family {(Fi, Ai) |i ∈ I} is also a soft topological hypergroupoid over H with the topology τ.

Theorem 3.3. Let {(F_i, A_i) |i ∈ I} be a non-empty family of soft topological hypergroupoids over H with the topology τ.

i . The extended union of the family {(F_i, A_i) |i ∈ I} is a soft topological hypergroupoid over H if F_i (x) ⊆ F_j (x) or F_j (x) ⊆ F_i (x) for all i, j ∈ I, x ∈ ⋃ _i∈IA_i;

ii . The restricted union of the family {(F_i, A_i) ∥ i ∈ I} is a soft topological hypergroupoid over H if F_i (x) ⊆ F_j (x) or F_j (x) ⊆ F_i (x) for all i, j ∈ I, x ∈ ∩ _i∈IA_i with ⋂_i∈IA_i≠ ∅.

Proof. i . Assume $(F, A) = {⋃^{˜}}_{i \in I} (F_{i}, A_{i})$ as the extended union of the family {(Fi, Ai) |i ∈ I} with ⋂_i∈IA_i≠ ∅. Let F_i (x) ⊆ F_j (x) or F_j (x) ⊆ F_i (x) for all i, j ∈ I, x ∈ ⋃ _i∈IA_i. Take a ∈ Supp (F, A). Since each (F_i, A_i) is non-null soft sets over H, it follows that ⋃_i∈I (F_i, A_i) is also a non-null soft set over H for all i ∈ I. By assumption, F_i (x) ⊆ F_j (x) or F_j (x) ⊆ F_i (x) for all i, j ∈ I, x ∈ ⋂ _i∈IA_i with ⋂_i∈IA_i≠ ∅ such that F_i (x) and F_j (x) are the topological subhypergroupoids of H and therefore their union must be non-null too. Thus, F (x) is a topological subhypergroup of H. Therefore, it can be concluded that (F, A) is a soft topological hypergroupoid over H with the topology τ.

ii . The proof is similar to i .

Corollary 3.4. Let {(F_i, A_i) |i ∈ I} be a non-empty family of soft topological hypergroupoids over H with the topology τ. Then the extended union of the family {(F_i, A_i) | i ∈ I} is a soft topological hypergroupoid over H with the topology τ if A_i∩ A_j ≠ ∅ for all i, j ∈ I, i ≠ j.

Proof. It is obvious from the above proposition.

Example 3.5. Suppose that $ℕ$ is be the set of natural numbers. The hyperoperation · on $ℕ$ is given by x · y = {x, y}, for all $x, y \in ℕ$ and so $(ℕ, \cdot)$ is a topological hypergroupoid with the topology $τ = {\emptyset, ℕ, {0}}$ . On the other hand, let the family {(F_i, A_i) |i ∈ I} on the topological hypergroupoid $ℕ$ be defined as follows: $\begin{matrix} F_{1} : A_{1} = {1} & ⟶ & P^{*} (ℕ) \\ 1 & \mapsto & F_{1} (1) = {0, 1} \\ F_{2} : A_{2} = {1, 2} & ⟶ & P^{*} (ℕ) \\ 1 & \mapsto & F_{2} (1) = {0, 1} \\ 2 & \mapsto & F_{2} (2) = {0, 2} \\ ⋮ \\ F_{i} : A_{i} = {1, 2, \dots, i} & ⟶ & P^{*} (ℕ) \\ a & \mapsto & F_{i} (a) = {0, a} . \end{matrix}$ Clearly, A_i∩ A_j ≠ ∅ for all i, j ∈ I, i ≠ j. Also, it is not difficult to see that the extended union of the family {(F_i, A_i) |i ∈ I} is the pair (F, A) with the topology τ such that $A = ⋂_{i \in I} A_{i} = ℕ$ and F (a) = ⋃ _i∈IF_i (a) = {0, a} for all a ∈ A_i which means that (F, A) is a soft topological hypergroupoid over $ℕ$ .

In a similar way, the restricted union of the family {(F_i, A_i) |i ∈ I} is a soft set $(F, A) = (⋃_{R})_{i \in I} (F_{i}, A_{i})$ such that A = ⋂ _i∈IA_i = {1} and F (1) = {0, 1} for all a ∈ A_i. This shows that (F, A) is also a soft topological hypergroupoid over $ℕ$ with the topology $τ = {\emptyset, ℕ, {0}}$ .

Theorem 3.5. Let {(F_i, A_i) |i ∈ I} be a non-empty family of soft topological hypergroupoids over H with the topology τ.

i . The ∧-intersection ${\tilde{⋀}}_{i \in I} (F_{i}, A_{i})$ is a soft topological hypergroupoid over H if it is non-null;

ii . The ∨-union ${\tilde{⋁}}_{i \in I} (F_{i}, A_{i})$ is a soft topological hypergroupoid over H if F_i (x_i) ⊆ F_j (x_j) or F_j (x_j) ⊆ F_i (x_i) for all i, j ∈ I, x_i ∈ A_i.

Proof. i . Consider $(F, A) = {\tilde{⋀}}_{i \in I} (F_{i}, A_{i})$ for a non-empty family {(F_i, A_i) |i ∈ I} of soft topological hypergroupoids over H with the topology τ. Let a ∈ Supp (F, A). By the hypothesis, ⋂_i∈IF_i (a_i)≠ ∅ so that F_i (a_i)≠ ∅ for all i ∈ I and (a_i) _i∈I ∈ A_i. Thus F_i (a_i) is a topological subhypergroupoid of H for all i ∈ I so that their intersection must be a topological subhypergroupoid of H too. Hence, (F, A) is a soft topological hypergroupoid over H with the topology τ.

ii . The proof is similar to i .

Example 3.6. Let H = {x, y, z, t} be a hypergroupoid with the following multiplication table:

∘	x	y	z	t
x	{x, y}	x	t	{z, t}
y	x	y	z	t
z	t	z	y	x
t	{z, t}	t	x	{x, y}

If τ₁ = {∅ , H, {y, z} }, ∘ is continuous related to τ₁. Hence, (H, τ₁) is a topological hypergroupoid. Now, suppose that

\begin{matrix} A_{1} = {1, 2, 3, 4, 5} \\ A_{2} = {0, 2, 4, 6, 8} . \end{matrix}

We define

F_{1} (x) = {\begin{matrix} {y} & if x = 1 \\ {x, y} & if x = 2 \\ {y, z} & if x = 3 \\ H & if x = 4 \\ \emptyset & if x = 5 \end{matrix}

F_{2} (x) = {\begin{matrix} {y} & if x = 8 \\ {y, z} & if x = 2 \\ {x, y} & if x = 4 \\ H & if x = 6 \\ \emptyset & if x = 8 \end{matrix}

Then, (F₁, A₁, τ₁), (F₁, A₂, τ₁), (F₂, A₁, τ₁), and (F₂, A₂, τ₁).

Definition 3.2. Let {(F_i, A_i) |i ∈ I} be a non-empty family of soft topological hypergroupoids over H_i with the topologies τ_i. Then the cartesian product of the family {(F_i, A_i) |i ∈ I} over Π_i∈IH_i with the product topology $\prod_{i \in I} τ_{i}$ is denoted by Π_i∈I (Fi, Ai), is defined as Π_i∈I (F_i, A_i) = (F, A) where A = Π_i∈IA_i and F (x_i) = Π_i∈IF_i (x_i) for all (x_i) _i∈I ∈ A.

Theorem 3.6. The cartesian product of the family {(F_i, A_i) | i ∈ I} is a soft topological hypergroupoid over Π_i∈IH_i with the product topology Π_i∈Iτ_i.

Proof. Suppose that (F_i, A_i) is a soft topological hypergroupoid over H_i with the topology τ_i for all i ∈ I. Then, F_i (a)≠ ∅ and F_i (a_i) a topological subhypergroupoid of H_i for all (a_i) _i∈I ∈ Supp (F_i, A_i). As such, Π_i∈IF_i (a_i)≠ ∅ and Π_i∈IF_i (a_i) a topological subhypergroupoid of Π_i∈IH_i with the product topology Π_i∈Iτ_i. Therefore, Π_i∈I (Fi, Ai) is a soft topological hypergroupoid over Π_i∈IH_i and the proof is completed.

3.1 Soft topological hypergroupoid homomorphisms

Here, we introduce the concept of soft topological hypergroupoid homomorphisms and obtain several important results.

Definition 3.3. Let (F, A) and (K, B) be soft topological hypergroupoids over H and H′ with the topologies τ and τ′, respectively. Let g : A ⟶ B and f : H ⟶ H′ be two mappings. Then the pair (f, g) is called a soft good topological homomorphism if the following conditions are satisfied:

i . f is a good homomorphism;

ii . f (F (a)) = K (g (a)) for all a ∈ Supp (F, A);

iii . $f_{a} : (F (a), τ_{F (a)}) ⟶ (K (g (a)), τ_{K (g (a))}^{'})$ continuous and open for all a ∈ Supp (F, A).

From the definition, it follows that a soft good topological homomorphism (f, g) is a mapping of soft topological hypergroupoids. Thus, we obtain a new category whose objects are soft topological hypergroupoids and whose arrows are soft good topological homomorphisms.

Also, we say that (F, A) is soft topologically isomorphic to (K, B) if the mappings f and g are one to one and onto.

Example 3.7. Let (K, B) be a soft topological subhypergroupoid of (F, A) over H. Together with the inclusion map i : B ⟶ A and the identity map $I : H ⟶ H$ , the pair $(I, i)$ is a soft good topological homomorphism from (K, B) to (F, A).

Example 3.8. Let (F, A) and (K, B) be the two soft good homomorphic hypergroupoids defined over H and H′, respectively. Then (F, A) is soft good topological homomorphic to (K, B) with discrete or anti-discrete topology. So any soft good homomorphic hypergroupoids can be considered as soft good topological homomorphic hypergroupoids in the discrete or anti-discrete topology.

Theorem 3.7. Let the pair (f, g) be a soft good topological homomorphism from (F, A) to (K, B), where (F, A) and (K, B) are two soft topological hypergroupoids over H and H′, respectively. Then (f (F) , B) is a soft topological hypergroupoid over H′ if g : A ⟶ B be an injective mapping.

Proof. Let (F, A) and (K, B) be two soft topological hypergroupoids over H and H′ with the topologies τ and τ′, respectively. Then, F (a) is a topological subhypergroup of H for all a ∈ Supp (F, A). Since (f, g) : (F, A) ⟶ (K, B) is a soft good topological homomorphism, we have g (Supp (F, A)) = Supp (f (F) , B). Take b ∈ Supp (f (F) , B). So there exist a ∈ Supp (F, A) such that g (a) = b, hence we have F (a)≠ ∅. Moreover, F (a) is a topological subhypergroupoid of H with respect to the topology induced by τ. Since f is a good topological homomorphism, it follows that f (F (x)) is a topological subhypergroupoid of H′ with respect to the topology induced by τ′. Thus, (f (F) , B) is a soft topological hypergroupoid over H′ with the topology τ′.

Example 3.9. Let $ℕ$ and $ℤ$ be two hypergroupoid with the hyperoperation x · y = {x, y} for all $x, y \in ℤ$ . Consider a soft topological hypergroupoid (F, A) over $ℕ$ with the topology $τ = {\emptyset, ℕ}$ defined by $\begin{matrix} F : A = {1, 3, 5} & ⟶ & P^{*} (ℕ) \\ a & \mapsto & F (a) = {\begin{matrix} 2 ℕ & , 2 ∤ a \\ ℕ & , 2 ∣ a \end{matrix} \end{matrix}$ and consider a soft topological hypergroupoid (K, B) over $ℤ$ with the topology $τ^{'} = {\emptyset, ℤ}$ given by $\begin{matrix} K : B = {2, 4, 6} & ⟶ & P^{*} (ℤ) \\ b & \mapsto & F (x) = {\begin{matrix} - 2 ℤ & , 2 ∣ b \\ \emptyset & , 2 ∤ b \end{matrix} \end{matrix}$ On the other hand, define the following mappings $\begin{matrix} f : ℕ & ⟶ & ℤ \\ x & \mapsto & f (x) = - x \end{matrix}$ and $\begin{matrix} g : A & ⟶ & B \\ a & \mapsto & g (a) = a + 1 . \end{matrix}$ Then, it is clear that f (F (a)) = K (g (a)) and $f_{a} : (F (a), {\emptyset, 2 ℕ}) ⟶ (K (g (a)), {\emptyset, - 2 ℕ})$ continuous and open for all a ∈ Supp (F, A). This implies that the pair (f, g) is a soft good topological homomorphism. Also, since g is an injective mapping, there exists a ∈ Supp (F, A) such that g (a) = b. So, f (F (a)) is a topological subhypergroupoid with the topology ${\emptyset, - ℕ}$ for a ∈ {1, 3, 5}. Therefore, (f (F) , B) is a soft topological hypergroupoid over $ℤ$ .

Theorem 3.9. Let the pair (f, g) be a soft topological good homomorphism from (F, A) to (K, B), where (F, A) and (K, B) are two soft topological hypergroupoids over H and H′, respectively. Then (f^-1 (K) , A) is a soft topological hypergroupoid over H if it is non-null.

Proof. Suppose that (F, A) and (K, B) are two soft topological hypergroupoids over H and H′ with the topologies τ and τ′, respectively. So for all b ∈ Supp (K, B), It is easy to verify that $g (Supp (f^{- 1} (K), A)) = g^{- 1} (Supp (K, B)) .$ Let a ∈ Supp (f^-1 (K) , A), hence g (a) ∈ Supp (K, B). Therefore, the nonempty set K (g (a)) is a topological subhypergroup of H′ with respect to the topology induced by τ′. Since f is a good topological homomorphism, it follows that f^-1 (K (g (b))) = f^-1 (K (a)) is a topological subhypergroupoid of H with respect to the topology induced by τ. Thus, the pair (f^-1 (K) , A) is obtained as a soft topological hypergroupoid over H with the topology τ.

Example 3.10. Consider the pair (f, g) given in Example 3.9 as a soft topological good homomorphism from (F, A) to (K, B). Then, it is easy to verify that (f^-1 (K) , A) is non-null and a soft topological hypergroupoid over H too.

Theorem 3.9. Let (F, A), (K, B) and (N, C) be soft topological hypergroupoids over H, H′ and H″ with the topologies τ, τ′ and τ″, respectively. Then (f′ ∘ f, g′ ∘ g) : (F, A) ⟶ (N, C) is a soft good topological homomorphism if (f, g) : (F, A) ⟶ (K, B) and (f′, g′) : (K, B) ⟶ (N, C) are two soft good topological homomorphisms.

Proof. Assume that (f, g) : (F, A) ⟶ (K, B) and (f′, g′) : (K, B) ⟶ (N, C) are two soft good topological homomorphisms. Then, f : H ⟶ H′ and f′ : H′ ⟶ H″ are two good topological homomorphisms, and g : A ⟶ B and g′ : B ⟶ C are two mappings such that the conditions f (F (a)) = K (g (a)) and f′ (K (b)) = N (g′ (b)) are satisfied for all a ∈ Supp (F, A), b ∈ Supp (K, B). Clearly, f′ ∘ f : H ⟶ H″ is also good topological homomorphism and g′ ∘ g : A ⟶ C is a mapping so that the condition (f′ ∘ f) (F (a)) = f′ (f (F (a))) = f′ (K (g (a))) = N (g′ (g (a))) = N ((g′ ∘ g) (a))

holds for all a ∈ Supp (F, A). Therefore, the pair (f′ ∘ f, g′ ∘ g) is a soft good topological homomorphism from (F, A) to (N, C).

3.2 Soft topological subhypergroupoids

In this section, we will describe the concept of soft topological subhypergroupoids and exemplify it. Morever, we will present some characterizations about it.

Definition 3.4. Let (F, A) and (K, B) be soft topological hypergroupoids over H with the topology τ. Then the pair (K, B) is called a soft topological subhypergroupoid of (F, A) if the following conditions are satisfied:

i . B ⊆ A;

ii . K (b) is a subhypergroupoid of F (b) for all b ∈ Supp (K, B);

iii . The hyperoperation · : K (b) × K (b) ⟶ P^∗ (K (b)) is $τ_{b}^{*} -$ continuous for all b ∈ Supp (K, B).

Example 3.11. Consider a soft topological hypergroupoid (F, A) over H with the topology τ. Then, (F|_B, B) is a soft topological subhypergroupoid of (F, A) if B ⊆ A.

Theorem 3.10. If (K, B) is a soft topological subhypergroupoid of (F, A) and (N, C) is a soft topological subhypergroupoid of (K, B), then (N, C) is the soft topological subhypergroupoid of (F, A).

Proof. The proof follows immediately from Definition 3.4.

Theorem 3.11. Let (F, A) and (K, B) be two soft topological hypergroupoids over H with the topology τ. Then (K, B) is a soft topological subhypergroupoid of (F, A) if (K, B) is a soft subset of (F, A).

Proof. Assume that (F, A) and (K, B) are two soft topological hypergroupoids over H with the topology τ. Then, the nonempty sets F (x) and K (x) are the topological subhypergroupoid of H. By hypothesis, if (K, B) is a soft subset of (F, A), then B ⊆ A and K (b) ⊆ F (b) for all b ∈ Supp (K, B). Hence, K (b) is a topological subhypergroupoid of F (b) with respect to the topology induced by τ. Consequently, It is easy to see that (K, B) is a soft topological subhypergroupoid of (F, A) with the topology τ.

Theorem 3.12. Let (F, A) be a soft topological hypergroupoid over H with the topology τ and {(Fi, Ai) |i ∈ I} be a non-empty family of soft topological subhypergroupoids of (F, A).

i . The restricted intersection of the family {(F_i, A_i) |i ∈ I} with ⋂_i∈IA_i≠ ∅ is a soft topological subhypergroupoid of (F, A) if ${⋂^{˜}}_{i \in I} (F_{i}, A_{i}) \neq \emptyset$ .

ii . The extended intersection of the family {(F_i, A_i) |i ∈ I} is a soft topological subhypergroupoid of (F, A) if $(⋂_{E})_{i \in I} \neq \emptyset$ .

Proof. i . The restricted intersection of the family {(F_i, A_i) |i ∈ I} with ⋂_i∈IA_i≠ ∅ defined as the soft set ${⋂^{˜}}_{i \in I} (F_{i}, A_{i}) = (F, A)$ such that F (a) = ⋂ _i∈IF_i (a) for all a ∈ A. Let a ∈ Supp (F, A). Assume ⋂_i∈IF_i (a)≠ ∅ so that F_i (a)≠ ∅ for all i ∈ I. Since {(Fi, Ai) | i ∈ I} is a non-empty family of soft topological subhypergroupoids of (F, A), it follows that A_i ⊆ A and F_i (a) is a topological subhypergroupoid of F (a) with respect to the topology induced by τ for all i ∈ I. Thus, ⋂_i∈IA_i ⊆ A and ⋂_i∈IF_i (a) is a topological subhypergroupoid of F (a). Therefore, the family {(F_i, A_i) | i ∈ I} is a soft topological subhypergroupoid of (F, A)

ii . The proof is similar to i .

Theorem 3.13. Let {(F_i, A_i) |i ∈ I} be a non-empty family of soft topological subhypergroupoids of a soft topological hypergroupoid (F, A) over H with the topology τ.

i . The extended union of the family {(F_i, A_i) |i ∈ I} is a soft topological subhypergroupoid of (F, A) if f_i (x) ⊆ f_j (x) or f_j (x) ⊆ f_i (x) for all i, j ∈ I, x ∈ ⋃ _i∈IA_i.

ii . The restricted union of the family {(F_i, A_i) |i ∈ I} is a soft topological subhypergroupoid of (F, A) if f_i (x) ⊆ f_j (x) or f_j (x) ⊆ f_i (x) for all i, j ∈ I, x ∈ ⋂ _i∈IA_i with ⋂_i∈IA_i≠ ∅.

Proof. i . Assume that {(F_i, A_i) |i ∈ I} is a non-empty family of soft topological subhypergroupoids of a soft topological hypergroupoid (F, A) with ⋂_i∈IA_i≠ ∅. Let F_i (x) ⊆ F_j (x) or F_j (x) ⊆ F_i (x) for all i, j ∈ I, x ∈ ⋃ _i∈IA_i. Choose a ∈ Supp (F, A). Since each (F_i, A_i) is non-null soft sets over H, it follows that ${⋃^{˜}}_{i \in I} (F_{i}, A_{i})$ is also a non-null soft set over H for all i ∈ I. By assumption, F_i (a) ⊆ F_j (a) or F_j (a) ⊆ F_i (a) for all i, j ∈ I, a ∈ ⋂ _i∈IA_i with ⋂_i∈IA_i≠ ∅ such that F_i (a) and F_j (a) are the topological subhypergroupoids of F (a) with respect to the topology induced by τ and therefore their union must be non-null too. Thus, it can be concluded that the extended union of the family {(F_i, A_i) | i ∈ I} is a soft topological subhypergroupoid of (F, A) with the topology τ.

ii . The proof is similar to i .

Corollary 3.14. Let {(F_i, A_i) |i ∈ I} be a non-empty family of soft topological subhypergroupoids of a soft topological hypergroupoid (F, A) over H with the topology τ. Then the extended union of the family {(F_i, A_i) | i ∈ I} is a soft topological subhypergroupoid of (F, A) with the topology τ if A_i∩ A_i ≠ ∅ for all i, j ∈ I, i ≠ j.

Theorem 3.15. Let {(F_i, A_i) | i ∈ I} be a non-empty family of soft topological hypergroupoids over H with the topology τ and let (K_i, B_i) be a soft topological subhypergroupoid of (F_i, A_i) for all i ∈ I.

i . The ∧-intersection ${\tilde{⋀}}_{i \in I} (K_{i}, B_{i})$ is a soft topological subhypergroupoid of ${\tilde{⋀}}_{i \in I} (F_{i}, A_{i})$ if it is non-null.

ii . The ∨-union ${\tilde{⋁}}_{i \in I} (K_{i}, B_{i})$ is a soft topological subhypergroupoid of ${\tilde{⋁}}_{i \in I} (F_{i}, A_{i})$ if K_i (b_i) ⊆ K_j (b_j) or K_j (b_j) ⊆ K_i (b_i) for all i, j ∈ I, b_i ∈ B_i.

Proof. i . Choose {(F_i, A_i) |i ∈ I} as a non-empty family of soft topological hypergroupoids over H with the topology τ. By Theorem 3.5 (ii), ${\tilde{⋁}}_{i \in I} (F_{i}, A_{i})$ is also a soft topological hypergroupoid over H with the topology τ. Take b_i ∈ Supp (K_i, B_i). By the supposition, ⋂_i∈IK_i (b_i)≠ ∅ such that K_i (b_i)≠ ∅ for all i ∈ I and (b_i) _i∈I ∈ B_i. Further, B_i ⊆ A_i and K_i (b_i) is a topological subhypergroupoid of F_i (b_i) with respect to the topology induced by τ for all i ∈ I so that ⋂_i∈IB_i ⊆ ⋂ _i∈IA_i and ⋁_i∈I (K_i (b_i)) must be a topological subhypergroupoid of ⋁_i∈I (F_i (b_i)) too. Therefore, ${\tilde{⋀}}_{i \in I} (K_{i}, B_{i})$ is a soft topological subhypergroupoid of ${\tilde{⋀}}_{i \in I} (F_{i}, A_{i})$ with the topology τ

ii . The proof is similar to i .

Theorem 3.16. Let (F, A) be a soft topological hypergroupoid over H with the topology τ and (K, B) be a soft topological subhypergroupoid of (F, A).

i . The restricted intersection of (F, A) and (K, B) is a soft topological subhypergroupoid of (F, A) if it is non-null.

ii . The restricted union of (F, A) and (K, B) is a soft topological subhypergroupoid of (F, A) if it is non-null.

Proof. i . Suppose that (K, B) is a soft topological subhypergroupoid of (F, A) over H with the topology τ. If it is non-null, then B ⊆ A and K (b) is a topological subhypergroupoid of F (b) with respect to the topology induced by τ for all b ∈ Supp (K, B). So, it is straighforward to see that A ∩ B ⊆ A and K (b) ∩ F (b) is also a topological subhypergroupoid of F (b) with respect to the topology induced by τ for all b ∈ Supp (K, B). Thus, the restricted intersection $(F, A) \tilde{\cap} (K, B)$ is a soft topological subhypergroupoid of (F, A) with the topology τ.

The second claim can be proved similarly.

Theorem 3.17. Let f : H ⟶ H′ be a good homomorphism of topological hypergroupoids with the topologies τ and τ′, respectively, and let (F, A) and (K, B) be two soft topological hypergroupoids over H′. Then (f^-1 (K) , B) is a soft topological subhypergroupoid of (f^-1 (F) , A) if (K, B) is a soft topological subhypergroupoid of (F, A) with the topology τ.

Proof. Suppose (K, B) be a soft topological subhypergroupoid of (F, A) over H with the topology τ′. Choose b ∈ Supp (f^-1 (K) , B). Since (K, B) is a soft topological subhypergroupoid of (F, A), then B ⊆ A and (K (b)) is a topological subhypergroupoid of (F (b) with respect to the topology induced by τ′ for all b ∈ Supp (f^-1 (K) , B). Also, since f : H ⟶ H′ be a good topological homomorphism, so f^-1 (F) (b) = f^-1 (F (b)) is a topological subhypergroupoid of f^-1 (K) (b) = f^-1 (K (b)) with respect to the topology induced by τ for all b ∈ Supp (f (K) , B). Therefore,, it conclude that (f^-1 (K) , B) is a soft topological subhypergroupoid of (f^-1 (F) , A) with the topology τ, and the proof is completed.

Theorem 3.18. Let f : H ⟶ H′ be a good homomorphism of topological hypergroupoids with the topologies τ and τ′, respectively, and let (F, A) and (K, B) be two soft topological hypergroupoids over H. Then (f (K) , B) is a soft topological subhypergroupoid of (f (F) , A) over H′ with the topology τ′ if (K, B) is a soft topological subhypergroupoid of (F, A) with the topology τ.

Proof. Assume that (K, B) is a soft topological subhypergroupoid of (F, A) over H with the topology τ. If (K, B) is a soft topological subhypergroupoid of (F, A), then B ⊆ A and (K (b)) is a topological subhypergroupoid of (F (b) with respect to the topology induced by τ for all b ∈ Supp (K, B). Morever, since f : H ⟶ H′ be a good topological homomorphism, it follows that f (F) (b) = f (F (b)) is a topological subhypergroupoid of f (K) (b) = f (K (b)) with respect to the topology induced by τ′ for all b ∈ Supp (f (K) , B). Thus, it conclude that (f (K) , B) is a soft topological subhypergroupoid of (f (F) , A).

4 Conclusion

The notion of soft topological hypergrupoid is introduced from the topological point of view in this manuscript. In addition, the relation between soft topological hypergroupoids and soft hypergroupoids is studied and some theoretical results are obtained. By introducing the concept of soft good topological homomorphism, the category of soft topological hypergrupoids is constructed. At last, the definition of soft topological subhypergrupoid is presented and some related properties are discussed. It is important to note that combining topological hypergrupoids with soft set theory, this study is a prelude to the examination of other soft topological hyperstructures in the future.

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