On the Algebraic Characteristics of Fuzzy Sub e-Groups

Bhunia, Supriya; Ghorai, Ganesh; Kutbi, Marwan Amin; Gulzar, Muhammad; Alam, Md Ashraful

doi:https://doi.org/10.1155/2021/5253346

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Developments in Functions and Operators of Complex Variables

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Volume 2021 | Article ID 5253346 | https://doi.org/10.1155/2021/5253346

On the Algebraic Characteristics of Fuzzy Sub e-Groups

Supriya Bhunia,¹Ganesh Ghorai,¹Marwan Amin Kutbi,²Muhammad Gulzar,³and Md Ashraful Alam⁴

Academic Editor: Sarfraz Nawaz Malik

Received24 Aug 2021

Revised16 Sept 2021

Accepted17 Sept 2021

Published20 Oct 2021

Abstract

Fuzzy set is a modern tool for depicting uncertainty. This paper introduces the concept of fuzzy sub e-group as an extension of fuzzy subgroup. The concepts of identity and inverse are generalized in fuzzy sub e-groups. Every fuzzy subgroup is proven to be a fuzzy sub e-group, but the converse is not true. Various properties of fuzzy sub e-groups are established. Moreover, the concepts of proper fuzzy sub e-group and super fuzzy sub e-group are discussed. Further, the concepts of fuzzy e-coset and normal fuzzy sub e-group are presented. Finally, we describe the effect of e-group homomorphism on normal fuzzy sub e-groups.

1. Introduction

Many decades ago, researchers developed an algebraic structure and named it as a group. Various properties of groups are proposed later on. In group theory, every group contains a unique identity element and every element has a unique inverse. In 2018, Saeid et al. [1] generalized the notion of groups to a new algebraic structure as e-groups. They generalized the notion of the identity of a group. Instead of choosing a single element as an identity element, Saeid et al. [1] considered a subset of the main set as an identity set. So, in an e-group, the identity element needs not to be unique. They proved that every group is an e-group, but the converse is not true. They defined homomorphism on e-groups in a different manner. E-group is an important tool for classifying isotopes. It is also a physical background in the unified Gauge theory.

Uncertainty is a massive component in the life of a person. In 1965, in his pioneer paper, Zadeh [2] first defined fuzzy set to handle uncertainty in real-life problems. In 1971, utilizing the concept of fuzzy set, Rosenfeld [3] first defined fuzzy subgroup. In 1979, using the -norm concept of the fuzzy subgroup was restructured by Anthony and Sherwood [4, 5]. In 1981, the idea of fuzzy level subgroup was introduced by Das [6]. In 1988, Choudhury et al. [7] proved various properties of fuzzy homomorphism. In 1990, Dixit et al. [8] discussed the union of fuzzy subgroups and fuzzy level subgroups. The concept of antifuzzy subgroups was proposed by Biswas [9]. In 1992, Ajmal and Prajapati [10] developed fuzzy cosets and fuzzy normal subgroups. Chakraborty and Khare [11] studied various properties of fuzzy homomorphism. Ajmal [12] also studied the homomorphism of fuzzy subgroups. Later, many researchers studied various properties of fuzzy subgroups [13–16]. In 2015, Tarnauceanu [17] classified fuzzy normal subgroups of finite groups. In 2016, Onasanya [18] reviewed some antifuzzy properties of fuzzy subgroups. In 2018, Shuaib and Shaheryar [19] introduced omicron fuzzy subgroups. In 2018, Addis [20] developed fuzzy homomorphism theorems on groups. In 2019, Bhunia and Ghorai [21] studied -Pythagorean fuzzy subgroups. In 2021, Bhunia et al. [22, 23] developed Pythagorean fuzzy subgroups. Abuhijleh et al. [24] worked on complex fuzzy subgroups in 2021. Alolaiyan et al. [25] studied algebraic structure of -complex fuzzy subgroups. Alolaiyan et al. [26] developed bipolar fuzzy subrings in 2021. In 2021, Talafha et al. [27] studied fuzzy fundamental groups and fuzzy folding of fuzzy Minkowski space.

In a fuzzy subgroup, the identity element of the group has the highest membership value. Also, in a fuzzy subgroup, the membership value of an element and its inverse are equal. But in an e-group, there is no unique identity element and no direct concept of the inverse of elements. Till now, no fuzzification has been done for e-groups. So, it is a challenge for us to fuzzify e-groups. In this study, we construct the concept of fuzzy sub e-groups. Here, we show that every fuzzy subgroup is a fuzzy sub e-group, but the converse is not true. So, the idea of fuzzy sub e-group is a much more generalized concept. In the study of isotopes, we notice that isotopes decay through neutron emission. So, the membership degree of these unstable neutrons must lie in . Therefore, fuzzy sub e-group will be more efficient in studying these unstable isotopes rather than a crisp e-group.

This paper is arranged in the following order. In Section 2, we recall some important concepts. In Section 3, utilizing the concept of e-groups, we generalize fuzzy subgroups. We develop the concept of fuzzy sub e-group and show that any fuzzy subgroup is also a fuzzy sub e-group. We also prove many algebraic properties of fuzzy sub e-groups. Further, we define fuzzy e-coset and normal fuzzy sub e-group in Section 4. Moreover, in Section 5, we show that after e-group homomorphism, a fuzzy sub e-group remains a fuzzy sub e-group. Finally, the conclusion is given in Section 6.

2. Preliminaries

Here, we will go over some basic definitions and concepts, which will be useful in the following sections.

Definition 1 (see [2]). A fuzzy set (FS) on a crisp set is an object having the form , where is the membership function.

Definition 2 (see [3]). Let be a FS on a group . Then, is referred to be a fuzzy subgroup (FSG) of if the following conditions hold: (i)(ii)

Definition 3 (see [2]). Let be a FS on . Then, for any , the set is called a-cut of .

Clearly, is a subset of .

Proposition 4 (see [28]). Let be a mapping from into . Let and be the two FSs on and , respectively. Then, and are FSs on and , respectively, where for all and for all , .

Definition 5 (see [1]). Let be a nonempty crisp set and . Then, is an e-group, where is the binary operation on , which meets the following criteria: (i)(ii)For every , an element such that (iii)For every , an element such that and

Definition 6 (see [1]). Let and be the two e-groups. If a mapping meets the following criteria, it is referred to as a homomorphism: (i)(ii)

3. Fuzzy Sub e-Group and Its Properties

In this section, fuzzy sub e-group is briefly described as a generalization of fuzzy subgroup. The notions of identity and inverse are generalized in fuzzy sub e-group. We investigate its properties. We define super fuzzy sub e-group. We check whether union and intersections of fuzzy sub e-group are fuzzy sub e-groups.

Definition 7. A FS is referred to be a fuzzy sub e-group of an e-group if the following conditions hold: (i)(ii) and where is the membership function.

Example 8. Let and . Define a binary operation on as below. Then, is an e-group.
Here, we assign membership degrees to the elements of by and .
Now, .
Similarly, we can check for other elements of .
Therefore, for all , .
Also, and .
Thus, forms a fuzzy sub e-group of the e-group .

Theorem 9. Let stand for an e-group and be a FS on . Then, is referred to be a fuzzy sub e-group of if for all and , for some such that , .

Proof. Let stand for a fuzzy sub e-group of .
Then, for all and in , .
Let , . Then, some such that and .
Therefore, .
Conversely, assume that , for some such that and . Let , .
Here, , then .
Therefore, , where and .
Let for all , such that and .
Therefore, .
Hence, the e-group has a fuzzy sub e-group .

Remark 10. The above theorem gives the necessary and sufficient condition for a FS of an e-group to be a fuzzy sub e-group.

Now, we will demonstrate that any FSG within a group is also a fuzzy sub e-group of the e-group , where is the group’s identity element. But the converse needs not to be true.

Theorem 11. Any fuzzy subgroup of a group is a fuzzy sub e-group.

Proof. Let stand for a FSG of a group .
Then, and for all and .
So, the first condition of fuzzy sub e-group is satisfied.
As is a group, then there is a unique identity element in .
Since is a FSG of , .
Now, we take ; then, .
Therefore, , where and
Hence, the FSG of is a fuzzy sub e-group of the e-group .

Example 12. Let and .
Define on as binary operation by the following: Then, is an e-group.
Now, we assign a membership value to each of the elements of by the following: Now, we can verify that is a fuzzy sub e-group of the e-group .
But . So, is not a group. Hence, the FS is not a FSG.

Remark 13. A fuzzy sub e-group of an e-group is not necessarily a FSG.

Definition 14. A fuzzy sub e-group of an e-group which is not a FSG is said to be a proper fuzzy sub e-group.
The fuzzy sub e-group in Example 12. is a proper fuzzy sub e-group.
Now, we will check about union and intersection of fuzzy sub e-groups.

Theorem 15. Intersection of fuzzy sub e-groups of an e-group is also a fuzzy sub e-group of that e-group.

Proof. Let and be the two fuzzy sub e-groups of an e-group .
Then, , and , where and .
Also, , and , where and .
Let be the intersection of and , where is given by .
Now for all , Therefore, for all .
Again for and , we have the following: Therefore, is a fuzzy sub e-group of the e-group .
Hence, the intersection of two fuzzy sub e-groups of an e-group is also a fuzzy sub e-group of that e-group.

Corollary 16. Intersection of any fuzzy sub e-groups of an e-group is also a fuzzy sub e-group of that e-group .

Remark 17. Union of two fuzzy sub e-groups of an e-group may not be a fuzzy sub e-group of that e-group.

Example 18. Let us take the e-group , where , , and is the addition of integers.
Let and be the two fuzzy sub e-groups of the e-group , where and are presented by the following: Let be the union of and , where is given by .
Therefore, Now, , but .
So, .
Hence, is not a fuzzy sub e-group of the e-group .

Definition 19. Let and be the two fuzzy sub e-groups of an e-group such that for all and for all , then is referred to be a super fuzzy sub e-group of .

Example 20. In Example 12., we take another FS on , where Then, we can simply verify that has a fuzzy sub e-group .
Now, we can see that for all , and for all , .
Hence, is a super fuzzy sub e-group of .

Theorem 21. Let stand for a fuzzy sub e-group of an e-group . Then, the set forms a sub e-group of the e-group , where .

Proof. Given , where .
To show that the e-group has a sub e-group , we have to show that itself forms an e-group.
Since is an e-group, the associative law holds.
Clearly, is a subset of . Then, associative law also holds in . Instead of showing the other two conditions of e-group, we will show that for all and , a such that and .
Let , , and . Then, .
Since is a fuzzy sub e-group of , by Theorem 9, we have the following: Similarly, we can show that .
Since , and .
Hence, forms a sub e-group of .

4. Normal Fuzzy Sub e-Group and Level Fuzzy Sub e-Group

This section will describe fuzzy e-cosets and normal fuzzy sub e-groups. We will also introduce the concept of level fuzzy sub e-groups.

Definition 22. Let stand for a fuzzy sub e-group of an e-group . Then, , the left fuzzy e-coset is defined by and the right fuzzy e-coset is defined by , where is any element of and such that and .

If a left fuzzy e-coset is also a right fuzzy e-coset, then we will simply call it is a fuzzy e-coset.

Definition 23. Let stand for a fuzzy sub e-group of an e-group . Then, forms a normal fuzzy sub e-group of the e-group if every left fuzzy e-coset of is a right fuzzy e-coset of in .
Equivalently, for all .

Example 24. Let us take the e-group . Now, forms a fuzzy sub e-group on , where is presented by the following: Let us take .
Then, , the left fuzzy e-coset is presented by and the right fuzzy e-coset is presented by .
Since addition is commutative on , then . Similarly, we can check for other elements of . Hence, forms a normal fuzzy sub e-group of .

In the next theorem, we represent the necessary and sufficient condition for a fuzzy sub e-group to be a normal fuzzy sub e-group.

Theorem 25. Let stand for a fuzzy sub e-group of an e-group . Then, forms a normal fuzzy sub e-group of the e-group iff .

Proof. Let stand for a normal fuzzy sub e-group of the e-group .
Then, every left fuzzy e-coset of is also a right fuzzy e-coset of in .
Therefore, for all . That is .
This suggests that such that .
Now, we put . Therefore, .
Conversely, let .
Let . Since is an e-group, then a such that .
Suppose . Then, .
Therefore, . Thus, .
Hence, forms a normal fuzzy sub e-group of .

Theorem 26. Let stand for a fuzzy sub e-group of an e-group . Then, the a-cut of forms a sub e-group of the e-group , where .

Proof. We have , where and .
Clearly, is nonempty as .
To show that is an e-subgroup of , we need to prove that for , , where .
Let and . Then, and .
Since forms a fuzzy sub e-group of the e-group , Therefore, , where . Hence, forms a sub e-group of the e-group .

Definition 27. The sub e-group of the e-group is referred to as a level fuzzy sub e-group of .

Example 28. We consider the fuzzy sub e-group of the e-group in Example 12..
Choose . Then, and .
Clearly, .
We can easily check that is a sub e-group of the e-group .
Therefore, is a level fuzzy sub e-group of .

5. Homomorphism of Fuzzy Sub e-Groups

We shall demonstrate some important theorems on fuzzy sub e-group homomorphism in this section.

Theorem 29. Let and be the two e-groups. Let be a bijective homomorphism from to and be a fuzzy sub e-group of . Then, forms a fuzzy sub e-group of .

Proof. Let and be the two elements of . Now, Therefore, for all and .
Let and .
Since is a homomorphism, as . Now, ☐

Therefore, for all and .

Hence, forms a fuzzy sub e-group of .

Theorem 30. Let and be the two e-groups. Let be a homomorphism from to and be a fuzzy sub e-group of . Then, forms a fuzzy sub e-group of .

Proof. Let and be the two elements of .
If either or then, Suppose and for some , . Now, Therefore for all and .
Let and .
Since is a homomorphism, . Now, Therefore, for all and .
Hence, forms a fuzzy sub e-group of .

Theorem 31. Let and be the two e-groups. Let be a bijective homomorphism from to and be a normal fuzzy sub e-group of . Then, forms a normal fuzzy sub e-group of .

Proof. From Theorem 29, we can say that forms a fuzzy sub e-group of .
Since forms a normal fuzzy sub e-group of , for all , .
Let and be the two elements of . Then, Therefore, for all and .
Hence, forms a normal fuzzy sub e-group of .

Theorem 32. Let and be the two e-groups. Let be a bijective homomorphism from to and be a normal fuzzy sub e-group of . Then, forms a normal fuzzy sub e-group of .

Proof. From Theorem 30, we can say that is a fuzzy sub e-group of .
Since forms a normal fuzzy sub e-group of , for all , .
Let and be the two elements of .
Suppose that there are unique and , such that and . Now, Therefore, .
Hence, forms a normal fuzzy sub e-group of .

6. Conclusion

In this paper, we presented a brief demonstration of fuzzy sub e-groups and its properties. A condition is given for a FS of an e-group to be a fuzzy sub e-group. We have demonstrated that any fuzzy sub e-group forms a fuzzy subgroup. However, the reverse is not always true. Therefore, fuzzy sub e-group is the generalization of fuzzy subgroup. We have presented the difference between FSG and fuzzy sub e-group. We have discussed about the idea of normal fuzzy sub e-groups and level fuzzy e-subgroups. Finally, we have explained the effect of e-group homomorphism on fuzzy sub e-groups. In future, we will work on important theorems like Lagrange’s theorem and Sylow theorem in fuzzy sub e-groups.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this article.

Acknowledgments

The third author expresses his thanks and gratefulness to King Abdulaziz University (Jeddah, Saudi Arabia) for the unlimited support during this research.

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Copyright © 2021 Supriya Bhunia et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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