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International Journal of Fuzzy Logic and Intelligent Systems 2021; 21(1): 29-37

Published online March 25, 2021

https://doi.org/10.5391/IJFIS.2021.21.1.29

© The Korean Institute of Intelligent Systems

On Soft Topological Polygroups and Their Examples

Rasoul Mousarezaei and Bijan Davvaz

Department of Mathematics, Yazd University, Yazd, Iran

Correspondence to :
Bijan Davvaz (davvaz@yazd.ac.ir)

Received: November 10, 2020; Revised: February 11, 2021; Accepted: February 22, 2021

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Polygroups are a special subclass of hypergroups that satisfies group-like axioms. A polygroup is a multigroup that is completely regular and reversible in itself. A topological polygroup is a polygroup P with a topology on P that satisfies certain conditions. Moreover, the concept of soft sets is a general mathematical tool for dealing with uncertainty. In this study, we investigate soft topological polygroups over a polygroup. The ideas presented in this article can be used to build more polygroups and soft topological polygroups.

Keywords: Soft set, Topological polygroup, Soft polygroup, Soft topological polygroups

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. [35] 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 [1316].

Let U be an initial universe, and ℘(U) denote the power set of U. Suppose that E is a set of parameters, AE, 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 aA, 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 AB and for all aA. 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 aA, then ( ) is an absolute soft set, denoted by Â. Otherwise, if (null set) for all aA, then ( ) is a null soft set, denoted by ^̂. Symbol denotes the bi-intersection of two soft sets, which is defined as , where C = AB and for all aC. The soft set (ℍ, C) defines a union of ( ) and ( ), which is shown by , where C = AB and

(a)={F(a),if aA-B;G(a),if aB-A;F(a)G(a),if aAB.

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 hH, we have hH = H = Hh, and it is called a semihypergroup if for every t, u,wH, we have t○(uw) = (tu)○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 eP such that for all p, q, rP, the following conditions hold: (1) (pq) ○ r = p ○ (qr); (2) ep = pe = p; (3) if pqr, then qpr−1 and rq−1p. The following results about polygroups follow easily from the definition: epp−1p−1p, e−1 = e, (p−1)−1 = p, and (pq)−1 = q−1p−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 × GG defined by p(g, h) = gh is continuous when G×G is endowed with the product topology;

(2) Mapping inv : GG 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 aA;

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

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 aA. 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 SV = {U ∈ ℘*(H) | UV,Uτ} 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) ↦ xy, from H × H ↦ ℘*(H), is continuous; (2) mapping (x, y) ↦ x/y, from H × H ↦ ℘*(H), is continuous, where x/y = {zH|xzy}; (3) mapping (x, y) ↦ y\x, from H × H to ℘*(H), is continuous, where y/x = {zH | xyz}. 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 : PP 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, yP and Uτ such that xyU there exist V,Wτ such that xV and yW and VWU.

Example 2 ( [11])

Let P be a polygroup and β* be the fundamental relation of P. Then, τ = {∪uU β*(u) | UP}∪{∅︀} 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 aA, and (b) mapping (x, y) ↦ x ○ (−y) of the topological space onto is continuous for all aA.

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, ...,Hn. Let A be an arbitrary set such that a1, a2, ..., anA. We define as follows:

F(x)={Hi,if x=ai(i=1,,n);,otherwise.

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, gG}, ○,H,−1) is a polygroup with hyperoperation (HaH) ○ (HbH) = {HahbH | hH} and the unitary function (HgH)−1 = {Hg−1H}. Let π : GG//H be a function, where π(g) = HgH. We define topology ΘG//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//HG//H) is a topological polygroup. Now, if H1,H2, ...,Hn are subpolygroups of (G//HG//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.

Proof

Suppose that ( ) is a soft polygroup over P and (P, Θ) is a topological polygroup. Therefore, for all aA, is a subpolygroup of P. Mapping to is continuous. Note that P is a topological polygroup, and mapping (a, b) ↦ ab−1 of the topological space P × P to ℘(P) is continuous. Hence, ( ) is a soft topological polygroup over P.

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 : PP is continuous because −1 is the identity function (x−1 = x for all xP), and identity is continuous with every topology. However, hyperoperation ⋇: P × P ↦ ℘(P) is not continuous with Θ2 = {∅︀, P, {2}}. Therefore, P with Θ1, Θdisndis is a topological polygroup. Subpolygroups of P are ∅︀, P, {1}. Let A be an arbitrary set, and let a1, a2A. We define the soft set as follows:

F(x)={{1},if x=a1;P,if x=a2;,otherwise.

Therefore, ( ) is a soft topological polygroup.

Example 8

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

eab
eeab
aaeb
bbb{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, a2A. We define a soft set by

F(x)={{e},if x=a1;{e,a},if x=a2;,otherwise.

Therefore, is a soft topological polygroup because the restriction of topologies Θ34 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

eabc
eeabc
aa{e, a}c{b, c}
bbcea
cc{b, c}a{e, a}

Hyperoperation ○ : P × P ↦ ℘(P) is continuous with the following topologies: Θdis, Θndis, Θ1 = {∅︀, P, {e, b}}, Θ2 = {∅︀, P, {e}, {b}}. As x−1 = x for all xP, the inverse operation is identity, and the identity function is continuous with every topology. This means that P with topologies Θ12 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, a4A. We define the soft set :

F(x)={{e},if x=a1;{e,a},if x=a2;{e,b},if x=a3;P,if x=a4;,otherwise.

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

F(x)={{e},if x=a1;{e,a},if x=a2;,otherwise.

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

F(x)={{e},if x=a1;{e,a},if x=a2;{e,b},if x=a3;,otherwise.

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:

eabc
eeabc
aaaPc
bb{e, a, b}b{b, c}
cc{a, c}cP

By Theorem 2, hyperoperation ○ : P × P ↦ ℘(P) is continuous with the following topologies: Θdis, Θndis, 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 Θ2345 and continuous with Θ1. This means that P with topologies Θ1dis, and Θndis 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, a2A. We define a soft set :

F(x)={{e},if x=a1;P,if x=a2;,otherwise.

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

F(x)={{e},if x=a1;,otherwise.

Hyperoperation and the inverse operation −1 are continuous with (Θi)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.

Proof

Suppose that (P, Θ) is a topological polygroup, and ( ) is a soft polygroup over P. In this case, for all aA, is a subpolygroup of P. By contrast, P is a topological polygroup, and the mapping (a, b) ↦ ab−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).

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.

Proof

(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 = AB. For all cC, we have . In addition, both and are subpolygroups. Thus, ℍ(c) is a subpolygroup of P for all cAB. In contrast, and and condition (b) of Definition 1 hold for and . Thus, they also hold for ℍ(c) for all cC. 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 cC, where C = AB.

Theorem 6

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

Proof

Suppose that C = ∩iIAi 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 .

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.

Proof

(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 = AB, , or then it is clear that conditions (a) and (b) of Definition 1 are established on ℍ(c).

Theorem 8

Suppose that, iI 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 Ai is disjoint, is a soft topological polygroup over P.

Proof

The proof is straightforward.

Example 11

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

C1C2C3C4C5
C1C1C2C3C4C5
C2C2C1C3C4C5
C3C3C3C1, C2C5C4
C4C4C4C5C1, C2C3
C5C5C5C4C3C1, 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} = C1C2. Thus, C3C3 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¯, Θdis) and (D4¯, Θndis) 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, a4A. We define a soft set by

F(x)={{C1},if x=a1;{C1,C2},if x=a2;{C1,C2,C3}if x=a3;{C1,C2,C4}if x=a4;,otherwise.

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:

*12345
112345
221435
334125
443215
55555{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^, Θdis), and (D4^, Θndis) 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, a5A. Then, we define a soft set as follows:

F(x)={{1},if x=a1;{1,2},if x=a2;{1,3},if x=a3;{1,4},if x=a4;{1,2,3,4},if x=a5;,otherwise.

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) = (xz, yt), 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.

Proof

Suppose that ( ) on P1 and ( ) on P2 are two soft topological polygroups. P1×P2 is a polygroup, and is a subpolygroup of P1 × P2. Maps and −1 : are continuous because maps and are continuous. Thus, is a soft topological polygroup on P1 × P2.

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 Z2, and let 3 denote polygroup S3//〈(12)〉 ≅= Z3, where θ is the special conjugation with blocks {0}, {1, 2}. The multiplication table for 3 is as follows:

01
002
11{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:

012
0012
11{0, 2}{1, 2}
22{1, 2}{0, 1}

Example 13

Consider polygroup 2[ℛ]:

0a12
00a12
aa012
111{0, a, 2}{1, 2}
222{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],Θdis), (2[ℛ],Θndis), (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, a3A. We define a soft set by

F(x)={{0},if x=a1;{0,a},if x=a2;2[],if x=a3;otherwise.

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

Example 14

Consider polygroup 3[ℛ]:

0a12
00a12
aa{0, a}12
111{0, a, 2}{1, 2}
222{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], (Θi)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

F(x)={{0},if x=0;{0,a},if x=a;3[],if x=1;,if x=2.

Then, ( ) is a soft topological polygroup. Now, let A be an arbitrary set, and a1, a2A. We consider a soft set :

F(x)={,if x=a1;{0,a},if x=a2;{0},otherwise.

In this case, ( ) are soft topological polygroups.

Example 15

Consider polygroup ℛ[2]:

012a
0012a
11{0, 2}{1, 2}a
22{1, 2}{0, 1}a
aaaa{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], Θdis), (ℛ[2],Θndis), 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, a3A. We define a soft set by

F(x)={{0},if x=a1;{0,1,2},if x=a2;,if x=a3;[2],otherwise.

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

F(x)={{0},if x=a;{0,1,2},if x=2;,otherwise,

then ( ) is a soft topological polygroup.

Example 16

Consider polygroup ℛ[3]:

012a
0012a
11{0, 2}{1, 2}a
22{1, 2}{0, 1}a
aaaa{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],Θdis), (ℛ[3],Θndis), and (ℛ[3], (Θi)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, a3A. We consider a soft set as

F(x)={{0},if x=a1;{0,1,2},if x=a2;[3],if x=a3;,otherwise.

In this case, ( ) is a soft topological polygroup. Let A be an arbitrary set, and let a1, a2A. We define a soft set as

F(x)={{0,1,2},if x=a1;,if x=a2;{0},otherwise.

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

Example 17

Consider ℛ[ℛ]:

012ab
0012ab
11{0, 2}{1, 2}ab
22{1, 2}{0, 1}ab
aaaa{0, 1, 2, b}{a, b}
bbbb{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],Θdis), (ℛ[3],Θndis), and (ℛ[3], (Θi)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, a3A. We define a soft set by

F(x)={{0},if x=a1;{0,1,2},if x=a2;[],if x=a3;,otherwise.

Then, ( ) is a soft topological polygroup. Now, let A be an arbitrary set, and let a1, a2A. We define a soft set as

F(x)={{0},if x=a1;,if x=a2;{0,1,2},otherwise..

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.

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Article

Original Article

International Journal of Fuzzy Logic and Intelligent Systems 2021; 21(1): 29-37

Published online March 25, 2021 https://doi.org/10.5391/IJFIS.2021.21.1.29

Copyright © The Korean Institute of Intelligent Systems.

On Soft Topological Polygroups and Their Examples

Rasoul Mousarezaei and Bijan Davvaz

Department of Mathematics, Yazd University, Yazd, Iran

Correspondence to:Bijan Davvaz (davvaz@yazd.ac.ir)

Received: November 10, 2020; Revised: February 11, 2021; Accepted: February 22, 2021

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Polygroups are a special subclass of hypergroups that satisfies group-like axioms. A polygroup is a multigroup that is completely regular and reversible in itself. A topological polygroup is a polygroup P with a topology on P that satisfies certain conditions. Moreover, the concept of soft sets is a general mathematical tool for dealing with uncertainty. In this study, we investigate soft topological polygroups over a polygroup. The ideas presented in this article can be used to build more polygroups and soft topological polygroups.

Keywords: Soft set, Topological polygroup, Soft polygroup, Soft topological polygroups

1. Introduction

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. [35] 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 [1316].

2. Preliminaries

Let U be an initial universe, and ℘(U) denote the power set of U. Suppose that E is a set of parameters, AE, 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 aA, 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 AB and for all aA. 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 aA, then ( ) is an absolute soft set, denoted by Â. Otherwise, if (null set) for all aA, then ( ) is a null soft set, denoted by ^̂. Symbol denotes the bi-intersection of two soft sets, which is defined as , where C = AB and for all aC. The soft set (ℍ, C) defines a union of ( ) and ( ), which is shown by , where C = AB and

(a)={F(a),if aA-B;G(a),if aB-A;F(a)G(a),if aAB.

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 hH, we have hH = H = Hh, and it is called a semihypergroup if for every t, u,wH, we have t○(uw) = (tu)○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 eP such that for all p, q, rP, the following conditions hold: (1) (pq) ○ r = p ○ (qr); (2) ep = pe = p; (3) if pqr, then qpr−1 and rq−1p. The following results about polygroups follow easily from the definition: epp−1p−1p, e−1 = e, (p−1)−1 = p, and (pq)−1 = q−1p−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 .

3. Topological Hyperstructure

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

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

(2) Mapping inv : GG 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 aA;

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

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 aA. 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 SV = {U ∈ ℘*(H) | UV,Uτ} 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) ↦ xy, from H × H ↦ ℘*(H), is continuous; (2) mapping (x, y) ↦ x/y, from H × H ↦ ℘*(H), is continuous, where x/y = {zH|xzy}; (3) mapping (x, y) ↦ y\x, from H × H to ℘*(H), is continuous, where y/x = {zH | xyz}. 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 : PP 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, yP and Uτ such that xyU there exist V,Wτ such that xV and yW and VWU.

Example 2 ( [11])

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

4. Soft Topological Polygroups

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 aA, and (b) mapping (x, y) ↦ x ○ (−y) of the topological space onto is continuous for all aA.

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, ...,Hn. Let A be an arbitrary set such that a1, a2, ..., anA. We define as follows:

F(x)={Hi,if x=ai(i=1,,n);,otherwise.

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, gG}, ○,H,−1) is a polygroup with hyperoperation (HaH) ○ (HbH) = {HahbH | hH} and the unitary function (HgH)−1 = {Hg−1H}. Let π : GG//H be a function, where π(g) = HgH. We define topology ΘG//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//HG//H) is a topological polygroup. Now, if H1,H2, ...,Hn are subpolygroups of (G//HG//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.

Proof

Suppose that ( ) is a soft polygroup over P and (P, Θ) is a topological polygroup. Therefore, for all aA, is a subpolygroup of P. Mapping to is continuous. Note that P is a topological polygroup, and mapping (a, b) ↦ ab−1 of the topological space P × P to ℘(P) is continuous. Hence, ( ) is a soft topological polygroup over P.

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 : PP is continuous because −1 is the identity function (x−1 = x for all xP), and identity is continuous with every topology. However, hyperoperation ⋇: P × P ↦ ℘(P) is not continuous with Θ2 = {∅︀, P, {2}}. Therefore, P with Θ1, Θdisndis is a topological polygroup. Subpolygroups of P are ∅︀, P, {1}. Let A be an arbitrary set, and let a1, a2A. We define the soft set as follows:

F(x)={{1},if x=a1;P,if x=a2;,otherwise.

Therefore, ( ) is a soft topological polygroup.

Example 8

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

eab
eeab
aaeb
bbb{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, a2A. We define a soft set by

F(x)={{e},if x=a1;{e,a},if x=a2;,otherwise.

Therefore, is a soft topological polygroup because the restriction of topologies Θ34 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

eabc
eeabc
aa{e, a}c{b, c}
bbcea
cc{b, c}a{e, a}

Hyperoperation ○ : P × P ↦ ℘(P) is continuous with the following topologies: Θdis, Θndis, Θ1 = {∅︀, P, {e, b}}, Θ2 = {∅︀, P, {e}, {b}}. As x−1 = x for all xP, the inverse operation is identity, and the identity function is continuous with every topology. This means that P with topologies Θ12 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, a4A. We define the soft set :

F(x)={{e},if x=a1;{e,a},if x=a2;{e,b},if x=a3;P,if x=a4;,otherwise.

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

F(x)={{e},if x=a1;{e,a},if x=a2;,otherwise.

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

F(x)={{e},if x=a1;{e,a},if x=a2;{e,b},if x=a3;,otherwise.

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:

eabc
eeabc
aaaPc
bb{e, a, b}b{b, c}
cc{a, c}cP

By Theorem 2, hyperoperation ○ : P × P ↦ ℘(P) is continuous with the following topologies: Θdis, Θndis, 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 Θ2345 and continuous with Θ1. This means that P with topologies Θ1dis, and Θndis 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, a2A. We define a soft set :

F(x)={{e},if x=a1;P,if x=a2;,otherwise.

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

F(x)={{e},if x=a1;,otherwise.

Hyperoperation and the inverse operation −1 are continuous with (Θi)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.

Proof

Suppose that (P, Θ) is a topological polygroup, and ( ) is a soft polygroup over P. In this case, for all aA, is a subpolygroup of P. By contrast, P is a topological polygroup, and the mapping (a, b) ↦ ab−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).

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.

Proof

(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 = AB. For all cC, we have . In addition, both and are subpolygroups. Thus, ℍ(c) is a subpolygroup of P for all cAB. In contrast, and and condition (b) of Definition 1 hold for and . Thus, they also hold for ℍ(c) for all cC. 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 cC, where C = AB.

Theorem 6

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

Proof

Suppose that C = ∩iIAi 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 .

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.

Proof

(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 = AB, , or then it is clear that conditions (a) and (b) of Definition 1 are established on ℍ(c).

Theorem 8

Suppose that, iI 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 Ai is disjoint, is a soft topological polygroup over P.

Proof

The proof is straightforward.

Example 11

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

C1C2C3C4C5
C1C1C2C3C4C5
C2C2C1C3C4C5
C3C3C3C1, C2C5C4
C4C4C4C5C1, C2C3
C5C5C5C4C3C1, 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} = C1C2. Thus, C3C3 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¯, Θdis) and (D4¯, Θndis) 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, a4A. We define a soft set by

F(x)={{C1},if x=a1;{C1,C2},if x=a2;{C1,C2,C3}if x=a3;{C1,C2,C4}if x=a4;,otherwise.

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:

*12345
112345
221435
334125
443215
55555{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^, Θdis), and (D4^, Θndis) 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, a5A. Then, we define a soft set as follows:

F(x)={{1},if x=a1;{1,2},if x=a2;{1,3},if x=a3;{1,4},if x=a4;{1,2,3,4},if x=a5;,otherwise.

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) = (xz, yt), 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.

Proof

Suppose that ( ) on P1 and ( ) on P2 are two soft topological polygroups. P1×P2 is a polygroup, and is a subpolygroup of P1 × P2. Maps and −1 : are continuous because maps and are continuous. Thus, is a soft topological polygroup on P1 × P2.

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 Z2, and let 3 denote polygroup S3//〈(12)〉 ≅= Z3, where θ is the special conjugation with blocks {0}, {1, 2}. The multiplication table for 3 is as follows:

01
002
11{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:

012
0012
11{0, 2}{1, 2}
22{1, 2}{0, 1}

Example 13

Consider polygroup 2[ℛ]:

0a12
00a12
aa012
111{0, a, 2}{1, 2}
222{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],Θdis), (2[ℛ],Θndis), (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, a3A. We define a soft set by

F(x)={{0},if x=a1;{0,a},if x=a2;2[],if x=a3;otherwise.

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

Example 14

Consider polygroup 3[ℛ]:

0a12
00a12
aa{0, a}12
111{0, a, 2}{1, 2}
222{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], (Θi)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

F(x)={{0},if x=0;{0,a},if x=a;3[],if x=1;,if x=2.

Then, ( ) is a soft topological polygroup. Now, let A be an arbitrary set, and a1, a2A. We consider a soft set :

F(x)={,if x=a1;{0,a},if x=a2;{0},otherwise.

In this case, ( ) are soft topological polygroups.

Example 15

Consider polygroup ℛ[2]:

012a
0012a
11{0, 2}{1, 2}a
22{1, 2}{0, 1}a
aaaa{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], Θdis), (ℛ[2],Θndis), 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, a3A. We define a soft set by

F(x)={{0},if x=a1;{0,1,2},if x=a2;,if x=a3;[2],otherwise.

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

F(x)={{0},if x=a;{0,1,2},if x=2;,otherwise,

then ( ) is a soft topological polygroup.

Example 16

Consider polygroup ℛ[3]:

012a
0012a
11{0, 2}{1, 2}a
22{1, 2}{0, 1}a
aaaa{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],Θdis), (ℛ[3],Θndis), and (ℛ[3], (Θi)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, a3A. We consider a soft set as

F(x)={{0},if x=a1;{0,1,2},if x=a2;[3],if x=a3;,otherwise.

In this case, ( ) is a soft topological polygroup. Let A be an arbitrary set, and let a1, a2A. We define a soft set as

F(x)={{0,1,2},if x=a1;,if x=a2;{0},otherwise.

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

Example 17

Consider ℛ[ℛ]:

012ab
0012ab
11{0, 2}{1, 2}ab
22{1, 2}{0, 1}ab
aaaa{0, 1, 2, b}{a, b}
bbbb{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],Θdis), (ℛ[3],Θndis), and (ℛ[3], (Θi)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, a3A. We define a soft set by

F(x)={{0},if x=a1;{0,1,2},if x=a2;[],if x=a3;,otherwise.

Then, ( ) is a soft topological polygroup. Now, let A be an arbitrary set, and let a1, a2A. We define a soft set as

F(x)={{0},if x=a1;,if x=a2;{0,1,2},otherwise..

Then, ( ) is a soft topological polygroup.

5. Conclusion

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.

eab
eeab
aaeb
bbb{e, a}

eabc
eeabc
aa{e, a}c{b, c}
bbcea
cc{b, c}a{e, a}

eabc
eeabc
aaaPc
bb{e, a, b}b{b, c}
cc{a, c}cP

C1C2C3C4C5
C1C1C2C3C4C5
C2C2C1C3C4C5
C3C3C3C1, C2C5C4
C4C4C4C5C1, C2C3
C5C5C5C4C3C1, C2

*12345
112345
221435
334125
443215
55555{1, 2, 3, 4}

01
002
11{0, 1}

012
0012
11{0, 2}{1, 2}
22{1, 2}{0, 1}

0a12
00a12
aa012
111{0, a, 2}{1, 2}
222{1, 2}{0, a, 1}

0a12
00a12
aa{0, a}12
111{0, a, 2}{1, 2}
222{1, 2}{0, a, 1}

012a
0012a
11{0, 2}{1, 2}a
22{1, 2}{0, 1}a
aaaa{0, 1, 2}

012a
0012a
11{0, 2}{1, 2}a
22{1, 2}{0, 1}a
aaaa{0, 1, 2, a}

012ab
0012ab
11{0, 2}{1, 2}ab
22{1, 2}{0, 1}ab
aaaa{0, 1, 2, b}{a, b}
bbbb{a, b}{0, 1, 2, a}

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