The real world is full of uncertainties that are not supported by classical mathematical structures. Therefore, it is necessary to redefine these structures to include uncertainties. Theory of fuzzy sets [1], vague set theory, rough set theory, interval mathematics theory, and other mathematical tools can help us implement these ideas. Much effort has been made on this subject in previous research. For example, Molodstov [2] introduced the concept of soft sets, Feng et al. [3–5] extended soft sets and combined them with fuzzy sets and rough sets, Aktas and Cagman [6] studied soft groups, and Acar et al. [7] presented soft rings. Polygroups were studied by Comer [8, 9]. Abbasizadeh and Davvaz [10] presented the concept of fuzzy topological polygroups and proved some results. Soft topological polygroups were based on work by Hidari et al. [11]. Recently, in [12], the author introduced the notion of soft topological polygroups by applying soft set theory to topological polygroups. In addition, see [13–16].

Let U be an initial universe, and ℘(U) denote the power set of U. Suppose that E is a set of parameters, A ⊆ E, and A ≠ ∅︀. A pair ( ) is said to be a soft set over U, where denotes a map. Furthermore, a soft set over U is a parameterized family of subsets of universe U. For a ∈ A, may be considered as the set of approximate elements of the soft set ( ). Note that a soft set is not a set [6, 7]. Suppose that ( ) and ( ) are soft sets over U. In this case, we say that ( ) is a soft subset of ( )(i.e., ) if A ⊆ B and for all a ∈ A. If , then we say that ( ) is a soft superset of ( ), which is denoted by . Soft sets ( ) and ( ) are said to be soft equal and are denoted by ( ) if ( ) and .

If for all a ∈ A, then ( ) is an absolute soft set, denoted by Â. Otherwise, if (null set) for all a ∈ A, then ( ) is a null soft set, denoted by ^̂. Symbol denotes the bi-intersection of two soft sets, which is defined as , where C = A ∩ B and for all a ∈ C. The soft set (ℍ, C) defines a union of ( ) and ( ), which is shown by , where C = A ∪ B and

In addition, ( ) AND ( ) is denoted by and defined by , where for all (a, b) ∈ A × B. Similarly, ( ) OR ( ) is denoted by and defined by , where for all (a, b) ∈ A × B. The support of the soft set ( ) is defined as . If the support of the soft set ( ) is not equal to the empty set, we say that ( ) is non-null.

Let H be a non-empty set. Pair (H, ○) is called a hyper-groupoid if ○ : H × H ↦ ℘(H) is a map, where ℘(H) is a family of non-empty subsets of H. A hypergroupoid (H, ○) is called a quasihypergroup if, for every h ∈ H, we have h ○ H = H = H ○ h, and it is called a semihypergroup if for every t, u,w ∈ H, we have t○(u○w) = (t○u)○w. Pair (H, ○) is called a hypergroup if it is a quasihypergroup and a semihypergroup. A special type of hypergroups is a polygroup. A hypergroup P = (P, ○, e,⁻¹ ) is a polygroup if there exists a unitary operation ⁻¹ on P and e ∈ P such that for all p, q, r ∈ P, the following conditions hold: (1) (p ○ q) ○ r = p ○ (q ○ r); (2) e○p = p○e = p; (3) if p ∈ q○r, then q ∈ p○r⁻¹ and r ∈ q⁻¹○p. The following results about polygroups follow easily from the definition: e ∈ p ○ p⁻¹ ∩ p⁻¹ ○ p, e⁻¹ = e, (p⁻¹)⁻¹ = p, and (p ○ q)⁻¹ = q⁻¹ ○ p⁻¹ [9, 17]. Let P be a polygroup and ( ) be a non-null soft set over P. Then, ( ) is called a (normal) soft polygroup over P if is a (normal) subpolygroup of P for all .

A topological group is a group G with a topology on G that satisfies the following two properties:

(1) Mapping p : G × G ↦ G defined by p(g, h) = gh is continuous when G×G is endowed with the product topology;

(2) Mapping inv : G ↦ G defined by inv(g) = g⁻¹ is continuous [18].

Let (F,A) be a non-null soft set defined over G. Then, triplet (F, A, τ ) is called a soft topological group over G if the following conditions are satisfied:

(a) F(a) is a subgroup of G for all a ∈ A;

(b) Mapping (x, y) ↦ xy of the topological space F(a) × F(a) to F(a) and mapping x ↦ x⁻¹ of the topological space F(a) to F(a) are continuous for all a ∈ A.

Example 1

Take G=S₃={e, (12), (13), (23), (123), (132)}, A = {e₁, e₂, e₃}, and let the base for topology τ be B = {{e}, {(12)}, {(123)}, {(132)}}. The set-valued function F is defined by F(e₁) = {e}, F(e₂) = {e, (12)}, and F(e₃) = {e, (123), (132)}. Clearly, F(a) is a subgroup of G, for all a ∈ A. In addition, condition (b) was satisfied. Hence, (F, A, τ ) is a soft topological group over G.

Let (H, τ) be a topological space. The following theorem provides a topology on ℘^*(H) that is induced by τ.

Theorem 1 ( [19])

Let (H, τ) be a topological space. Then, family β consisting of all sets S_V = {U ∈ ℘^*(H) | U ⊆ V,U ∈ τ} is a base for a topology on ℘^*(H). This topology is denoted by τ^*. Let (H, ○) be a hypergroup and (H, τ) be a topological space. Then, system (H, ○, τ) is called a topological hypergroup if the following conditions are satisfied: (1) mapping (x, y) ↦ x ○ y, from H × H ↦ ℘^*(H), is continuous; (2) mapping (x, y) ↦ x/y, from H × H ↦ ℘^*(H), is continuous, where x/y = {z ∈ H|x ∈ z ○ y}; (3) mapping (x, y) ↦ y\x, from H × H to ℘^*(H), is continuous, where y/x = {z ∈ H | x ∈ y ○ z}. Let ⟨P, ○, e,⁻¹⟩ be a polygroup and (P, τ) be a topological space. Then, system ⟨P, ○, e,⁻¹, τ⟩ is called a topological polygroup if mappings ○ : P × P ↦ ℘^*(P) and ⁻¹ : P ↦ P are continuous. Obviously, every topological group is a topological polygroup. The following theorem helps us to recognize continuous hyperoperations.

Theorem 2 ( [11])

Let P be a polygroup. Then, hyperoperation ○ : P ×P ↦ ℘^*(P) is continuous if and only if for every x, y ∈ P and U ∈ τ such that x ○ y ⊆ U there exist V,W ∈ τ such that x ∈ V and y ∈ W and V ○W ⊆ U.

Example 2 ( [11])

Let P be a polygroup and β^* be the fundamental relation of P. Then, τ = {∪_u∈U β^*(u) | U ⊆ P}∪{∅︀} is a topology on P, and (P, ○, e,⁻¹, τ) is a topological polygroup.

In [12], the author defined the concept of soft topological polygroups and achieved several results by establishing important characterizations of this concept. We recall the definition of soft topological polygroups as follows.

Definition 1 ( [12])

Let Θ be a topology defined on a polygroup P. Let ( ) be a soft set defined over P. Then, system ( ) is called a soft topological polygroup over P if (a) is a subgroup of P for all a ∈ A, and (b) mapping (x, y) ↦ x ○ (−y) of the topological space onto is continuous for all a ∈ A.

Topology Θ on P induces topologies on , , and .

Example 3

Let P be a polygroup, and let Θ be a discrete or anti-discrete topology. Suppose that ( ) is a soft polygroup on P; in this case, ( ) is a soft topological polygroup.

It is easy to verify that any soft polygroup satisfies condition (b) of Definition 1 with both topologies.

Example 4

Every soft topological group is a soft topological polygroup.

Example 5

Let (P, Θ) be a topological polygroup, and let all subpolygroups of P be H₁,H₂, ...,H_n. Let A be an arbitrary set such that a₁, a₂, ..., a_n ∈ A. We define as follows:

With the above conditions, ( ) is a soft topological polygroup over P. In fact, every topological polygroup becomes a soft topological polygroup using this method.

Example 6

Suppose that (G, Θ) is a topological group, where H is a compact subgroup of G. It is well-known that system G//H = ({HgH, g ∈ G}, ○,H,−1) is a polygroup with hyperoperation (HaH) ○ (HbH) = {HahbH | h ∈ H} and the unitary function (HgH)⁻¹ = {Hg⁻¹H}. Let π : G ↦ G//H be a function, where π(g) = HgH. We define topology Θ_G//H on G//H as follows. A subset U of G//H is open if π⁻¹(U) is an open subset of G. Therefore, (G//H,Θ_G//H) is a topological polygroup. Now, if H₁,H₂, ...,H_n are subpolygroups of (G//H,Θ_G//H), then, using the method from Example 5, we can construct a soft topological polygroup ( ), where A is an arbitrary non-empty subset of the parameter set E.

Theorem 3

Every soft polygroup over a topological polygroup (anti-discrete) is a soft topological polygroup.

Example 7

Let P = {1, 2}, and let hyperoperation ⋇ be defined by 1⋇1 = 1, 1⋇2 = 2⋇1 = 2, and 2⋇2 = {1, 2}. Let Θ₁ be topology {∅︀, P, {1}}. By Theorem 2, hyperoperation ⋇ : P × P ↦ ℘(P) is continuous, and the inverse operation ⁻¹ : P ↦ P is continuous because ⁻¹ is the identity function (x⁻¹ = x for all x ∈ P), and identity is continuous with every topology. However, hyperoperation ⋇: P × P ↦ ℘(P) is not continuous with Θ₂ = {∅︀, P, {2}}. Therefore, P with Θ₁, Θ_dis,Θ_ndis is a topological polygroup. Subpolygroups of P are ∅︀, P, {1}. Let A be an arbitrary set, and let a₁, a₂ ∈ A. We define the soft set as follows:

Therefore, ( ) is a soft topological polygroup.

Example 8

Let P be {e, a, b} and the multiplication table be

○	e	a	b
e	e	a	b
a	a	e	b
b	b	b	{e, a}

By Theorem 2, hyperoperation ○ : P × P ↦ ℘(P) is not continuous with topologies Θ₁ = {∅︀, P, {e}}, Θ₂ = {∅︀, P, {a}}, Θ₃ = {∅︀, P, {b}}, Θ₄ = {∅︀, P, {e, a}}, Θ₅ = {∅︀, P, {e, b}}, and Θ₆ = {∅︀, P, {a, b}}. Subpolygroups of P are ∅︀, P, {e}, {e, a}. Let A be an arbitrary set, and a₁, a₂ ∈ A. We define a soft set by

Therefore, is a soft topological polygroup because the restriction of topologies Θ₃,Θ₄ to subspaces {e}, {e, a} are discrete or anti-discrete topologies.

Example 9

Let P be {e, a, b, c} and the multiplication table be

○	e	a	b	c
e	e	a	b	c
a	a	{e, a}	c	{b, c}
b	b	c	e	a
c	c	{b, c}	a	{e, a}

Hyperoperation ○ : P × P ↦ ℘(P) is continuous with the following topologies: Θ_dis, Θ_ndis, Θ₁ = {∅︀, P, {e, b}}, Θ₂ = {∅︀, P, {e}, {b}}. As x⁻¹ = x for all x ∈ P, the inverse operation is identity, and the identity function is continuous with every topology. This means that P with topologies Θ₁,Θ₂ is a topological polygroup. Hyperoperation ○ : P × P ↦ ℘(P) with the following topologies is not continuous: Θ₃ = {∅︀, P, {e}}, Θ₄ = {∅︀, P, {a}}, Θ₅ = {∅︀, P, {b}}, Θ₆ = {∅︀, P, {c}}, Θ₇ = {∅︀, P, {e, a}}, Θ₈ = {∅︀, P, {e, c}}, Θ₉ = {∅︀, P, {a, b}}, Θ₁₀ = {∅︀, P, {a, c}}, Θ₁₁ = {∅︀, P, {b, c}}, Θ₁₂ = {∅︀, P, {e, a, b}}, Θ₁₃ = {∅︀, P, {e, a, c}}, Θ₁₄ = {∅︀, P, {e, b, c}}, Θ₁₅ = {∅︀, P, {a, b, c}}, Θ₁₆ = {∅︀, P, {e}, {a}}. Subpolygroups of P are ∅︀, P, {e}, {e, a}, {e, b}. Let A be an arbitrary set, and a₁, a₂, a₃, a₄ ∈ A. We define the soft set :

Therefore, is a soft topological polygroup. Now, we consider Θ₅ = {∅︀, P, {b}} and define a soft set :

Hyperoperation is continuous with Θ₅. Therefore, ( ) is a soft topological polygroup. Suppose that Θ₆ = {∅︀, P, {c}}. We define a soft set as follows:

Hyperoperation is continuous with Θ₆. Hence, ( ) is a soft topological polygroup. We can construct additional examples using this method.

Example 10

Consider the non-abelian polygroup P = {e, a, b, c} with the following multiplication table:

○	e	a	b	c
e	e	a	b	c
a	a	a	P	c
b	b	{e, a, b}	b	{b, c}
c	c	{a, c}	c	P

By Theorem 2, hyperoperation ○ : P × P ↦ ℘(P) is continuous with the following topologies: Θ_dis, Θ_ndis, Θ₁ = {∅︀, P, {e}}, Θ₂ = {∅︀, P, {e, a}}, Θ₃ = {∅︀, P, {e, b}}, Θ₄ = {∅︀, P, {e}, {a}}, Θ₅ = {∅︀, P, {e}, {b}}. The inverse operation ⁻¹ is continuous if the inverse of any open set is open. The inverse operation ⁻¹ is not continuous with the topologies Θ₂,Θ₃,Θ₄,Θ₅ and continuous with Θ₁. This means that P with topologies Θ₁,Θ_dis, and Θ_ndis is a topological polygroup. Hyperoperation ○ : P ×P ↦ ℘(P) with the following topologies is not continuous: Θ₆ = {∅︀, P, {a}}, Θ₇ = {∅︀, P, {b}}, Θ₈ = {∅︀, P, {c}}, Θ₉ = {∅︀, P, {e, c}}, Θ₁₀ = {∅︀, P, {a, b}}, Θ₁₁ = {∅︀, P, {a, c}}, Θ₁₂ = {∅︀, P, {b, c}}, Θ₁₃ = {∅︀, P, {e, a, b}}, Θ₁₄ = {∅︀, P, {e, a, c}}, Θ₁₅ = {∅︀, P, {e, b, c}}, Θ₁₆ = {∅︀, P, {a, b, c}}. Subpolygroups of P are ∅︀, P, {e}. Let A be an arbitrary set, and a₁, a₂ ∈ A. We define a soft set :

Therefore, are soft topological polygroups. We define a soft set as follows:

Hyperoperation and the inverse operation ⁻¹ are continuous with (Θ_i)_i=1,...,16. Therefore, ( ) is a soft topological polygroup.

Using Theorems 4–9, we can build many other examples using the examples given in this article.

Theorem 4

Every soft polygroup over a topological polygroup (non-discrete) is a soft topological polygroup.

Theorem 5

Suppose that ( ) and ( ) are soft topological polygroups over P.

(1)is a soft topological polygroup over P.

(2)is a soft topological polygroup over P.

Theorem 6

Bi-intersectionis a soft topological polygroup over P, where, i ∈ I is a nonempty family of soft topological polygroups over P.

Theorem 7

Let Θ be a topology defined over P, and let ( ) and ( ) be soft topological polygroups over P.

The following statements are true:

(1)is a soft topological polygroup over P.

(2)is a soft topological polygroup over P, if A and B are disjoint.

Theorem 8

Suppose that, i ∈ I is a nonempty family of soft topological polygroups over P. Then, the following statements hold:

(1)is a soft topological polygroup over P.

(2) If A_i is disjoint, is a soft topological polygroup over P.

Example 11

By referring to [9, 17], we can construct polygroup D4¯ as follows:

○	C1	C2	C3	C4	C5
C1	C1	C2	C3	C4	C5
C2	C2	C1	C3	C4	C5
C3	C3	C3	C1, C2	C5	C4
C4	C4	C4	C5	C1, C2	C3
C5	C5	C5	C4	C3	C1, C2

As a sample of how to calculate the table entries, consider C₃ · C₃. To determine this product, we compute the element-wise product of the conjugacy classes {r, t}{r, t} = {s, 1} = C₁ ∪ C₂. Thus, C₃ ○ C₃ consists of the two conjugacy classes C₁, C₂. Hyperoperation ∘:D4¯×D4¯↦P(D4¯) is not continuous with the following topologies: Θ1={∅,D4¯,{C1}},Θ2={∅,D4¯,{C2}},Θ3={∅,D4¯,{C3}},Θ4={∅,D4¯,{C4}},Θ5={∅,D4¯,{C5}},Θ6={∅,D4¯,{C1,C2}},Θ7={∅,D4¯,{C1,C3}},Θ8={∅,D4¯,{C1,C4}},Θ9={∅,D4¯,{C1,C5}},Θ10={∅,D4¯,{C2,C3}},Θ11={∅,D4¯,{C2,C4}},Θ12={∅,D4¯,{C2,C5}},Θ13={∅,D4¯,{C3,C4}},Θ14={∅,D4¯,{C3,C5}},Θ15={∅,D4¯,{C4,C5}},Θ16={∅,D4¯,{C1,C2,C3}},Θ17={∅,D4¯,{C1,C2,C4}},Θ18={∅,D4¯,{C1,C2,C5}},Θ19={∅,D4¯,{C2,C3,C4}},Θ20={∅,D4¯,{C2,C3,C5}},Θ21={∅,D4¯,{C3,C4,C5}},Θ22={∅,D4¯,{C1,C2,C3,C4}},Θ23={∅,D4¯,{C1,C2,C3,C5}},Θ24={∅,D4¯,{C2,C3,C4,C5}},. This means that (D4¯, Θ_dis) and (D4¯, Θ_ndis) are topological polygroups. Subpolygroups of D4¯ are ∅︀, D4¯, {C₁}, {C₁, C₂}, {C₁, C₂, C₃}, {C₁, C₂, C₄}, {C₁, C₂, C₅}. Let A be an arbitrary set, and a₁, a₂, a₃, a₄ ∈ A. We define a soft set by

Consider Θ5={∅,D4¯,{C5}}. In this case, ( ) is a soft topological polygroup. This example is a good template for creating more polygroups and soft topological polygroups.

Example 12

By referring to [9, 17], we can construct polygroup D4^ as follows:

*	1	2	3	4	5
1	1	2	3	4	5
2	2	1	4	3	5
3	3	4	1	2	5
4	4	3	2	1	5
5	5	5	5	5	{1, 2, 3, 4}

Hyperoperation *:D4^×D4^↦P(D4^) is not continuous with the following topologies: Θ1={∅,D4^,{1}},Θ2={∅,D4^,{2}},Θ3={∅,D4^,{3}},Θ4={∅,D4^,{4}},Θ5={∅,D4^,{5}},Θ6={∅,D4^,{1,2}},Θ7={∅,D4^,{1,3}},Θ8={∅,D4^,{1,4}},Θ9={∅,D4^,{1,5}},Θ10={∅,D4^,{2,3}},Θ11={∅,D4^,{2,4}},Θ12={∅,D4^,{2,5}},Θ13={∅,D4^,{3,4}},Θ14={∅,D4^,{3,5}},Θ15={∅,D4^,{4,5}},Θ16={∅,D4^,{1,2,3}},Θ17={∅,D4^,{1,2,4}},Θ18={∅,D4^,{1,2,5}},Θ19={∅,D4^,{2,3,4}},Θ20={∅,D4^,{2,3,5}},Θ21={∅,D4^,{2,4,5}},Θ22={∅,D4^,{3,4,5}},Θ23={∅,D4^,{1,2,3,4}},Θ24={∅,D4^,{1,2,3,5}},Θ25={∅,D4^,{2,3,4,5}}.

This means that (D4^, Θ_dis), and (D4^, Θ_ndis) are topological polygroups. Subpolygroups of D4^ are ∅︀, D4^, {1}, {1, 2}, {1, 3}, {1, 4}, {1, 2, 3, 4}. Let A be an arbitrary set, and let a₁, a₂, a₃, a₄, a₅ ∈ A. Then, we define a soft set as follows:

If we consider Θ5={∅,D4^,{5}}, then ( ) is a soft topological polygroup. This horizontal approach creates an opportunity for us to build more examples.

Definition 2

Let ( ) on P₁ and ( ) on P₂ be two soft topological polygroups. The product of ( ) and ( ) is denoted by , where Θ₁ ×Θ₂ induce topology on P₁ ×P₂ and , such that , and , where (x, y) * (z, t) = (x ○ z, y ⋄ t), such that and . On the other hand, , where (x, y) ⇝ (x⁻¹, y⁻¹).

Theorem 9

A product of two soft topological polygroups is a soft topological polygroup.

Example 13

Consider polygroup 2[ℛ]:

○	0	a	1	2
0	0	a	1	2
a	a	0	1	2
1	1	1	{0, a, 2}	{1, 2}
2	2	2	{1, 2}	{0, a, 1}

Hyperoperation ○ : 2[ℛ]×2[ℛ] ↦ ℘(2[ℛ]) is not continuous with the following topologies: Θ₁ = {∅︀, 2[ℛ], {0}}, Θ₂ = {∅︀, 2[ℛ], {a}}, Θ₃ = {∅︀, 2[ℛ], {1}}, Θ₄ = {∅︀, 2[ℛ], {2}}, Θ₅ = {∅︀, 2[ℛ], {0, 1}}, Θ₆ = {∅︀, 2[ℛ], {0, 2}}, Θ₇ = {∅︀, 2[ℛ], {a, 1}}, Θ₈ = {∅︀, 2[ℛ], {a, 2}}, Θ₉ = {∅︀, 2[ℛ], {1, 2}}, Θ₁₀ = {∅︀, 2[ℛ], {0, a, 1}}, Θ₁₁ = {∅︀, 2[ℛ], {0, a, 2}}, Θ₁₂ = {∅︀, 2[ℛ], {a, 1, 2}}, Θ₁₃ = {∅︀, 2[ℛ], {0, 1, 2}}. However, ○ : 2[ℛ] × 2[ℛ] ↦ ℘(2[ℛ]) is continuous with Θ₁₄ = {∅︀, 2[ℛ], {0, a}}, Θ₁₅ = {∅︀, 2[ℛ], {0}, {a}}. This means that (2[ℛ],Θ_dis), (2[ℛ],Θ_ndis), (2[ℛ],Θ₁₄), and (2[ℛ],Θ₁₅) are topological polygroups. Subpolygroups of 2[ℛ] are ∅︀, 2[ℛ], {0}, {0, a}. Let A be an arbitrary set, and let a₁, a₂, a₃ ∈ A. We define a soft set by

In this case, ( ) and ( ) are soft topological polygroups.

Example 14

Consider polygroup 3[ℛ]:

○	0	a	1	2
0	0	a	1	2
a	a	{0, a}	1	2
1	1	1	{0, a, 2}	{1, 2}
2	2	2	{1, 2}	{0, a, 1}

Hyperoperation ○ : 3[ℛ]×3[ℛ] ↦ ℘(3[ℛ]) is not continuous with the following topologies: Θ₁ = {∅︀, 3[ℛ], {a}}, Θ₂ = {∅︀, 3[ℛ], {1}}, Θ₃ = {∅︀, 3[ℛ], {2}}, Θ₄ = {∅︀, 3[ℛ], {0, 1} }, Θ₅ = {∅︀, 3[ℛ], {0, 2}}, Θ₆ = {∅︀, 3[ℛ], {a, 1}}, Θ₇ = {∅︀, 3[ℛ], {a, 2}}, Θ₈ = {∅︀, 3[ℛ], {1, 2}}, Θ₉ = {∅︀, 3[ℛ], {0, a, 1}}, Θ₁₀ = {∅︀, 3[ℛ], {0, a, 2}}, Θ₁₁ = {∅︀, 3[ℛ], {a, 1, 2}}. However, ○ : 3[ℛ] × 3[ℛ] ↦ ℘(3[ℛ]) is continuous with Θ₁₂ = {∅︀, 3[ℛ], {0}}, Θ₁₃ = {∅︀, 3[ℛ], {0, a}}, Θ₁₄ = {∅︀, 3[ℛ], {0}, {a}}. Therefore, (3[ℛ], (Θ_i)_i=12,13,14) are topological polygroups. Subpolygroups of 3[ℛ] are ∅︀, 3[ℛ], {0}, and {0, a}. Let A be 3[ℛ]. We define a soft set as

Then, ( ) is a soft topological polygroup. Now, let A be an arbitrary set, and a₁, a₂ ∈ A. We consider a soft set :

In this case, ( ) are soft topological polygroups.

Example 15

Consider polygroup ℛ[2]:

○	0	1	2	a
0	0	1	2	a
1	1	{0, 2}	{1, 2}	a
2	2	{1, 2}	{0, 1}	a
a	a	a	a	{0, 1, 2}

Hyperoperation ○ : ℛ[2]×ℛ[2] ↦ ℘(ℛ[2]) is not continuous with the following topologies: Θ₁ = {∅︀, ℛ[2], {1}}, Θ₂ = {∅︀, ℛ[2], {2}}, Θ₃ = {∅︀, ℛ[2], {a}}, Θ₄ = {∅︀, ℛ[2], {0, 1}}, Θ₅ = {∅︀, ℛ[2], {0, 2}}, Θ₆ = {∅︀, ℛ[2], {0, a}}, Θ₇ = {∅︀, ℛ[2], {1, 2}}, Θ₈ = {∅︀, ℛ[2], {1, a}}, Θ₉ = {∅︀, ℛ[2], {2, a}}, Θ₁₀ = {∅︀, ℛ[2], {0, 1, 2}}, Θ₁₁ = {∅︀, ℛ[2], {0, 1, a}}, Θ₁₂ = {∅︀, ℛ[2], {0, 2, a}}. However, ○ : ℛ[2]×ℛ[2] ↦ ℘(ℛ[2]) is continuous with Θ₁₃ = {∅︀, ℛ[2], {0}}. Therefore, (ℛ[2], Θ_dis), (ℛ[2],Θ_ndis), and (ℛ[2], Θ₁₃) are soft topological polygroups. Subpolygroups of ℛ[2] are ∅︀,ℛ[2], {0}, {0, 1, 2}.

Let A be an arbitrary set, and a₁, a₂, a₃ ∈ A. We define a soft set by

Then, ( ) is a soft topological polygroup. If A = ℛ[2], we define a soft set by

then ( ) is a soft topological polygroup.

Example 16

Consider polygroup ℛ[3]:

○	0	1	2	a
0	0	1	2	a
1	1	{0, 2}	{1, 2}	a
2	2	{1, 2}	{0, 1}	a
a	a	a	a	{0, 1, 2, a}

Hyperoperation ○ : ℛ[3]×ℛ[3] ↦ ℘(ℛ[3]) is not continuous with the following topologies: Θ₁ = {∅︀, ℛ[3], {1}}, Θ₂ = {∅︀, ℛ[3], {2}}, Θ₃ = {∅︀, ℛ[3], {a}}, Θ₄ = {∅︀, ℛ[3], {0, 1} }, Θ₅ = {∅︀, ℛ[3], {0, 2}}, Θ₆ = {∅︀, ℛ[3], {0, a}}, Θ₇ = {∅︀, ℛ[3], {1, 2}}, Θ₈ = {∅︀, ℛ[3], {1, a}}, Θ₉ = {∅︀, ℛ[3], {2, a}}, Θ₁₀ = {∅︀, ℛ[3], {0, 1, a}}, Θ₁₁ = {∅︀, ℛ[3], {0, 2, a} }, Θ₁₂ = {∅︀, ℛ[3], {1, 2, a}}. However, ○ : ℛ[3] × ℛ[3] ↦ ℘(ℛ[3]) is continuous, with Θ₁₃ = {∅︀, ℛ[3], {0}}, Θ₁₄ = {∅︀, ℛ[3], {0, 1, 2}}. Consequently, (ℛ[3],Θ_dis), (ℛ[3],Θ_ndis), and (ℛ[3], (Θ_i)_i=13,14) are soft topological polygroups. Subpolygroups of ℛ[3] are ∅︀,ℛ[3], {0}, {0, 1, 2}. Let A be an arbitrary set, and let a₁, a₂, a₃ ∈ A. We consider a soft set as

In this case, ( ) is a soft topological polygroup. Let A be an arbitrary set, and let a₁, a₂ ∈ A. We define a soft set as

Then, ( ) is a soft topological polygroup. We can construct many examples using this method.

Example 17

Consider ℛ[ℛ]:

○	0	1	2	a	b
0	0	1	2	a	b
1	1	{0, 2}	{1, 2}	a	b
2	2	{1, 2}	{0, 1}	a	b
a	a	a	a	{0, 1, 2, b}	{a, b}
b	b	b	b	{a, b}	{0, 1, 2, a}

According to Theorem 2, hyperoperation ○ : ℛ[ℛ] × ℛ[ℛ] to ℘(ℛ[ℛ]) is not continuous with the following topologies: Θ₁ = {∅︀, ℛ[ℛ], {1}}, Θ₂ = {∅︀, ℛ[ℛ], {2}}, Θ₃ = {∅︀, ℛ[ℛ], {a}}, Θ₄ = {∅︀, ℛ[ℛ], {b}}, Θ₅ = {∅︀, ℛ[ℛ], {0, 1}}, Θ₆ = {∅︀,ℛ[ℛ], {0, 2} }, Θ₇ = {∅︀,ℛ[ℛ], {0, a}}, Θ₈ = {∅︀, ℛ[ℛ], {0, b}}, Θ₉ = {∅︀, ℛ[ℛ], {1, 2}}, Θ₁₀ = {∅︀, ℛ[ℛ], {1, a}}, Θ₁₁ = {∅︀, ℛ[ℛ], {1, b}}, Θ₁₂ = {∅︀, ℛ[ℛ], {2, a}}, Θ₁₃ = {∅︀, ℛ[ℛ], {2, b}}, Θ₁₄ = {∅︀, ℛ[ℛ], {0, 1, a}}, Θ₁₅ = {∅︀, ℛ[ℛ], {0, 1, and b}}, Θ₁₆ = {∅︀, ℛ[ℛ], {1, 2, a}}, Θ₁₇ = {∅︀, ℛ[ℛ], {1, 2, b}}, Θ₁₈ = {∅︀, ℛ[ℛ], {2, a, b}}, Θ₁₉ = {∅︀, ℛ[ℛ], {0, 1, 2, a}}, Θ₂₀ = {∅︀, ℛ[ℛ], {0, 1, 2, b}}, Θ₂₁ = {∅︀, ℛ[ℛ], {1, 2, a, b}}. However, ○ : ℛ[ℛ] × ℛ[ℛ] ↦ ℘(ℛ[ℛ]) is continuous with Θ₂₂ = {∅︀, ℛ[ℛ], {0}},Θ₂₃ = {∅︀, ℛ[ℛ], {0, 1, 2}}. Therefore, (ℛ[3],Θ_dis), (ℛ[3],Θ_ndis), and (ℛ[3], (Θ_i)_i=22,23) are topological polygroups. Subpolygroups of ℛ[ℛ] are ∅︀, ℛ[ℛ], {0}, and {0, 1, 2}. Let A be an arbitrary set, and let a₁, a₂, a₃ ∈ A. We define a soft set by

Then, ( ) is a soft topological polygroup. Now, let A be an arbitrary set, and let a₁, a₂ ∈ A. We define a soft set as

Then, ( ) is a soft topological polygroup.

Polygroups, which are a certain subclass of hypergroups, were investigated in this study. In particular, we combined the notions of polygroups, topologies, and soft sets. Moreover, we constructed several examples of soft topological polygroups. The idea presented in this work can be applied to other algebraic hyperstructures.