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.
Take G=S_{3}={e, (12), (13), (23), (123), (132)}, A = {e_{1}, e_{2}, e_{3}}, and let the base for topology τ be B = {{e}, {(12)}, {(123)}, {(132)}}. The set-valued function F is defined by F(e_{1}) = {e}, F(e_{2}) = {e, (12)}, and F(e_{3}) = {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.
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.
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.
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.
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 .
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.
Every soft topological group is a soft topological polygroup.
Let (P, Θ) be a topological polygroup, and let all subpolygroups of P be H_{1},H_{2}, ...,H_{n}. Let A be an arbitrary set such that a_{1}, a_{2}, ..., 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.
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^{−1}H}. 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 H_{1},H_{2}, ...,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.
Every soft polygroup over a topological polygroup (anti-discrete) is a soft topological polygroup.
Suppose that ( ) is a soft polygroup over P and (P, Θ) is a topological polygroup. Therefore, for all a ∈ A, is a subpolygroup of P. Mapping to is continuous. Note that P is a topological polygroup, and mapping (a, b) ↦ a ○ b^{−1} of the topological space P × P to ℘(P) is continuous. Hence, ( ) is a soft topological polygroup over P.
Let P = {1, 2}, and let hyperoperation ⋇ be defined by 1⋇1 = 1, 1⋇2 = 2⋇1 = 2, and 2⋇2 = {1, 2}. Let Θ_{1} 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 Θ_{2} = {∅︀, P, {2}}. Therefore, P with Θ_{1}, Θ_{dis},Θ_{ndis} is a topological polygroup. Subpolygroups of P are ∅︀, P, {1}. Let A be an arbitrary set, and let a_{1}, a_{2} ∈ A. We define the soft set as follows:
Therefore, ( ) is a soft topological polygroup.
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 Θ_{1} = {∅︀, P, {e}}, Θ_{2} = {∅︀, P, {a}}, Θ_{3} = {∅︀, P, {b}}, Θ_{4} = {∅︀, P, {e, a}}, Θ_{5} = {∅︀, P, {e, b}}, and Θ_{6} = {∅︀, P, {a, b}}. Subpolygroups of P are ∅︀, P, {e}, {e, a}. Let A be an arbitrary set, and a_{1}, a_{2} ∈ A. We define a soft set by
Therefore, is a soft topological polygroup because the restriction of topologies Θ_{3},Θ_{4} to subspaces {e}, {e, a} are discrete or anti-discrete topologies.
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}, Θ_{1} = {∅︀, P, {e, b}}, Θ_{2} = {∅︀, 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 Θ_{1},Θ_{2} is a topological polygroup. Hyperoperation ○ : P × P ↦ ℘(P) with the following topologies is not continuous: Θ_{3} = {∅︀, P, {e}}, Θ_{4} = {∅︀, P, {a}}, Θ_{5} = {∅︀, P, {b}}, Θ_{6} = {∅︀, P, {c}}, Θ_{7} = {∅︀, P, {e, a}}, Θ_{8} = {∅︀, P, {e, c}}, Θ_{9} = {∅︀, P, {a, b}}, Θ_{10} = {∅︀, P, {a, c}}, Θ_{11} = {∅︀, P, {b, c}}, Θ_{12} = {∅︀, P, {e, a, b}}, Θ_{13} = {∅︀, P, {e, a, c}}, Θ_{14} = {∅︀, P, {e, b, c}}, Θ_{15} = {∅︀, P, {a, b, c}}, Θ_{16} = {∅︀, P, {e}, {a}}. Subpolygroups of P are ∅︀, P, {e}, {e, a}, {e, b}. Let A be an arbitrary set, and a_{1}, a_{2}, a_{3}, a_{4} ∈ A. We define the soft set :
Therefore, is a soft topological polygroup. Now, we consider Θ_{5} = {∅︀, P, {b}} and define a soft set :
Hyperoperation is continuous with Θ_{5}. Therefore, ( ) is a soft topological polygroup. Suppose that Θ_{6} = {∅︀, P, {c}}. We define a soft set as follows:
Hyperoperation is continuous with Θ_{6}. Hence, ( ) is a soft topological polygroup. We can construct additional examples using this method.
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}, Θ_{1} = {∅︀, P, {e}}, Θ_{2} = {∅︀, P, {e, a}}, Θ_{3} = {∅︀, P, {e, b}}, Θ_{4} = {∅︀, P, {e}, {a}}, Θ_{5} = {∅︀, 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 Θ_{2},Θ_{3},Θ_{4},Θ_{5} and continuous with Θ_{1}. This means that P with topologies Θ_{1},Θ_{dis}, and Θ_{ndis} is a topological polygroup. Hyperoperation ○ : P ×P ↦ ℘(P) with the following topologies is not continuous: Θ_{6} = {∅︀, P, {a}}, Θ_{7} = {∅︀, P, {b}}, Θ_{8} = {∅︀, P, {c}}, Θ_{9} = {∅︀, P, {e, c}}, Θ_{10} = {∅︀, P, {a, b}}, Θ_{11} = {∅︀, P, {a, c}}, Θ_{12} = {∅︀, P, {b, c}}, Θ_{13} = {∅︀, P, {e, a, b}}, Θ_{14} = {∅︀, P, {e, a, c}}, Θ_{15} = {∅︀, P, {e, b, c}}, Θ_{16} = {∅︀, P, {a, b, c}}. Subpolygroups of P are ∅︀, P, {e}. Let A be an arbitrary set, and a_{1}, a_{2} ∈ 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.
Every soft polygroup over a topological polygroup (non-discrete) is a soft topological polygroup.
Suppose that (P, Θ) is a topological polygroup, and ( ) is a soft polygroup over P. In this case, for all a ∈ A, is a subpolygroup of P. By contrast, P is a topological polygroup, and the mapping (a, b) ↦ a ○ b^{−1} of the topological space P×P to P is continuous. Thus, its restriction from to is also continuous. Therefore, ( ) is a soft topological polygroup over (P, Θ).
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.
(1) Note that ( ) and ( ) are soft topological polygroups over P. Therefore, their bi-intersection over P is the soft topological set (ℍ, C, Θ), where C = A ∩ B. For all c ∈ C, we have . In addition, both and are subpolygroups. Thus, ℍ(c) is a subpolygroup of P for all c ∈ A ∩ B. In contrast, and and condition (b) of Definition 1 hold for and . Thus, they also hold for ℍ(c) for all c ∈ C. Hence, ( ) is a soft topological polygroup over P.
(2) , and are subpolygroups of P, and condition (b) of Definition 1 is established for , and . Thus, it is also established for ℍ(c) for all c ∈ C, where C = A ∪ B.
Bi-intersectionis a soft topological polygroup over P, where, i ∈ I is a nonempty family of soft topological polygroups over P.
Suppose that C = ∩_{i}_{∈}_{I}A_{i} and . We have that are subpolygroups of P. Therefore, are subpolygroups of P as well, and condition (b) of Definition 1 holds for and is also established for .
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.
(1) If C = A × B, , then ℍ(a, b) is a subpolygroup of P. Condition (b) of Definition 1 holds on . Thus, is established on .
(2) If C = A ∪ B, , or then it is clear that conditions (a) and (b) of Definition 1 are established on ℍ(c).
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.
The proof is straightforward.
By referring to [9, 17], we can construct polygroup
○ | C_{1} | C_{2} | C_{3} | C_{4} | C_{5} |
---|---|---|---|---|---|
C_{1} | C_{1} | C_{2} | C_{3} | C_{4} | C_{5} |
C_{2} | C_{2} | C_{1} | C_{3} | C_{4} | C_{5} |
C_{3} | C_{3} | C_{3} | C_{1}, C_{2} | C_{5} | C_{4} |
C_{4} | C_{4} | C_{4} | C_{5} | C_{1}, C_{2} | C_{3} |
C_{5} | C_{5} | C_{5} | C_{4} | C_{3} | C_{1}, C_{2} |
As a sample of how to calculate the table entries, consider C_{3} · C_{3}. To determine this product, we compute the element-wise product of the conjugacy classes {r, t}{r, t} = {s, 1} = C_{1} ∪ C_{2}. Thus, C_{3} ○ C_{3} consists of the two conjugacy classes C_{1}, C_{2}. Hyperoperation
Consider
By referring to [9, 17], we can construct polygroup
* | 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
This means that (
If we consider
Let ( ) on P_{1} and ( ) on P_{2} be two soft topological polygroups. The product of ( ) and ( ) is denoted by , where Θ_{1} ×Θ_{2} induce topology on P_{1} ×P_{2} 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}).
A product of two soft topological polygroups is a soft topological polygroup.
Suppose that ( ) on P_{1} and ( ) on P_{2} are two soft topological polygroups. P_{1}×P_{2} is a polygroup, and is a subpolygroup of P_{1} × P_{2}. Maps and ^{−1} : are continuous because maps and are continuous. Thus, is a soft topological polygroup on P_{1} × P_{2}.
Extensions of polygroups by polygroups were investigated in [9]. By referring to [8, 17], we can construct . Several special cases of algebra are useful. Before describing them, we need to assign names to the two 2-element polygroups. Let 2 denote group Z_{2}, and let 3 denote polygroup S_{3}//〈(12)〉 ≅= Z_{3}/θ, where θ is the special conjugation with blocks {0}, {1, 2}. The multiplication table for 3 is as follows:
0 | 1 | |
---|---|---|
0 | 0 | 2 |
1 | 1 | {0, 1} |
System 3[ℳ] is the result of adding a new identity to polygroup [ℳ]. System 2[ℳ] is almost as good. For example, suppose that ℛ is the system with the following table:
0 | 1 | 2 | |
---|---|---|---|
0 | 0 | 1 | 2 |
1 | 1 | {0, 2} | {1, 2} |
2 | 2 | {1, 2} | {0, 1} |
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: Θ_{1} = {∅︀, 2[ℛ], {0}}, Θ_{2} = {∅︀, 2[ℛ], {a}}, Θ_{3} = {∅︀, 2[ℛ], {1}}, Θ_{4} = {∅︀, 2[ℛ], {2}}, Θ_{5} = {∅︀, 2[ℛ], {0, 1}}, Θ_{6} = {∅︀, 2[ℛ], {0, 2}}, Θ_{7} = {∅︀, 2[ℛ], {a, 1}}, Θ_{8} = {∅︀, 2[ℛ], {a, 2}}, Θ_{9} = {∅︀, 2[ℛ], {1, 2}}, Θ_{10} = {∅︀, 2[ℛ], {0, a, 1}}, Θ_{11} = {∅︀, 2[ℛ], {0, a, 2}}, Θ_{12} = {∅︀, 2[ℛ], {a, 1, 2}}, Θ_{13} = {∅︀, 2[ℛ], {0, 1, 2}}. However, ○ : 2[ℛ] × 2[ℛ] ↦ ℘(2[ℛ]) is continuous with Θ_{14} = {∅︀, 2[ℛ], {0, a}}, Θ_{15} = {∅︀, 2[ℛ], {0}, {a}}. This means that (2[ℛ],Θ_{dis}), (2[ℛ],Θ_{ndis}), (2[ℛ],Θ_{14}), and (2[ℛ],Θ_{15}) are topological polygroups. Subpolygroups of 2[ℛ] are ∅︀, 2[ℛ], {0}, {0, a}. Let A be an arbitrary set, and let a_{1}, a_{2}, a_{3} ∈ A. We define a soft set by
In this case, ( ) and ( ) are soft topological polygroups.
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: Θ_{1} = {∅︀, 3[ℛ], {a}}, Θ_{2} = {∅︀, 3[ℛ], {1}}, Θ_{3} = {∅︀, 3[ℛ], {2}}, Θ_{4} = {∅︀, 3[ℛ], {0, 1} }, Θ_{5} = {∅︀, 3[ℛ], {0, 2}}, Θ_{6} = {∅︀, 3[ℛ], {a, 1}}, Θ_{7} = {∅︀, 3[ℛ], {a, 2}}, Θ_{8} = {∅︀, 3[ℛ], {1, 2}}, Θ_{9} = {∅︀, 3[ℛ], {0, a, 1}}, Θ_{10} = {∅︀, 3[ℛ], {0, a, 2}}, Θ_{11} = {∅︀, 3[ℛ], {a, 1, 2}}. However, ○ : 3[ℛ] × 3[ℛ] ↦ ℘(3[ℛ]) is continuous with Θ_{12} = {∅︀, 3[ℛ], {0}}, Θ_{13} = {∅︀, 3[ℛ], {0, a}}, Θ_{14} = {∅︀, 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 a_{1}, a_{2} ∈ A. We consider a soft set :
In this case, ( ) are soft topological polygroups.
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: Θ_{1} = {∅︀, ℛ[2], {1}}, Θ_{2} = {∅︀, ℛ[2], {2}}, Θ_{3} = {∅︀, ℛ[2], {a}}, Θ_{4} = {∅︀, ℛ[2], {0, 1}}, Θ_{5} = {∅︀, ℛ[2], {0, 2}}, Θ_{6} = {∅︀, ℛ[2], {0, a}}, Θ_{7} = {∅︀, ℛ[2], {1, 2}}, Θ_{8} = {∅︀, ℛ[2], {1, a}}, Θ_{9} = {∅︀, ℛ[2], {2, a}}, Θ_{10} = {∅︀, ℛ[2], {0, 1, 2}}, Θ_{11} = {∅︀, ℛ[2], {0, 1, a}}, Θ_{12} = {∅︀, ℛ[2], {0, 2, a}}. However, ○ : ℛ[2]×ℛ[2] ↦ ℘(ℛ[2]) is continuous with Θ_{13} = {∅︀, ℛ[2], {0}}. Therefore, (ℛ[2], Θ_{dis}), (ℛ[2],Θ_{ndis}), and (ℛ[2], Θ_{13}) are soft topological polygroups. Subpolygroups of ℛ[2] are ∅︀,ℛ[2], {0}, {0, 1, 2}.
Let A be an arbitrary set, and a_{1}, a_{2}, a_{3} ∈ 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.
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: Θ_{1} = {∅︀, ℛ[3], {1}}, Θ_{2} = {∅︀, ℛ[3], {2}}, Θ_{3} = {∅︀, ℛ[3], {a}}, Θ_{4} = {∅︀, ℛ[3], {0, 1} }, Θ_{5} = {∅︀, ℛ[3], {0, 2}}, Θ_{6} = {∅︀, ℛ[3], {0, a}}, Θ_{7} = {∅︀, ℛ[3], {1, 2}}, Θ_{8} = {∅︀, ℛ[3], {1, a}}, Θ_{9} = {∅︀, ℛ[3], {2, a}}, Θ_{10} = {∅︀, ℛ[3], {0, 1, a}}, Θ_{11} = {∅︀, ℛ[3], {0, 2, a} }, Θ_{12} = {∅︀, ℛ[3], {1, 2, a}}. However, ○ : ℛ[3] × ℛ[3] ↦ ℘(ℛ[3]) is continuous, with Θ_{13} = {∅︀, ℛ[3], {0}}, Θ_{14} = {∅︀, 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 a_{1}, a_{2}, a_{3} ∈ A. We consider a soft set as
In this case, ( ) is a soft topological polygroup. Let A be an arbitrary set, and let a_{1}, a_{2} ∈ A. We define a soft set as
Then, ( ) is a soft topological polygroup. We can construct many examples using this method.
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: Θ_{1} = {∅︀, ℛ[ℛ], {1}}, Θ_{2} = {∅︀, ℛ[ℛ], {2}}, Θ_{3} = {∅︀, ℛ[ℛ], {a}}, Θ_{4} = {∅︀, ℛ[ℛ], {b}}, Θ_{5} = {∅︀, ℛ[ℛ], {0, 1}}, Θ_{6} = {∅︀,ℛ[ℛ], {0, 2} }, Θ_{7} = {∅︀,ℛ[ℛ], {0, a}}, Θ_{8} = {∅︀, ℛ[ℛ], {0, b}}, Θ_{9} = {∅︀, ℛ[ℛ], {1, 2}}, Θ_{10} = {∅︀, ℛ[ℛ], {1, a}}, Θ_{11} = {∅︀, ℛ[ℛ], {1, b}}, Θ_{12} = {∅︀, ℛ[ℛ], {2, a}}, Θ_{13} = {∅︀, ℛ[ℛ], {2, b}}, Θ_{14} = {∅︀, ℛ[ℛ], {0, 1, a}}, Θ_{15} = {∅︀, ℛ[ℛ], {0, 1, and b}}, Θ_{16} = {∅︀, ℛ[ℛ], {1, 2, a}}, Θ_{17} = {∅︀, ℛ[ℛ], {1, 2, b}}, Θ_{18} = {∅︀, ℛ[ℛ], {2, a, b}}, Θ_{19} = {∅︀, ℛ[ℛ], {0, 1, 2, a}}, Θ_{20} = {∅︀, ℛ[ℛ], {0, 1, 2, b}}, Θ_{21} = {∅︀, ℛ[ℛ], {1, 2, a, b}}. However, ○ : ℛ[ℛ] × ℛ[ℛ] ↦ ℘(ℛ[ℛ]) is continuous with Θ_{22} = {∅︀, ℛ[ℛ], {0}},Θ_{23} = {∅︀, 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 a_{1}, a_{2}, a_{3} ∈ A. We define a soft set by
Then, ( ) is a soft topological polygroup. Now, let A be an arbitrary set, and let a_{1}, a_{2} ∈ 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.
No potential conflict of interest relevant to this article was reported.
○ | |||
---|---|---|---|
{ |
○ | ||||
---|---|---|---|---|
{ |
{ |
|||
{ |
{ |
○ | ||||
---|---|---|---|---|
{ |
{ |
|||
{ |
○ | |||||
---|---|---|---|---|---|
* | 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} |
0 | 1 | |
---|---|---|
0 | 0 | 2 |
1 | 1 | {0, 1} |
0 | 1 | 2 | |
---|---|---|---|
0 | 0 | 1 | 2 |
1 | 1 | {0, 2} | {1, 2} |
2 | 2 | {1, 2} | {0, 1} |
○ | 0 | 1 | 2 | |
---|---|---|---|---|
0 | 0 | 1 | 2 | |
0 | 1 | 2 | ||
1 | 1 | 1 | {0, |
{1, 2} |
2 | 2 | 2 | {1, 2} | {0, |
○ | 0 | 1 | 2 | |
---|---|---|---|---|
0 | 0 | 1 | 2 | |
{0, |
1 | 2 | ||
1 | 1 | 1 | {0, |
{1, 2} |
2 | 2 | 2 | {1, 2} | {0, |
○ | 0 | 1 | 2 | |
---|---|---|---|---|
0 | 0 | 1 | 2 | |
1 | 1 | {0, 2} | {1, 2} | |
2 | 2 | {1, 2} | {0, 1} | |
{0, 1, 2} |
○ | 0 | 1 | 2 | |
---|---|---|---|---|
0 | 0 | 1 | 2 | |
1 | 1 | {0, 2} | {1, 2} | |
2 | 2 | {1, 2} | {0, 1} | |
{0, 1, 2, |
○ | 0 | 1 | 2 | ||
---|---|---|---|---|---|
0 | 0 | 1 | 2 | ||
1 | 1 | {0, 2} | {1, 2} | ||
2 | 2 | {1, 2} | {0, 1} | ||
{0, 1, 2, |
{ |
||||
{ |
{0, 1, 2, |
E-mail:
E-mail: davvaz@yazd.ac.ir