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De-yuan Li, Chengyi Zhang, Sheng-quan Ma (2008)
The Intuitionistic Anti-fuzzy Subgroup in Group G
L. Zadeh (1975)
The concept of a linguistic variable and its application to approximate reasoning - IInf. Sci., 8
Katy Ahmad, Mikail Bal, Malath Aswad (2021)
The Kernel Of Fuzzy and Anti-Fuzzy GroupsJournal of Neutrosophic and Fuzzy Systems
P. Das (1981)
Fuzzy groups and level subgroupsJournal of Mathematical Analysis and Applications, 84
P. Maji (2013)
Neutrosophic Soft SetviXra
P. Majumdar, S. Samanta (2010)
Generalised fuzzy soft setsComput. Math. Appl., 59
P. Maji (2009)
More on Intuitionistic Fuzzy Soft Sets
N. Mukherjee, P. Bhattacharya (1984)
Fuzzy normal subgroups and fuzzy cosetsInf. Sci., 34
K. Atanassov (2019)
On Interval Valued Intuitionistic Fuzzy SetsInterval-Valued Intuitionistic Fuzzy Sets
K. Kim, Y. Jun, Y. Yon (2005)
ON ANTI FUZZY IDEALS IN NEAR-RINGSIranian Journal of Fuzzy Systems, 2
R. Biswas (1990)
Fuzzy subgroups and anti fuzzy subgroupsFuzzy Sets and Systems, 35
K. Kalaiarasi, P. Sudha, N. Kausar, S. Kousar, D. Pamučar, N. Ide (2022)
The Characterization of Substructures of γ -Anti Fuzzy Subgroups with Application in GeneticsDiscrete Dynamics in Nature and Society
Robert Lin (2014)
NOTE ON FUZZY SETSYugoslav Journal of Operations Research, 24
Yuming Feng, Bingxue Yao (2012)
On (λ, μ)-anti-fuzzy subgroupsJournal of Inequalities and Applications, 2012
J. Anthony, H. Sherwood (1982)
A characterization of fuzzy subgroupsFuzzy Sets and Systems, 7
Š. Hošková-Mayerová, M. Tahan (2021)
Anti-Fuzzy Multi-Ideals of Near RingMathematics
F. Smarandache (2004)
A geometric interpretation of the neutrosophic set — A generalization of the intuitionistic fuzzy set2011 IEEE International Conference on Granular Computing
K. Atanassov (1986)
Intuitionistic fuzzy setsFuzzy Sets and Systems, 20
V. Çetkin, H. Aygün (2015)
An approach to neutrosophic subgroup and its fundamental propertiesJ. Intell. Fuzzy Syst., 29
N. Kausar (2019)
Direct Product of Finite Intuitionistic Anti Fuzzy Normal Subrings over Non-associative RingsEuropean Journal of Pure and Applied Mathematics
J. Anthony, H. Sherwood (1979)
Fuzzy groups redefinedJournal of Mathematical Analysis and Applications, 69
Hindawi Advances in Mathematical Physics Volume 2023, Article ID 4430103, 10 pages https://doi.org/10.1155/2023/4430103 Research Article 1 2 3 4 Sudipta Gayen , S. A. Edalatpanah , Sripati Jha , and Ranjan Kumar Centre for Data Science, Faculty of Engineering & Technology, Siksha ‘O’ Anusandhan (Deemed to be University), Odisha, India Department of Applied Mathematics, Ayandegan Institute of Higher Education, Tonekabon, Iran Department of Mathematics, National Institute of Technology Jamshedpur, Jharkhand, India School of Advanced Sciences, VIT-AP University, Amaravati AP, India Correspondence should be addressed to Ranjan Kumar; ranjank.nit52@gmail.com Received 11 July 2022; Revised 28 December 2022; Accepted 10 April 2023; Published 17 May 2023 Academic Editor: Mohammad Mirzazadeh 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. 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 The concept of the antifuzzy subgroup (AFSG) [17] is a kind of dual to FSG. It was defined and characterized by Bis- Fuzzy set (FS) [1] theory was introduced to handle uncertain was in 1990. He has mentioned some relationships between FSG and AFSG and studied several other properties. Simi- situations more precisely than crisp sets. But there may exist some complex uncertain situations for which even FS theory larly, there is notion of the intuitionistic antifuzzy subgroup (IAFSG) [18], which was developed by Li et al. in 2009. They is insufficient. As a result, intuitionistic fuzzy set (IFS) [2] and have also studied its homomorphic properties and estab- neutrosophic set (NS) [3] theories evolved, where the latter is more capable of dealing with uncertainties. Apart from these, lished some connections with its intuitionistic fuzzy coun- terpart. Table 1 contains some contributions of various there exist several byproducts of these set theories, like interval-valued versions [4–6]; type-I, type-II, and type-III ver- researchers involving different antialgebraic notions under uncertainty. sions; and soft [7–9] and hard versions. Presently, these theories Hence, it is obvious that antiversions of FSG, IFSG, etc. have been adopted by several researchers in different applied fields. Also, in several pure mathematical fields, these notions have been adopted by different researchers for the anticipa- tion of unique and impactful results. In neutrosophic group are being utilized. In abstract algebra, Rosenfeld [10] was the pioneertodoso. He defined and studied the characteristics of theory, so far, authors have discussed NSGs and some of their algebraic structures. But still, the antineutrosophic sub- a fuzzy subgroup (FSG). Thereafter, Das [11] presented the con- group (ANSG) is undefined. Also, the relationship between cept of the level subgroup of a FSG and showed several beautiful relationships between them. Afterward, Anthony and Sher- NSG and ANSG are still unexplored. Hence, this can be a fruitful area which can generate some scope of future wood [12, 13] redefined FSG by applying general T-norms and defined function generated FSG and subgroup generated research. Based on the aforementioned gaps, the objectives of this paper are as follows: FSG. In 1984, Mukherjee and Bhattacharya [14] introduced normal versions of FSG and cosets. Furthermore, Biswas [15] (i) to introduce ANSG and investigate its algebraic established the concept of intuitionistic fuzzy subgroup (IFSG). features Similarly, Çetkin and Aygün [16] developed the neutrosophic subgroup (NSG) and studied its homomorphic properties. They (ii) to define the antineutrosophic normal subgroup have also established some connections between an NSG and its (ANNSG) and explore its algebraic characteristics level subgroup. 2 Advances in Mathematical Physics Table 1: Desk research of different antialgebraic notions. Author & references Year Contributions in various fields Introduced the concept of antifuzzy ideals of near-rings and investigated some Kim et al. [19] 2005 of its properties. Feng & Yao [20] 2012 Introduced (λ, μ)-antifuzzy subgroups and studied its properties. Introduced intuitionistic fuzzy normal subrings and intuitionistic anti fuzzy normal Kausar [21] 2019 subrings over a nonassociative ring and studied their properties. Ejegwa et al. [22] 2021 Studied antifuzzy multigroup and its characteristics. Introduced different operations on fuzzy multi-ideals of near-rings and defined Hoskova-Mayerova & Al Tahan [23] 2021 antifuzzy multisubnear-rings of near-rings and study their properties. Ahmad et al. [24] 2021 Defined kernel subgroup of a FSG and AFSG and presented several results involving them. Studied the properties of γ-antifuzzy normal subgroup and γ-fuzzy normal subgroup Kalaiarasi et al. [25] 2022 and presented their application in gene mutation. Introduced and analyzed anti neutrosophic multifuzzy ideals of γ near-ring and Hemabala & Kumar [26] 2022 studied their product. (iii) to figure out the relationships between NSG and 2.1. Fuzzy, Intuitionistic Fuzzy, and Neutrosophic Subgroup ANSG Definition 6 (see [10]). For a classical group V,aFS ψ is (iv) to study several homomorphic attributes of ANSG denoted as a FSG iff ∀m, r ∈ V, the subsequent conditions and ANNSG are fulfilled: This article has been structured in the following manner. (i) ψðm ⋅ rÞ ≥ min fψðmÞ, ψðrÞg In Section 2, desk research of FSG, IFSG, and NSG and their −1 normal versions are given. Also, antiversions of FSG and (ii) ψðr Þ ≥ ψðrÞ IFSG are discussed. In Section 3, the notions of ANSG and ANNSG are introduced along with some other essential def- initions and theories are given. Finally, in Section 4, conclu- Theorem 7 (see [10]). ψ is a FSG of V iff ∀m, r ∈ V sion is given by mentioning some scopes of further research. −1 ψðmr Þ ≤ min fψðmÞ, ψðrÞg. 2. Preliminaries Proposition 8 (see [10]). Homomorphic image and preimage Here, some elementary set theories under uncertainties are of a FSG is a FSG. discussed which are required for our current study. Theorem 9 (see [11]). Let V be a classical group and ψ ∈ Definition 1 (see [1]). A FS λ of a crisp set V is defined as FSGðVÞ, then ∀t ∈ ½0, 1 with ψðeÞ ≥ t, ψ are classical sub- λ : V⟶ ½0, 1 . groups of V. Definition 2 (see [2]). An IFS γ of a crisp set V is defined as γ = fðr, t ðrÞ, f ðrÞÞ: r ∈ Vg, where t and f are, respec- Theorem 10 (see [11]). Let V be a classical group and ∀t ∈ γ γ γ γ ½0, 1 with ψðeÞ ≥ t, ψ are classical subgroups of V, then ψ tively, known as the membership and nonmembership ∈ FSGðVÞ. degrees. Definition 3 (see [3]). A NS η of a crisp set V is defined as Definition 11 (see [11]). Let ψ be a FSG of a classical group V. η = fðr, t ðrÞ, i ðrÞ, f ðrÞÞ: r ∈ Vg, where t , i , and f are, η η η η η η Then, ∀t ∈ ½0, 1 and ψðeÞ ≥ t the subgroups ψ are termed as respectively, known as the truth, indeterminacy, and falsity level subgroups of ψ. degrees. Definition 12 (see [15]). For a classical group V, an IFS γ Definition 4 (see [1]). Let ψ be a FS of V : Then, the set ψ = = fðr, t ðrÞ, f ðrÞÞ: r ∈ Vg is denoted an IFSG iff ∀m, r ∈ V, fr ∈ V : ψðrÞ ≥ tg∀t ∈ ½0, 1 is denoted as a level subset of ψ. γ γ (i) t ðm ⋅ rÞ ≥ min ft ðmÞ, t ðrÞg Definition 5 (see [17]). Let φ be a FS of V : Then, the set φ γ γ γ = fr ∈ V : φðrÞ ≤ tg∀t ∈ ½0, 1 is denoted as a lower level sub- −1 (ii) t ðr Þ ≥ t ðrÞ γ γ set of φ. (iii) f ðm ⋅ rÞ ≤ max ff ðmÞ, f ðrÞg γ γ γ Next, the notions of FSG, IFSG, NSG, and a few of their −1 essential properties are addressed. (iv) f ðr Þ ≤ f ðrÞ γ γ Advances in Mathematical Physics 3 Proposition 13 (see [15]). For a classical group V, an IFS γ Definition 22 (see [16]). For a classical group V, a neutro- sophic δ is called an NNSG of V iff ∀m, r ∈ V = fðm, t ðmÞ, f ðmÞÞ:m ∈ Vg is an IFSG iff ∀m, r ∈ V γ γ −1 ÀÁ (i) t ðmr Þ ≥ min ft ðmÞ, t ðrÞg −1 γ γ γ δ m ⋅ r ⋅ m ≤ δðÞ r , ð2Þ −1 (ii) f ðmr Þ ≤ max ff ðmÞ, f ðrÞg γ γ γ −1 −1 i.e., t ðm ⋅ r ⋅ m Þ ≤ t ðrÞ, i ðm ⋅ r ⋅ m Þ ≤ i ðrÞ, and f ðm ⋅ δ δ δ δ δ −1 r ⋅ m Þ ≥ f ðrÞ. Theorem 14 (see [27]). Let V and R be two classical groups and l : V⟶ R be a homomorphism. Also, let γ ∈ IFSGðVÞ The set of all NNSG of V will be signified as NNSGðVÞ. and γ ∈ IFSGðRÞ. Then, Also, notice that η ∈ NNSGðVÞ implies that t and i are δ δ fuzzy normal subgroups (FNSG) of V and f is the antifuzzy (i) If γ has the supremum property, then lðγÞ ∈ IFSGðRÞ normal subgroup (AFNSG) of V. −1 (ii) l ðγ Þ ∈ IFSGðVÞ Theorem 23 (see [16]). Homomorphic image and preimage of any NNSG is a NNSG. Definition 15 (see [27]). Let γ be an IFS of V and let s , s In the next segment, the notions of AFSG and IAFSG are 1 2 ∈ ½0, 1 with s + s ≤ 1. Then, the set γ = fm ∈ V : t ð 1 2 γ discussed. ðs ,s Þ 1 2 mÞ ≥ s &f ðmÞ ≤ s g is known as ðs , s Þ-level set of γ. 1 γ 2 1 2 2.2. Antifuzzy Subgroup and Intuitionistic Antifuzzy Subgroup Theorem 16 (see [27]). Let V be a classical group and γ ∈ IFSGðVÞ. Then, ∀s , s ∈ ½0, 1 with t ðeÞ ≥ s and f ðeÞ ≤ s , 1 2 γ 1 γ 2 Definition 24 (see [17]). For a classical group V,a FS φ is γ are classical subgroups of V. ðs ,s Þ denoted as an AFSG of V if ∀m, r ∈ V, the subsequent terms 1 2 are fulfilled: Theorem 17 (see [27]). Let V be a classical group and ∀s , s ∈ ½0, 1 with t ðeÞ ≥ s and f ðeÞ ≤ s , γ are classical (i) φðm ⋅ rÞ ≤ max fφðmÞ, φðrÞg 2 γ 1 2 γ ðs ,s Þ 1 2 subgroups of V. Then, γ ∈ IFSGðVÞ. −1 (ii) φðr Þ ≤ φðrÞ Definition 18 (see [16]). For a classical group V,aNS δ is defined as an NSG of V iff the subsequent terms are fulfilled: Theorem 25 (see [17]). φ is an AFSG of V iff ∀m, r ∈ V −1 φðmr Þ ≤ max fφðmÞ, φðrÞg. (i) δðm ⋅ rÞ ≥ min fδðmÞ, δðrÞg, i.e., t ðm ⋅ rÞ ≥ min f t ðmÞ, t ðrÞg, i ðm ⋅ rÞ ≥ min fi ðmÞ, i ðrÞg and f δ δ δ δ δ δ Proposition 26 (see [17]). φ is a FSG of the group V iff its ðm ⋅ rÞ ≤ max ff ðmÞ, f ðrÞg δ δ complement φ is an AFSG of V. −1 −1 −1 (ii) δðm Þ ≥ δðmÞ, i.e., t ðm Þ ≥ t ðrÞ, i ðm Þ ≥ i ðrÞ δ δ δ δ , Definition 27 (see [17]). Let φ be an AFSG of a group V. −1 and f ðm Þ ≤ f ðrÞ δ δ Then, ∀t ∈ ½0, 1 and φðeÞ ≤ t, the subgroups φ are called lower-level subgroups of φ. A set of all the NSGs will be signified as NSGðVÞ. Here, note Proposition 28 (see [17]). Let φ be an AFSG of V. Then, ∀ that t and i are following Definition 6, i.e., they are FSGs of δ δ t ∈ ½0, 1 such that t ≥ μðeÞ,φ are classical subgroups of V. V whereas, f is following Definition 24, i.e., it is an AFS of V. Proposition 29 (see [17]). Let φ be a FS of a classical group Theorem 19 (see [16]). For a classical group Vδ ∈ NSGðVÞ V such that φ is a classical subgroup of V∀t ∈ ½0, 1 with t iff ∀m, r ∈ V ≥ μðeÞ. Then, μ is an AFSG of V. ÀÁ −1 δ m ⋅ r ≥ min δ m , δ r , ð1Þ fg ðÞ ðÞ Definition 30 (see [28]). For a classical group V, an IFS γ −1 −1 = fðm, t ðmÞ, f ðmÞÞ: m ∈ Vg is called an IAFSG of V iff ∀ γ γ i.e., t ðm ⋅ r Þ ≥ min ft ðmÞ, t ðrÞg, i ðm ⋅ r Þ ≥ min fi ðm δ δ δ δ δ −1 m, r ∈ V Þ, i ðrÞg, and f ðm ⋅ r Þ ≤ max ff ðmÞ, f ðrÞg. δ δ δ −1 (i) t ðmr Þ ≤ max ft ðmÞ, t ðrÞg Theorem 20 (see [16]). δ ∈ NSGðVÞ iff the p-level sets ðt Þ , γ γ γ −1 ði Þ , and p-lower level set ðf Þ are classical subgroups of δ δ p p (ii) f ðmr Þ ≥ min ff ðmÞ, f ðrÞg γ γ γ V∀p ∈ ½0, 1 . Theorem 21 (see [16]). Homomorphic image and preimage Proposition 31 (see [28]). γ is a IFSG of the group V iff its of any NSG is a NSG. complement γ is an IAFSG of V. 4 Advances in Mathematical Physics Theorem 32 (see [28]). γ ∈ IFSGðVÞ iff ∀s , s ∈ ½0, 1 with Proof. 1 2 t ðeÞ ≥ s and f ðeÞ ≤ s , ðs , s Þ-level set of γ, i.e., γ are γ 1 γ 2 1 2 ðs ,s Þ 1 2 (i) Here, f is a FSG and both t and i are AFSGs of V, η η η classical subgroups of V. −1 −1 −1 by Definition 6. So, f ðrÞ = f ððr Þ Þ ≥ f ðr Þ and η η η −1 Theorem 33 (see [18]). Homomorphic image and preimage hence f ðr Þ = f ðrÞ. Again, from Definition 24, t η η η of any IAFSG is a IAFSG. −1 −1 −1 −1 ðr Þ ≤ t ðrÞ. So, t ðrÞ = t ððr Þ Þ ≤ t ðr Þ and η η η η −1 hence t ðr Þ = t ðrÞ: Similarly, using Definition 24, η η In the following section, the notions of ANSG and ANNSG −1 −1 we can prove i ðr Þ = i ðrÞ. So, ηðr Þ = ηðrÞ have been introduced and some of their fundamental prop- η η erties are discussed. −1 (ii) Using Definition 6, we have f ðeÞ = f ðr ⋅ r Þ ≥ min η η −1 ff ðrÞ, f ðr Þg = f ðrÞ.Again, using Definition 24, 3. Antineutrosophic Subgroup η η η ÀÁ ÈÉÀÁ −1 −1 Definition 34. For a classical group V, a neutrosophic set η is t ðÞ e = t r ⋅ r ≤ max t ðÞ r , t r = t ðÞ r : η η η η η called an ANSG of V iff the following terms are fulfilled: Similarly, using Definition 24, we have (i) ηðm ⋅ rÞ ≤ max fηðmÞ, ηðrÞg, i.e., t ðm ⋅ rÞ ≤ max f t ðmÞ, t ðrÞg, i ðm ⋅ rÞ ≤ max fi ðmÞ, i ðrÞg, and η η η η η ÀÁ ÈÉÀÁ −1 −1 i ðÞ e = i r ⋅ r ≤ max i ðÞ r , i r = i ðÞ r : f ðm ⋅ rÞ ≥ min ff ðmÞ, f ðrÞg η η η η η η η η −1 −1 −1 (ii) ηðr Þ ≤ ηðrÞ, i.e., t ðr Þ ≤ t ðrÞ, i ðr Þ ≤ i ðrÞ, and η η η η −1 Hence, ηðeÞ ≤ ηðrÞ f ðr Þ ≥ f ðrÞ η η The set of all ANSGs will be signified as ANSGðVÞ Theorem 39. η ∈ ANSGðVÞ iff ∀m, r ∈ V −1 Proposition 35. η ∈ ANSGðVÞ iff t and i are AFSGs of V η η ηðm ⋅ r Þ ≤ max fηðmÞ, ηðrÞg. and f is FSG of V. Proof. Let η ∈ ANSGðVÞ. Then, by Definition 34, we have η −1 −1 ðm ⋅ r Þ ≤ max fηðmÞ, ηðr Þg. Again, by Definition 34, Proof. Let η ∈ ANSGðVÞ then from Definition 34, it is evi- −1 ηðr Þ = ηðrÞ and hence dent that t and i are following Definition 24, i.e., they are η η AFSGs of V : Whereas f is following Definition 6, i.e., it is η ÀÁ ÈÉÀÁ −1 −1 η m ⋅ r ≤ max η m , η r = max η m , η r : ð3Þ ðÞ fg ðÞ ðÞ a FSG of V. Again, if t and i are AFSGs of V and f is a η η FSG of V then η ∈ ANSGðVÞ. −1 Conversely, let ηðm ⋅ r Þ ≤ max fηðmÞ, ηðrÞg. So, Example 36. Let V = f1, i,−1,−ig be a classical group of order ÀÁ ÈÉ −1 t m ⋅ r ≤ max t m , t r , ðÞ ðÞ 4 and η be a neutrosophic set of V, where the memberships η η η ÀÁ ÈÉ of truth (t ), indeterminacy (i ), and falsity (f ) of elements −1 η η η i m ⋅ r ≤ max i m , i r , ðÞ ðÞ η η η ð4Þ in η are given in Figure 1. no ÀÁ −1 Notice that t and i are following Definition 24, i.e., are η η f m ⋅ r ≥ min f m , f r : ðÞ ðÞ η η η AFSGs of V. Again, f is following Definition 6, i.e., is a FSG of V. Hence, η is an ANSG of V. Notice that, ÀÁ ÀÁ ÈÉ ÈÉ ÀÁ −1 −1 −1 Example 37. Let V = fa, eg be a classical group of order 2 t r = t e ⋅ r ≤ max t ðÞ e , t ðÞ r =max t r ⋅ r , t ðÞ r η η η η η η and η be a NS of V, where considering θ ∈ ½π/4, π/2 , let η ÈÉ ≤ max t r , t r , t r = t r : ðÞ ðÞ ðÞ ðÞ η η η η = fða, sin θ/2, sin θ/4, ðsin θ + cos θÞ/2Þ, ðe, cos θ/2, cos θ/4, ðsin θ + cos θÞ/2Þg. In Figures 2 and 3, memberships of a ð5Þ and e have been described graphically. −1 −1 Here, η is following Definition 34 and hence it is an Similarly, i ðr Þ ≤ i ðrÞ and f ðr Þ ≥ f ðrÞ. η η η η ANSG. Again, ÀÁ ÈÉÀÁ Theorem 38. Let η ∈ ANSG ðVÞ where V is a classical group. −1 −1 −1 t m ⋅ r = t m ⋅ r ≤ max i m , i r ðÞ ðÞ η η η η Then, ∀r ∈ V ð6Þ ÈÉ ≤ max i m , i r : ðÞ ðÞ η η −1 (i) ηðr Þ = ηðrÞ (ii) ηðeÞ ≤ ηðrÞ, where e is the neutral element of V Similarly, i ðm ⋅ rÞ ≤ max fi ðmÞ, i ðrÞg and f ðm ⋅ rÞ ≥ η η η η Advances in Mathematical Physics 5 0.8 0.6 0.4 0.2 1–1 i –i Truth Indeterminancy Falsity Figure 1: Memberships of elements in η. 0. 9 0. 8 0. 7 0. 6 0. 5 0. 4 0. 3 0. 2 0. 1 45 50 55 60 65 70 75 80 Figure 2: Memberships of elements in a. min ff ðmÞ, f ðrÞg can be proved. Hence, η satisfies Defini- other words, if η η tion 34, i.e., η ∈ ANSGðVÞ. no η = r, t ðÞ r , i ðÞ r , f ðÞ r : r ∈ V then η η η η Theorem 40. η ∈ ANSGðVÞ iff η ∈ NSGðVÞ. ð7Þ no = r, f r , i r , t r : r ∈ V : ðÞ ðÞ ðÞ η η η Proof. If we take the complement of η, i.e., η then corre- sponding degree of truth and degree of falsity will inter- Let η ∈ ANSGðVÞ then by Proposition 35 t and i are change their positions in η . Also, the degree of η η c c c indeterminacy will have its complement, i.e., i =1 − i .In AFSGs of V and f is FSG of V. So, in case of η , f and i η η η η 6 Advances in Mathematical Physics 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 45 50 55 60 65 70 75 80 Figure 3: Memberships of elements in e. will become FSGs and t will become AFS of V. Hence, they c Table 2: Membership values of elements belonging to η. will follow Definition 18, i.e., η ∈ NSGðVÞ. Similarly, the converse part can also be proved. η t i f η η η 0.66 0.31 0.78 Example 41. Let ðℤ ,+Þ be the group of integers modulo 4 1 0.85 0.35 0.59 with usual addition and η = fðr, t ðrÞ, i ðrÞ, f ðrÞÞ: r ∈ ℤ g η η η 4 0.72 0.32 0.67 is a NS of ℤ , where t , i and f are mentioned in Table 2. 4 η η 0.85 0.35 0.59 According to Definition 34, η is an ANSG of ℤ . Now η = fðr, t c ðrÞ, i c ðrÞ, f ðrÞÞ: r ∈ ℤ g, where t c, i c η η 4 η η , and f are mentioned in Table 3. Here, according to Definition 18, η is a NSG of ℤ . Table 3: Membership values of elements belonging to η . η t c i c f η η Theorem 42. η ∈ ANSGðVÞ iff the p-lower level sets ðt Þ , p 0 0.78 0.69 0.66 ði Þ and p-level set ðf Þ are classical subgroups of V η , η p p 0.59 0.65 0.85 ∀p ∈ ½0, 1 . 2 0.67 0.68 0.72 0.59 0.65 0.85 Proof. Let η ∈ ANSGðVÞ, p ∈ ½0, 1 and m, r ∈ ðt Þ . Then, t ðmÞ ≤ p and t ðrÞ ≤ p. Since η ∈ ANSGðVÞ, we have t ð η η η −1 −1 −1 m ⋅ r Þ ≤ max ft ðmÞ, t ðrÞg ≤ p and hence m ⋅ r ∈ ðt Þ . η η η Let p ≤ p . Then, m, r ∈ ðt Þ and hence mu ∈ ðt Þ : p 1 2 η η p p 2 2 −1 −1 −1 Similarly, it can be shown that m ⋅ r ∈ ði Þ and m ⋅ r So, t ðmr Þ ≤ p ≤ max ft ðmÞ, t ðrÞg, i.e., t is an AFSG p η 2 η η η of V. Similarly, it can be proved that i is an AFSG and f ∈ ðf Þ . So, ðt Þ , ði Þ , and ðf Þ are classical subgroups η η η η η η p p p p is a FSG of V : So, η ∈ ANSGðVÞ. of V. Conversely, let ∀p ∈ ½0, 1ð t Þ is a classical subgroup of V. Let m, r ∈ V such that t ðmÞ = p and t ðrÞ = p for some Theorem 43. Intersection of any two ANSG of any group is η 1 η 2 p , p ∈ ½0, 1 . Then, m ∈ ðt Þ and r ∈ ðt Þ . an ANSG. η η 1 2 p p 1 2 Advances in Mathematical Physics 7 Proof. Let η , η ∈ ANSGðVÞ. To prove this, using Theorem Let ∃n , n ∈ U such that n = sðm Þ and n = sðm Þ. 1 2 1 2 2 1 1 2 2 39, we can show that Now, as s is a group homomorphism, we have ÀÁ ÀÁ ÀÁ ÀÁ −1 −1 −1 η ∩ η m ⋅ r ≤ max η ∩ η m , η ∩ η r ,i:e:, st n ⋅ n = min t m ≤ t m ⋅ m ðÞ fg ðÞðÞðÞðÞ ðÞ 1 2 1 2 1 2 η 1 2 η η 1 2 −1 −1 m∈s n ⋅n ðÞ 1 no 2 ð13Þ ÀÁ −1 ÈÉ t m ⋅ r ≤ max t ðÞ m , t ðÞ r , η ∩η η ∩η η ∩η ðÞ ðÞ ðÞ 1 2 1 2 1 2 ≤ max t ðÞ m , t ðÞ m : η 1 η 2 no ÀÁ −1 i m ⋅ r ≤ max i ðÞ m , i ðÞ r , η ∩η η ∩η η ∩η ðÞ ðÞ ðÞ 1 2 1 2 1 2 Again, sðt Þðn Þ = min t ðmÞ ≤ t ðm Þ. Where-from η 1 η η 1 no −1 m∈s ðn Þ ÀÁ 1 −1 f m ⋅ r ≥ min f m , f r : ðÞ ðÞ η ∩η η ∩η η ∩η max sðt Þðn Þ = t ðm Þ and hence, ðÞ ðÞ ðÞ 1 2 1 2 1 2 η 1 η 1 ð8Þ ÀÁ ÀÁ ÈÉ −1 st n ⋅ n ≤ max t ðÞ m , t ðÞ m η 1 2 η 1 η 2 ÈÉ ÀÁ ÀÁ Here, = max max st n , max st n ðÞ ðÞ η 1 η 2 ÈÉ ÀÁ ÀÁ no = max st n ,st n : ðÞ ðÞ ÀÁ ÀÁ ÀÁ η 1 η 2 −1 −1 −1 t m ⋅ r = max t m ⋅ r , t m ⋅ r ðÞ η ∩η η η 1 2 1 2 ð14Þ no no no ≤ max max t m , t r , max t m , t r ðÞ ðÞ ðÞ ðÞ η η η η 1 1 2 2 no no no −1 = max max t ðÞ m , t ðÞ m , max t ðÞ r , t ðÞ r Similarly, it can be shown that sði Þðn ⋅ n Þ ≤ max fsð η η η η η 1 2 1 2 1 2 no i Þðn Þ, sði Þðn Þg. η 1 η 2 = max t m , t r : ðÞ ðÞ η ∩η η ∩η ðÞ ðÞ 1 2 1 2 Also, ð9Þ ÀÁ ÀÁ −1 −1 sf n ⋅ n = max f m ≥ f m ⋅ m ðÞ η 1 2 η η 1 2 −1 −1 Similarly, we can show that m∈s n ⋅n ðÞ ð15Þ no no ≥ min f ðÞ m , f ðÞ m : ÀÁ 1 2 η η −1 i m ⋅ r ≤ max i m , i r : ð10Þ ðÞ ðÞ ðÞ η ∩η ðÞ η ∩η ðÞ η ∩η 1 2 1 2 1 2 Again sðf Þðn Þ = max f ðmÞ ≥ t ðm Þ. Where-from η 1 η η 1 −1 m∈s ðn Þ Again, 1 min sðf Þðn Þ = f ðm Þ and hence η 1 η 1 no ÀÁ ÀÁ ÀÁ −1 −1 −1 f m ⋅ r = min f m ⋅ r , f m ⋅ r ðÞ η ∩η η η 1 2 1 2 no ÀÁ no no no −1 sf n ⋅ n ≥ min f m , f m ðÞ ðÞ η 1 2 η 1 η 2 ≥ min min f ðÞ m , f ðÞ r , min f ðÞ m , f ðÞ r η η η η 1 1 2 2 no no no no = min minsf ðÞ n , minsf ðÞ n = min min f ðÞ m , f ðÞ m , min f ðÞ r , f ðÞ r 1 2 η η η η η η 1 2 1 2 no no = min f ðÞ m , f ðÞ r : η ∩η η ∩η ðÞ ðÞ = min sf ðÞ n ,sf ðÞ n : 1 2 1 2 η 1 η 2 ð11Þ ð16Þ Hence, η ∩ η ∈ ANSGðVÞ. 1 2 So, sðηÞ is an ANSG of U . Theorem 44. Homomorphic image of any ANSG is an Theorem 45. Homomorphic preimage of any ANSG is an ANSG. ANSG. Proof. Let U and U be two classical groups and s : U 1 2 1 Proof. Let U and U be two classical groups and s : U 1 2 1 ⟶ U be a homomorphism. Let η ∈ ANSGðU Þ. Then, ∀ 2 1 ⟶ U be a homomorphism. Let δ ∈ ANSGðU Þ. Then, ∀ 2 2 m , m ∈ U , we have 1 2 1 n , n ∈ U , we have 1 2 2 ÀÁ ÈÉ −1 ÀÁ t m ⋅ m ≤ max t ðÞ m , t ðÞ m , −1 η 1 2 η 1 η 2 t n ⋅ n ≤ maxfg t ðÞ n , t ðÞ n , δ 1 2 δ 1 δ 2 ÀÁ ÈÉ −1 ÀÁ i m ⋅ m ≤ max i ðÞ m , i ðÞ m , −1 η 1 2 η 1 η 2 ð12Þ i n ⋅ n ≤ max i n , i n , ð17Þ fg ðÞ ðÞ δ 1 2 δ 1 δ 2 no ÀÁ ÀÁ −1 −1 f m ⋅ m ≥ min f ðÞ m , f ðÞ m : 1 2 1 2 f n ⋅ n ≥ minfg f ðÞ n , f ðÞ n : η η η δ 1 2 δ 1 δ 2 −1 Here, we have to show that sðηÞ is an ANSG of U . Here, we have to show that s ðδÞ is an ANSG of U . 2 1 8 Advances in Mathematical Physics Let m , m ∈ U . Since s is a group homomorphism, In the next segment, ANNSG has been introduced. Also, 1 2 1 its homomorphic characteristics are mentioned. ÀÁ ÀÁÀÁ ÀÁÀÁ −1 −1 −1 −1 s t m ⋅ m = t sm ⋅ m = t sm ⋅sm ðÞ ðÞ δ 1 2 δ 1 2 δ 1 2 ÀÁ −1 3.1. Antineutrosophic Normal Subgroup = t smðÞ ⋅smðÞ ≤ maxfg tðÞ smðÞ , tðÞ smðÞ δ 1 2 δ 1 δ 2 ÈÉ −1 −1 = max s t m , s t m : ðÞ ðÞ ðÞ ðÞ δ 1 δ 2 Definition 48. For a classical group V, a neutrosophic set η is −1 ð18Þ called an ANNSG of V iff ∀m, r ∈ Vηðm ⋅ r ⋅ m Þ ≤ ηðrÞ, i.e., −1 −1 t ðm ⋅ r ⋅ m Þ ≤ t ðrÞ, i ðm ⋅ r ⋅ m Þ ≤ i ðrÞ, and f ðm ⋅ r ⋅ η η η η Similarly, we can show that −1 m Þ ≥ f ðrÞ. ÀÁ ÈÉ The set of all ANNSGs of V will be signified as −1 −1 −1 −1 s ðÞ i m ⋅ m ≤ max s ðÞ i ðÞ m , s ðÞ i ðÞ m , δ 1 δ 1 δ 2 ANNSGðVÞ. ÀÁ ÈÉ −1 −1 −1 −1 s f m ⋅ m ≥ min s f m , s f m : ðÞ ðÞ ðÞ ðÞ ðÞ δ 1 2 δ 1 δ 2 Example 49. Let V = fe, m, r, mrg be the Klien’s 4-group and ð19Þ η = fðr, t ðrÞ, i ðrÞ, f ðrÞÞ: r ∈ Vg is a NS of V, where t , i , η η η η −1 and f are mentioned in Table 4. Hence, s ðδÞ is an ANSG of U . Here, η follows Definition 48, i.e., it is an ANNSG. Theorem 46. Let η ∈ ANSGðVÞ and l be a homomorphism −1 −1 on V. Let η : V⟶ ½0, 1 × ½0, 1 × ½0, 1 is defined as η Proposition 50. η ∈ ANNSGðVÞ iff t and i are AFNSs of V η η −1 −1 ðrÞ = ηðr Þ for any r ∈ V then η ∈ ANSGðVÞ and and f is FNS of V. −1 −1 ðlðηÞÞ = lðη Þ. Proof. Using Definition 48, this can be observed. Proof. Here, Theorem 51. Intersection of any two ANNSG of any group is ÀÁ ÀÁ ÀÁ −1 −1 −1 −1 −1 −1 −1 η m ⋅ r = η m ⋅ r = η r ⋅ m an ANNSG. ÀÁ ÈÉÀÁ −1 −1 = η r ⋅ m ≤ max η r , η m ðÞ Proof. Using Theorem 43, this can be proved. ÈÉ ÀÁ ÀÁ −1 −1 = max η r , η m ½ as η is an ANSG ÈÉ −1 −1 = max η m , η r : Theorem 52. Let η ∈ ANNSGðVÞ. Then, the subsequent con- ðÞ ðÞ ditions are equivalent: ð20Þ (i) η ∈ ANNSðUÞ −1 Hence, by Theorem 39, η ∈ ANSGðVÞ. −1 Again, notice that, (ii) ηðm ⋅ r ⋅ m Þ = ηðrÞ, ∀m, r ∈ V ÀÁ ÀÁ ÀÁ ÀÁ ÂÃ ÀÁ (iii) ηðm ⋅ rÞ = ηðV ⋅ mÞ, ∀m, r ∈ V −1 −1 lt ðÞ q =lt q =lt ðÞ q aslt is an ANSG η η η η ÀÁ −1 = min t m = min t m = min t m ðÞ −1ðÞ η η η −1 −1 −1 m∈l q m∈l q m∈l q ðÞ ðÞ ðÞ Proof. Let (i) be true. Then, by Definition 48, we have ηðm ÀÁ −1 −1 −1 ⋅ r ⋅ m Þ ≤ ηðrÞ, i.e., t ðm ⋅ r ⋅ m Þ ≤ t ðrÞ, i ðm ⋅ r ⋅ m Þ ≤ =lt −1 ðÞ q : η η η η −1 i ðrÞ, and f ðm ⋅ r ⋅ m Þ ≥ f ðrÞ. η η ð21Þ To prove (ii), we need to show −1 Similarly, it can be shown that lði Þ = lði −1 Þ and η η −1 ÀÁ lðf Þ = lðf Þ. −1 −1 η η t m ⋅ r ⋅ m ≥ t r , ðÞ η η −1 −1 Hence, ðlðηÞÞ = lðη Þ ÀÁ −1 i m ⋅ r ⋅ m ≥ i r , ð23Þ ðÞ η η ÀÁ Theorem 47. Let η ∈ ANSGðVÞ and l be an isomorphism on −1 f m ⋅ r ⋅ m ≤ f ðÞ r : −1 η η V, then l ðlðηÞÞ = η. Proof. Here In other words, we need to prove ÀÁÀÁ ÀÁ −1 l lt ðÞ p =lt ðÞ lp ðÞ = min t ðÞ m = t ðÞ p : ð22Þ η η η η −1 m∈l ðÞ lp ðÞ ÀÁ −1 t m ⋅ r ⋅ m = t r , ðÞ η η −1 −1 ÀÁ Similarly, it can be shown that l ðlði ÞÞ = i and l −1 η η i m ⋅ r ⋅ m = i r , ð24Þ ðÞ η η ðlðf ÞÞ = f . ÀÁ η η −1 −1 f m ⋅ r ⋅ m = f r : ðÞ η η Hence, l ðlðηÞÞ = η. Advances in Mathematical Physics 9 Table 4: Membership values of elements belonging to η. Theorem 55. Let η ∈ ANNSGðVÞ and l be a homomorphism −1 on V. Then, the homomorphic pre-image of η, i.e., l ðηÞ ∈ η t i f η η η ANNSGðVÞ. 0.1 0.5 0.9 −1 0.3 0.6 0.7 Proof. Using Theorem 44, we have l ðηÞ ∈ ANSGðVÞ. Then, −1 by Proposition 50, we can easily prove normality of l ðηÞ. 0.4 0.5 0.6 −1 Hence, l ðηÞ ∈ ANNSGðVÞ. ab 0.4 0.3 0.6 Theorem 56. Let η ∈ ANNSGðVÞ and l be a surjective homo- Notice that morphism on V : Then the homomorphic image of η, i.e., lðη Þ ∈ ANNSGðVÞ. ÀÁ ÀÁ −1 −1 −1 −1 t m ⋅ r ⋅ m = t m ⋅ r ⋅ m ≤ t r : ð25Þ ðÞ η η η Proof. Using Theorem 44, we have lðηÞ ∈ ANSGðVÞ. Again, by Proposition 50, the normality condition can easily be Again, proved. So, lðηÞ ∈ ANNSGðVÞ. ÀÁ ÀÁ ÀÁ −1 −1 −1 t r = t m ⋅ m ⋅ r ⋅ m ⋅ m ≤ t m ⋅ r ⋅ m : ð26Þ ðÞ η η η 4. Conclusion −1 The studies of ANSG and its normal version might open Hence, t ðm ⋅ r ⋅ m Þ = t ðrÞ. η η some new directions of research. Here, homomorphism −1 Similarly, it can be shown that i ðm ⋅ r ⋅ m Þ = i ðrÞ and η η has been introduced in ANSG and ANNSG to understand −1 f ðm ⋅ r ⋅ m Þ = f ðrÞ. Hence (i)⇒(ii). η η their algebraic characteristics. Moreover, connections with Let condition (ii) be true. In (ii), substituting r in place of their nonantiversions are provided. For these, numerous −1 r ⋅ m (iii) can easily be proved. So, (ii)⇒(iii). examples, theories, and propositions are given. In the future, Let condition (iii) be true. Applying ηðm ⋅ rÞ = ηðr ⋅ mÞ in these studies can be further extended by introducing various −1 t ðm ⋅ r ⋅ m Þ,wehave notions like the antineutrosophic ideal, antineutrosophic ring, antineutrosophic field, and antineutrosophic topologi- ÀÁ ÀÁ −1 −1 cal space. Furthermore, their interval-valued versions can t m ⋅ r ⋅ m = t r ⋅ m ⋅ m = t ðÞ r ≤ t ðÞ r : ð27Þ η η η η be introduced and studied. So, ðiiiÞ⇒ ðiÞ. Data Availability Theorem 53. η ∈ ANNSGðVÞ iff the p-lower level setsðt Þ , p This work is a contribution towards the theoretical develop- ði Þ , and p-level set ðf Þ are classical normal subgroups of ment of fuzzy algebra and its generalizations. The data that η η p p support the findings of this study are not publicly available V∀p ∈ ½0, 1 . due to the fact that they were created specifically for this study. We have not used any additional data set for drafting Proof. Using Theorem 42, this can be proved. this manuscript. Theorem 54. Let η ∈ ANNSGðVÞ. The set U = fm ∈ V : ηð Conflicts of Interest mÞ = ηðeÞg is a classical normal subgroup of V, where e is the identity element of V. The authors declare that they have no conflicts of interest. Proof. Since η ∈ ANNSGðVÞ, we have η ∈ ANSGðVÞ. Let m References , r ∈ U then by Theorem 39 [1] L. A. Zadeh, “Fuzzy sets,” Information and Control, vol. 8, ÀÁ −1 no. 3, pp. 338–353, 1965. η m ⋅ r ≤ max η m , η r = max η e , η e = η e : fg ðÞ ðÞ fg ðÞ ðÞ ðÞ [2] K. T. Atanassov, “Intuitionistic fuzzy sets,” Fuzzy Sets and Sys- ð28Þ tems, vol. 20, no. 1, pp. 87–96, 1986. [3] F. Smarandache, “Neutrosophic set - a generalization of the −1 Again, by Theorem 38, we have ηðm ⋅ r Þ ≥ ηðeÞ and intuitionistic fuzzy set,” International Journal of Pure and −1 −1 hence ηðm ⋅ r Þ = ηðeÞ, i.e., m ⋅ r ∈ U . Since η ∈ ANNSGð Applied Mathematics, vol. 24, no. 3, pp. 287–297, 2005. VÞ, we have [4] L. A. Zadeh, “The concept of a linguistic variable and its appli- cation to approximate reasoning–I,” Information Sciences, ÀÁ ÀÁ −1 −1 vol. 8, no. 3, pp. 199–249, 1975. η m ⋅ r ⋅ m = η r ⋅ m ⋅ m = η r = η e , ð29Þ ðÞ ðÞ [5] K. T. Atanassov, “Interval valued intuitionistic fuzzy sets,” in Intuitionistic Fuzzy Sets: Theory and Applications, pp. 139– −1 i.e., m ⋅ r ⋅ m ∈ U or U is a normal subgroup of V. η η 177, Springer, 1999. 10 Advances in Mathematical Physics [6] H. Wang, F. Smarandache, Y. Q. Zhang, and R. Sunderraman, [26] K. Hemabala and B. S. Kumar, “Anti neutrosophic multi fuzzy Interval Neutrosophic Sets and Logic: Theory and Applications ideals of near ring,” Neutrosophic Sets and Systems, vol. 48, in Computing, vol. 5, Infinite Study, 2005. 85, 2022. pp. 66– [7] P. Majumdar and S. K. Samanta, “Generalised fuzzy soft sets,” [27] K. Hur, H. W. Kang, and H. K. Song, “Intuitionistic fuzzy sub- Computers & Mathematics with Applications, vol. 59, no. 4, groups and subrings,” Honam Mathematical Journal, vol. 25, pp. 1425–1432, 2010. pp. 19–41, 2003. [8] P. K. Maji, “More on intuitionistic fuzzy soft sets,” in Interna- [28] P. K. Sharma, “On intuitionistic anti-fuzzy subgroup of a tional Workshop on Rough Sets, Fuzzy Sets, Data Mining, and group,” International Journal of Mathematics and Applied Sta- Granular-Soft Computing, Springer, 2009. tistics, vol. 3, pp. 147–153, 2012. [9] P. K. Maji, “Neutrosophic Soft Set,” Annals of Fuzzy Mathe- matics and Informatics, vol. 5, no. 1, pp. 157–168, 2013. [10] A. Rosenfeld, “Fuzzy groups,” Journal of Mathematical Analy- sis and Applications, vol. 35, no. 3, pp. 512–517, 1971. [11] P. S. Das, “Fuzzy groups and level subgroups,” Journal of Mathematical Analysis and Applications, vol. 84, no. 1, pp. 264–269, 1981. [12] J. M. Anthony and H. Sherwood, “Fuzzy groups redefined,” Journal of Mathematical Analysis and Applications, vol. 69, no. 1, pp. 124–130, 1979. [13] J. M. Anthony and H. Sherwood, “A characterization of fuzzy subgroups,” Fuzzy Sets and Systems, vol. 7, no. 3, pp. 297–305, [14] N. P. Mukherjee and P. Bhattacharya, “Fuzzy normal sub- groups and fuzzy cosets,” Information Sciences, vol. 34, no. 3, pp. 225–239, 1984. [15] R. Biswas, “Intuitionistic fuzzy subgroups,” Notes on IFS, vol. 3, no. 2, pp. 53–60, 1997. [16] V. Çetkin and H. Aygün, “An approach to neutrosophic sub- group and its fundamental properties,” Journal of Intelligent & Fuzzy Systems, vol. 29, no. 5, pp. 1941–1947, 2015. [17] R. Biswas, “Fuzzy subgroups and anti fuzzy subgroups,” Fuzzy Sets and Systems, vol. 35, no. 1, pp. 121–124, 1990. [18] D. Li, C. Zhang, and S. Ma, “The intuitionistic anti-fuzzy sub- group in group G,” in Fuzzy Information and Engineering, pp. 145–151, Springer, 2009. [19] K. H. Kim, Y. B. Jun, and Y. H. Yon, “On anti fuzzy ideals in near-rings,” Iranian Journal of Fuzzy Systems, vol. 2, no. 2, pp. 71–80, 2005. [20] Y. Feng and B. Yao, “On (λ, μ)-anti-fuzzy subgroups,” Journal of Inequalities and Applications, vol. 2012, Article ID 78, 2012. [21] N. Kausar, “Direct product of finite intuitionistic anti fuzzy normal subrings over non-associative rings,” European Jour- nal of Pure and Applied Mathematics, vol. 12, no. 2, pp. 622– 648, 2019. [22] P. A. Ejegwa, J. A. Awolola, J. M. Agbetayo, and I. M. Adamu, “On the characterisation of anti-fuzzy multigroups,” Annals of Fuzzy Mathematics and Informatics, vol. 21, pp. 307–318, [23] S. Hoskova-Mayerova and M. Al Tahan, “Anti-fuzzy multi- ideals of near ring,” Mathematics, vol. 9, no. 5, p. 494, 2021. [24] K. Ahmad, M. Bal, and M. Aswad, “The kernel of fuzzy and anti-fuzzy groups,” Journal of Neutrosophic and Fuzzy Systems, vol. 1, no. 1, pp. 48–54, 2021. [25] K. Kalaiarasi, P. Sudha, N. Kausar, S. Kousar, D. Pamucar, and N. A. D. Ide, “The characterization of substructures of γ-anti fuzzy subgroups with application in genetics,” Discrete Dynamics in Nature and Society, vol. 2022, Article ID 1252885, 8 pages, 2022.
Advances in Mathematical Physics – Hindawi Publishing Corporation
Published: May 17, 2023
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