On the Characterization of Antineutrosophic Subgroup

Gayen, Sudipta; Edalatpanah, S. A.; Jha, Sripati; Kumar, Ranjan

doi:https://doi.org/10.1155/2023/4430103

Advances in Mathematical Physics

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Application of Fuzzy Set and its Extensions in Engineering and Sciences: Theory, Models, and Simulations

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Research Article | Open Access

Volume 2023 | Article ID 4430103 | https://doi.org/10.1155/2023/4430103

On the Characterization of Antineutrosophic Subgroup

Sudipta Gayen,¹S. A. Edalatpanah,²Sripati Jha,³and Ranjan Kumar⁴

Academic Editor: Mohammad Mirzazadeh

Received11 Jul 2022

Revised28 Dec 2022

Accepted10 Apr 2023

Published17 May 2023

Abstract

This article gives some essential scopes to study the characterizations of the antineutrosophic subgroup and antineutrosophic normal subgroup. Again, several theories and properties have been mentioned which are essential for analyzing their mathematical framework. Moreover, their homomorphic properties have been discussed.

1. Introduction

Fuzzy set (FS) [1] theory was introduced to handle uncertain situations more precisely than crisp sets. But there may exist some complex uncertain situations for which even FS theory is insufficient. As a result, intuitionistic fuzzy set (IFS) [2] and neutrosophic set (NS) [3] theories evolved, where the latter is more capable of dealing with uncertainties. Apart from these, there exist several byproducts of these set theories, like interval-valued versions [4–6]; type-I, type-II, and type-III versions; and soft [7–9] and hard versions. Presently, these theories have been adopted by several researchers in different applied fields. Also, in several pure mathematical fields, these notions are being utilized. In abstract algebra, Rosenfeld [10] was the pioneer to do so. He defined and studied the characteristics of a fuzzy subgroup (FSG). Thereafter, Das [11] presented the concept of the level subgroup of a FSG and showed several beautiful relationships between them. Afterward, Anthony and Sherwood [12, 13] redefined FSG by applying general T-norms and defined function generated FSG and subgroup generated FSG. In 1984, Mukherjee and Bhattacharya [14] introduced normal versions of FSG and cosets. Furthermore, Biswas [15] established the concept of intuitionistic fuzzy subgroup (IFSG). Similarly, Çetkin and Aygün [16] developed the neutrosophic subgroup (NSG) and studied its homomorphic properties. They have also established some connections between an NSG and its level subgroup.

The concept of the antifuzzy subgroup (AFSG) [17] is a kind of dual to FSG. It was defined and characterized by Biswas in 1990. He has mentioned some relationships between FSG and AFSG and studied several other properties. Similarly, there is notion of the intuitionistic antifuzzy subgroup (IAFSG) [18], which was developed by Li et al. in 2009. They have also studied its homomorphic properties and established some connections with its intuitionistic fuzzy counterpart. Table 1 contains some contributions of various researchers involving different antialgebraic notions under uncertainty.

Hence, it is obvious that antiversions of FSG, IFSG, etc. have been adopted by different researchers for the anticipation of unique and impactful results. In neutrosophic group theory, so far, authors have discussed NSGs and some of their algebraic structures. But still, the antineutrosophic subgroup (ANSG) is undefined. Also, the relationship between NSG and ANSG are still unexplored. Hence, this can be a fruitful area which can generate some scope of future research. Based on the aforementioned gaps, the objectives of this paper are as follows: (i)to introduce ANSG and investigate its algebraic features(ii)to define the antineutrosophic normal subgroup (ANNSG) and explore its algebraic characteristics(iii)to figure out the relationships between NSG and ANSG(iv)to study several homomorphic attributes of ANSG and ANNSG

This article has been structured in the following manner. In Section 2, desk research of FSG, IFSG, and NSG and their normal versions are given. Also, antiversions of FSG and IFSG are discussed. In Section 3, the notions of ANSG and ANNSG are introduced along with some other essential definitions and theories are given. Finally, in Section 4, conclusion is given by mentioning some scopes of further research.

2. Preliminaries

Here, some elementary set theories under uncertainties are discussed which are required for our current study.

Definition 1 (see [1]). A FS of a crisp set is defined as .

Definition 2 (see [2]). An IFS of a crisp set is defined as , where and are, respectively, known as the membership and nonmembership degrees.

Definition 3 (see [3]). A NS of a crisp set is defined as , where are, respectively, known as the truth, indeterminacy, and falsity degrees.

Definition 4 (see [1]). Let be a FS of Then, the set is denoted as a level subset of .

Definition 5 (see [17]). Let be a FS of Then, the set is denoted as a lower level subset of .

Next, the notions of FSG, IFSG, NSG, and a few of their essential properties are addressed.

2.1. Fuzzy, Intuitionistic Fuzzy, and Neutrosophic Subgroup

Definition 6 (see [10]). For a classical group , a FS is denoted as a FSG iff , the subsequent conditions are fulfilled: (i)(ii)

Theorem 7 (see [10]). is a FSG of iff .

Proposition 8 (see [10]). Homomorphic image and preimage of a FSG is a FSG.

Theorem 9 (see [11]). Let be a classical group and , then with , are classical subgroups of .

Theorem 10 (see [11]). Let be a classical group and with , are classical subgroups of , then .

Definition 11 (see [11]). Let be a FSG of a classical group . Then, and the subgroups are termed as level subgroups of .

Definition 12 (see [15]). For a classical group , an IFS is denoted an IFSG iff , (i)(ii)(iii)(iv)

Proposition 13 (see [15]). For a classical group , an IFS is an IFSG iff (i)(ii)

Theorem 14 (see [27]). Let and be two classical groups and be a homomorphism. Also, let and . Then, (i)If has the supremum property, then (ii)

Definition 15 (see [27]). Let be an IFS of and let with . Then, the set is known as -level set of .

Theorem 16 (see [27]). Let be a classical group and . Then, with and , are classical subgroups of .

Theorem 17 (see [27]). Let be a classical group and with and , are classical subgroups of . Then, IFSG.

Definition 18 (see [16]). For a classical group , a NS is defined as an NSG of iff the subsequent terms are fulfilled: (i), i.e., , and (ii), i.e., , _, and

A set of all the NSGs will be signified as . Here, note that and are following Definition 6, i.e., they are FSGs of whereas, is following Definition 24, i.e., it is an AFS of .

Theorem 19 (see [16]). For a classical group iff i.e., , , and .

Theorem 20 (see [16]). iff the -level sets , , and -lower level set are classical subgroups of .

Theorem 21 (see [16]). Homomorphic image and preimage of any NSG is a NSG.

Definition 22 (see [16]). For a classical group , a neutrosophic is called an NNSG of iff i.e., , , and .

The set of all NNSG of will be signified as . Also, notice that implies that and are fuzzy normal subgroups (FNSG) of and is the antifuzzy normal subgroup (AFNSG) of .

Theorem 23 (see [16]). Homomorphic image and preimage of any NNSG is a NNSG.
In the next segment, the notions of AFSG and IAFSG are discussed.

2.2. Antifuzzy Subgroup and Intuitionistic Antifuzzy Subgroup

Definition 24 (see [17]). For a classical group , a FS is denoted as an AFSG of if , the subsequent terms are fulfilled: (i)(ii)

Theorem 25 (see [17]). is an AFSG of iff .

Proposition 26 (see [17]). is a FSG of the group iff its complement is an AFSG of .

Definition 27 (see [17]). Let be an AFSG of a group . Then, and , the subgroups are called lower-level subgroups of .

Proposition 28 (see [17]). Let be an AFSG of . Then, such that are classical subgroups of .

Proposition 29 (see [17]). Let be a FS of a classical group such that is a classical subgroup of with . Then, is an AFSG of .

Definition 30 (see [28]). For a classical group , an IFS is called an IAFSG of iff (i)(ii)

Proposition 31 (see [28]). is a IFSG of the group iff its complement is an IAFSG of .

Theorem 32 (see [28]). iff with and , -level set of , i.e., are classical subgroups of .

Theorem 33 (see [18]). Homomorphic image and preimage of any IAFSG is a IAFSG.

In the following section, the notions of ANSG and ANNSG have been introduced and some of their fundamental properties are discussed.

3. Antineutrosophic Subgroup

Definition 34. For a classical group , a neutrosophic set is called an ANSG of iff the following terms are fulfilled: (i), i.e., , , and (ii), i.e., , , and The set of all ANSGs will be signified as

Proposition 35. iff and are AFSGs of and is FSG of .

Proof. Let then from Definition 34, it is evident that and are following Definition 24, i.e., they are AFSGs of Whereas is following Definition 6, i.e., it is a FSG of . Again, if and are AFSGs of and is a FSG of then .

Example 36. Let be a classical group of order 4 and be a neutrosophic set of , where the memberships of truth (), indeterminacy (), and falsity () of elements in are given in Figure 1.

Notice that and are following Definition 24, i.e., are AFSGs of . Again, is following Definition 6, i.e., is a FSG of . Hence, is an ANSG of .

Example 37. Let be a classical group of order and be a NS of , where considering , let . In Figures 2 and 3, memberships of and have been described graphically.

Here, is following Definition 34 and hence it is an ANSG.

Theorem 38. Let where is a classical group. Then, (i)(ii), where is the neutral element of

Proof. (i)Here, is a FSG and both and are AFSGs of , by Definition 6. So, and hence . Again, from Definition 24, . So, and hence Similarly, using Definition 24, we can prove . So, (ii)Using Definition 6, we have . Again, using Definition 24, Similarly, using Definition 24, we have Hence,

Theorem 39. iff .

Proof. Let . Then, by Definition 34, we have . Again, by Definition 34, and hence Conversely, let . So, Notice that, Similarly, and .
Again, Similarly, and can be proved. Hence, satisfies Definition 34, i.e., .

Theorem 40. iff .

Proof. If we take the complement of , i.e., then corresponding degree of truth and degree of falsity will interchange their positions in . Also, the degree of indeterminacy will have its complement, i.e., . In other words, if Let then by Proposition 35 and are AFSGs of and is FSG of . So, in case of , and will become FSGs and will become AFS of . Hence, they will follow Definition 18, i.e., . Similarly, the converse part can also be proved.

Example 41. Let be the group of integers modulo with usual addition and is a NS of , where and are mentioned in Table 2.

According to Definition 34, is an ANSG of .

Now , where are mentioned in Table 3.

Here, according to Definition 18, is a NSG of .

Theorem 42. iff the -lower level sets , _, and -level set are classical subgroups of .

Proof. Let , and . Then, and . Since , we have and hence . Similarly, it can be shown that and . So, , , and are classical subgroups of .
Conversely, let is a classical subgroup of . Let such that and for some . Then, and .
Let . Then, and hence So, , i.e., is an AFSG of . Similarly, it can be proved that is an AFSG and is a FSG of So, .

Theorem 43. Intersection of any two ANSG of any group is an ANSG.

Proof. Let . To prove this, using Theorem 39, we can show that Here, Similarly, we can show that Again, Hence, .

Theorem 44. Homomorphic image of any ANSG is an ANSG.

Proof. Let and be two classical groups and be a homomorphism. Let . Then, , we have Here, we have to show that is an ANSG of .
Let such that and . Now, as is a group homomorphism, we have Again, . Where-from and hence, Similarly, it can be shown that .
Also, Again . Where-from and hence So, is an ANSG of .

Theorem 45. Homomorphic preimage of any ANSG is an ANSG.

Proof. Let and be two classical groups and be a homomorphism. Let . Then, , we have Here, we have to show that is an ANSG of .
Let . Since is a group homomorphism, Similarly, we can show that Hence, is an ANSG of .

Theorem 46. Let and be a homomorphism on . Let is defined as for any then and .

Proof. Here, Hence, by Theorem 39, .
Again, notice that, Similarly, it can be shown that and .
Hence,

Theorem 47. Let and be an isomorphism on , then .

Proof. Here Similarly, it can be shown that and .
Hence, .

In the next segment, ANNSG has been introduced. Also, its homomorphic characteristics are mentioned.

3.1. Antineutrosophic Normal Subgroup

Definition 48. For a classical group , a neutrosophic set is called an ANNSG of iff , i.e., , , and .
The set of all ANNSGs of will be signified as ANNSG.

Example 49. Let be the Klien’s 4-group and is a NS of , where are mentioned in Table 4.

Here, follows Definition 48, i.e., it is an ANNSG.

Proposition 50. iff and are AFNSs of and is FNS of .

Proof. Using Definition 48, this can be observed.

Theorem 51. Intersection of any two ANNSG of any group is an ANNSG.

Proof. Using Theorem 43, this can be proved.

Theorem 52. Let . Then, the subsequent conditions are equivalent: (i)(ii), (iii),

Proof. Let (i) be true. Then, by Definition 48, we have , i.e., , , and .
To prove (ii), we need to show In other words, we need to prove Notice that Again, Hence, .
Similarly, it can be shown that and . Hence (i)(ii).
Let condition (ii) be true. In (ii), substituting in place of (iii) can easily be proved. So, (ii)(iii).
Let condition (iii) be true. Applying in , we have So, .

Theorem 53. iff the -lower level sets, , and -level set are classical normal subgroups of .

Proof. Using Theorem 42, this can be proved.

Theorem 54. Let . The set is a classical normal subgroup of , where is the identity element of .

Proof. Since , we have . Let then by Theorem 39Again, by Theorem 38, we have and hence , i.e., . Since , we have i.e., or is a normal subgroup of .

Theorem 55. Let and be a homomorphism on . Then, the homomorphic pre-image of , i.e., .

Proof. Using Theorem 44, we have . Then, by Proposition 50, we can easily prove normality of . Hence, .

Theorem 56. Let and be a surjective homomorphism on Then the homomorphic image of , i.e., .

Proof. Using Theorem 44, we have . Again, by Proposition 50, the normality condition can easily be proved. So, .

4. Conclusion

The studies of ANSG and its normal version might open some new directions of research. Here, homomorphism has been introduced in ANSG and ANNSG to understand their algebraic characteristics. Moreover, connections with their nonantiversions are provided. For these, numerous examples, theories, and propositions are given. In the future, these studies can be further extended by introducing various notions like the antineutrosophic ideal, antineutrosophic ring, antineutrosophic field, and antineutrosophic topological space. Furthermore, their interval-valued versions can be introduced and studied.

Data Availability

This work is a contribution towards the theoretical development of fuzzy algebra and its generalizations. The data that support the findings of this study are not publicly available due to the fact that they were created specifically for this study. We have not used any additional data set for drafting this manuscript.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Copyright © 2023 Sudipta Gayen 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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