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
Rasoul Mousarezaei and Bijan Davvaz
Department of Mathematics, Yazd University, Yazd, Iran
Correspondence to :
Bijan Davvaz (davvaz@yazd.ac.ir)
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. [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
If for all
In addition, ( ) AND ( ) is denoted by and defined by , where for all (
Let
A topological group is a group
(1) Mapping
(2) Mapping
Let (
(a)
(b) Mapping (
Take
Let (
Let
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
Topology Θ on
Let
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 (
With the above conditions, ( ) is a soft topological polygroup over
Suppose that (
Suppose that ( ) is a soft polygroup over
Let
Therefore, ( ) is a soft topological polygroup.
Let
○ | |||
---|---|---|---|
{ |
By Theorem 2, hyperoperation ○ :
Therefore, is a soft topological polygroup because the restriction of topologies Θ_{3},Θ_{4} to subspaces {
Let
○ | ||||
---|---|---|---|---|
{ | { | |||
{ | { |
Hyperoperation ○ :
Therefore, is a soft topological polygroup. Now, we consider Θ_{5} = {∅︀,
Hyperoperation is continuous with Θ_{5}. Therefore, ( ) is a soft topological polygroup. Suppose that Θ_{6} = {∅︀,
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
○ | ||||
---|---|---|---|---|
{ | { | |||
{ |
By Theorem 2, hyperoperation ○ :
Therefore, are soft topological polygroups. We define a soft set as follows:
Hyperoperation and the inverse operation ^{−1} are continuous with (Θ
Using Theorems 4–9, we can build many other examples using the examples given in this article.
Suppose that (
(
(
(1) Note that ( ) and ( ) are soft topological polygroups over
(2) , and are subpolygroups of
Suppose that
(
(
(1) If
(2) If
(
(
The proof is straightforward.
By referring to [9, 17], we can construct polygroup
○ | |||||
---|---|---|---|---|---|
As a sample of how to calculate the table entries, consider
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
Suppose that ( ) on
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
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 | 1 | 2 | |
---|---|---|---|---|
0 | 0 | 1 | 2 | |
0 | 1 | 2 | ||
1 | 1 | 1 | {0, | {1, 2} |
2 | 2 | 2 | {1, 2} | {0, |
Hyperoperation ○ : 2[ℛ]×2[ℛ] ↦ ℘(2[ℛ]) is not continuous with the following topologies: Θ_{1} = {∅︀, 2[ℛ], {0}}, Θ_{2} = {∅︀, 2[ℛ], {
In this case, ( ) and ( ) are soft topological polygroups.
Consider polygroup 3[ℛ]:
○ | 0 | 1 | 2 | |
---|---|---|---|---|
0 | 0 | 1 | 2 | |
{0, | 1 | 2 | ||
1 | 1 | 1 | {0, | {1, 2} |
2 | 2 | 2 | {1, 2} | {0, |
Hyperoperation ○ : 3[ℛ]×3[ℛ] ↦ ℘(3[ℛ]) is not continuous with the following topologies: Θ_{1} = {∅︀, 3[ℛ], {
Then, ( ) is a soft topological polygroup. Now, let
In this case, ( ) are soft topological polygroups.
Consider polygroup ℛ[2]:
○ | 0 | 1 | 2 | |
---|---|---|---|---|
0 | 0 | 1 | 2 | |
1 | 1 | {0, 2} | {1, 2} | |
2 | 2 | {1, 2} | {0, 1} | |
{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], {
Let
Then, ( ) is a soft topological polygroup. If
then ( ) is a soft topological polygroup.
Consider polygroup ℛ[3]:
○ | 0 | 1 | 2 | |
---|---|---|---|---|
0 | 0 | 1 | 2 | |
1 | 1 | {0, 2} | {1, 2} | |
2 | 2 | {1, 2} | {0, 1} | |
{0, 1, 2, |
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], {
In this case, ( ) is a soft topological polygroup. Let
Then, ( ) is a soft topological polygroup. We can construct many examples using this method.
Consider ℛ[ℛ]:
○ | 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, |
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], {
Then, ( ) is a soft topological polygroup. Now, let
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.
E-mail:
E-mail: davvaz@yazd.ac.ir
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.
Rasoul Mousarezaei and Bijan Davvaz
Department of Mathematics, Yazd University, Yazd, Iran
Correspondence to:Bijan Davvaz (davvaz@yazd.ac.ir)
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. [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
If for all
In addition, ( ) AND ( ) is denoted by and defined by , where for all (
Let
A topological group is a group
(1) Mapping
(2) Mapping
Let (
(a)
(b) Mapping (
Take
Let (
Let
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
Topology Θ on
Let
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 (
With the above conditions, ( ) is a soft topological polygroup over
Suppose that (
Suppose that ( ) is a soft polygroup over
Let
Therefore, ( ) is a soft topological polygroup.
Let
○ | |||
---|---|---|---|
{ |
By Theorem 2, hyperoperation ○ :
Therefore, is a soft topological polygroup because the restriction of topologies Θ_{3},Θ_{4} to subspaces {
Let
○ | ||||
---|---|---|---|---|
{ | { | |||
{ | { |
Hyperoperation ○ :
Therefore, is a soft topological polygroup. Now, we consider Θ_{5} = {∅︀,
Hyperoperation is continuous with Θ_{5}. Therefore, ( ) is a soft topological polygroup. Suppose that Θ_{6} = {∅︀,
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
○ | ||||
---|---|---|---|---|
{ | { | |||
{ |
By Theorem 2, hyperoperation ○ :
Therefore, are soft topological polygroups. We define a soft set as follows:
Hyperoperation and the inverse operation ^{−1} are continuous with (Θ
Using Theorems 4–9, we can build many other examples using the examples given in this article.
Suppose that (
(
(
(1) Note that ( ) and ( ) are soft topological polygroups over
(2) , and are subpolygroups of
Suppose that
(
(
(1) If
(2) If
(
(
The proof is straightforward.
By referring to [9, 17], we can construct polygroup
○ | |||||
---|---|---|---|---|---|
As a sample of how to calculate the table entries, consider
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
Suppose that ( ) on
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
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 | 1 | 2 | |
---|---|---|---|---|
0 | 0 | 1 | 2 | |
0 | 1 | 2 | ||
1 | 1 | 1 | {0, | {1, 2} |
2 | 2 | 2 | {1, 2} | {0, |
Hyperoperation ○ : 2[ℛ]×2[ℛ] ↦ ℘(2[ℛ]) is not continuous with the following topologies: Θ_{1} = {∅︀, 2[ℛ], {0}}, Θ_{2} = {∅︀, 2[ℛ], {
In this case, ( ) and ( ) are soft topological polygroups.
Consider polygroup 3[ℛ]:
○ | 0 | 1 | 2 | |
---|---|---|---|---|
0 | 0 | 1 | 2 | |
{0, | 1 | 2 | ||
1 | 1 | 1 | {0, | {1, 2} |
2 | 2 | 2 | {1, 2} | {0, |
Hyperoperation ○ : 3[ℛ]×3[ℛ] ↦ ℘(3[ℛ]) is not continuous with the following topologies: Θ_{1} = {∅︀, 3[ℛ], {
Then, ( ) is a soft topological polygroup. Now, let
In this case, ( ) are soft topological polygroups.
Consider polygroup ℛ[2]:
○ | 0 | 1 | 2 | |
---|---|---|---|---|
0 | 0 | 1 | 2 | |
1 | 1 | {0, 2} | {1, 2} | |
2 | 2 | {1, 2} | {0, 1} | |
{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], {
Let
Then, ( ) is a soft topological polygroup. If
then ( ) is a soft topological polygroup.
Consider polygroup ℛ[3]:
○ | 0 | 1 | 2 | |
---|---|---|---|---|
0 | 0 | 1 | 2 | |
1 | 1 | {0, 2} | {1, 2} | |
2 | 2 | {1, 2} | {0, 1} | |
{0, 1, 2, |
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], {
In this case, ( ) is a soft topological polygroup. Let
Then, ( ) is a soft topological polygroup. We can construct many examples using this method.
Consider ℛ[ℛ]:
○ | 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, |
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], {
Then, ( ) is a soft topological polygroup. Now, let
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.
○ | |||
---|---|---|---|
{ |
○ | ||||
---|---|---|---|---|
{ | { | |||
{ | { |
○ | ||||
---|---|---|---|---|
{ | { | |||
{ |
○ | |||||
---|---|---|---|---|---|
* | 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, |
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