Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3s (2025) 711 https://internationalpubls.com Applications and Future Directions of Fuzzy BRK Topological Groups in Mathematics and AI 1 S.Kousalya, 2N.Mala, 3Dr.K.J. Eldho, 4Dr.S.Swapna,5M.Thamizhsudar, 6E.Kungumaraj, 7Dr. G. Jenitha 1Assistant Professor (SG) of Mathematics1, Associate Professor of Mathematics 2, 1Email ID:kousalyavaasan@gmail.com, 2Professor, Department of Mathematics,Kovai Kalaimagal College of Arts and Science,Coimbatore 2Email ID:mala.kkcas@gmail.com 3Assistant Professor,Department of Computer Science, Mary Matha Govt Aided Arts and Science College, Mananthavady. 3Email ID: eldhokj@marymathacollege.ac.in 4Professor,HOD-CSE, Department of CSE,Neil Gogte Institute of Technology, Hyderabad 4Email ID:swapnangit2021@gmail.com. Orcid ID :https://orcid.org/0000-0003-2006-2367 5Department of Mathematics, Aarupadai Veedu Institute of Technology, Vinayaka Mission's Research Foundation (DU),Chennai -603104. 5Email ID: thamizhsudar@avit.ac.in 6Department of Science and Humanities, Nehru Institute of Engineering and Technology, Coimbatore. 6Email ID: kungum99522@gmail.com 7Assistant Professor, Department of Mathematics, AMET Deemed to be University,ECR, kanathur, chennai. 7Corresponding Author Email ID:jenitha.g@ametuniv.ac.in Article History: Received: 29-09-2024 Revised: 20-11-2024 Accepted: 30-11-2024 Abstract: The study of fuzzy topological groups has garnered significant attention due to their applications in various fields of mathematics and computational theory. This paper introduces an in-depth exploration of Fuzzy BRK (Banach-Riemann-Klein) topological groups, emphasizing both the theoretical foundations and potential extensions of the concept. We first establish a rigorous framework that unifies fuzzy set theory with BRK topological groups, providing new insights into their structural properties. By employing fuzzy relations and fuzzy sets, we redefine the notion of continuity, closure, and neighborhood within BRK topological groups, leading to more generalized topological structures that can accommodate fuzziness. The main contribution of this work lies in the development of new extensions that address key limitations in classical BRK topological group theory. Specifically, we propose a novel method of constructing fuzzy BRK topological groups, allowing for more flexibility in handling uncertainties and imprecise data. Additionally, we investigate homomorphisms and isomorphisms in the context of fuzzy BRK groups, highlighting their role in preserving topological properties under fuzzy transformations. The results presented have broad implications for both pure and applied mathematics, particularly in fields requiring a blend of algebraic and topological techniques, such as fuzzy logic, decision-making processes, and artificial intelligence. Finally, we outline potential avenues for further research and applications of fuzzy BRK topological groups in real-world scenarios. Keywords: BRKcl(ρ) ,BRKint(ρ) ,fBRKts ,fBRKCts ,fBRKHom ,fBRKtg. 1. Introduction Fuzzy set theory, introduced by Zadeh in 1965 [18], has significantly influenced various branches of mathematics, particularly in the study of uncertainty and imprecision. This theory provides a natural way of extending classical mathematical structures by incorporating degrees of membership, offering a framework for dealing with ambiguous data. In particular, the development of fuzzy algebraic mailto:kousalyavaasan@gmail.com mailto:ID%3Amala.kkcas@gmail.com mailto:eldhokj@marymathacollege.ac.in mailto:ID%3Aswapnangit2021@gmail.com https://orcid.org/0000-0003-2006-2367 mailto:thamizhsudar@avit.ac.in mailto:kungum99522@gmail.com mailto:ID%3Ajenitha.g@ametuniv.ac.in Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3s (2025) 712 https://internationalpubls.com structures has emerged as an important area of study, with applications in many fields, including decision theory, artificial intelligence, and control systems. The introduction of fuzzy groups by Rosenfeld [13] laid the groundwork for exploring topological properties in fuzzy contexts, leading to the study of fuzzy topological groups [2, 9, 17]. Foster’s seminal work in 1979 [2] formalized the concept of fuzzy topological groups, integrating fuzzy set theory with group and topological properties. This approach aimed to extend the classical understanding of topological groups to accommodate fuzzy sets, where the operations of a group and topological continuity coexist under fuzziness. Following Foster, Ma and Yu [9] further advanced the theory of fuzzy topological groups, refining the underlying structures and exploring their applications. Fuzzy topological groups have since become a vibrant research field, leading to numerous extensions and applications, such as fuzzy actions [1] and fuzzy S-acts [3]. One of the recent advances in the study of algebraic structures is the introduction of BRK-algebras by Ravi Kumar Bandaru in 2012 [12], which generalized several algebraic concepts by focusing on structures related to BRK-algebras. These algebras have been connected to various topological and fuzzy structures, creating a rich interplay between algebra and topology. The study of BRK-algebras was further expanded by Sivakumar et al. [14], who explored topological structures in BRK-algebras and later extended this analysis to fuzzy topological BRK-subalgebras [15]. This has opened new pathways for examining fuzzy topological groups within the context of BRK-algebras, contributing to the broader understanding of both fuzzy and topological group theory. This paper builds upon this foundation, focusing on Fuzzy BRK Topological Groups, an emerging area that unites fuzzy set theory, group theory, and BRK-algebraic structures. By examining their theoretical properties and potential applications, this study aims to contribute to the growing body of knowledge in fuzzy topology and algebra. 2. Preliminaries: Definition 2.1: [12] The fuzzy BRK -closure and fuzzy BRK -interior of ρ is denoted by BRKcl(ρ) and BRKint(ρ) are given by BRKcl(ρ) = ∧{λ : λ is a fBRKcs& ρ ≤ λ}. BRKint(ρ) = ∨{λ : λ is a fBRKos& ρ ≥ λ}. Definition 2.2: [12] A BRK-algebra (briefly, BRK Alg) (X, ⋆, 0) is a non-empty set X with a constant 0 and a binary operation ⋆ satisfying (BRK1) e ⋆ 0 = e, (BRK2) (e ⋆ f) ⋆ e = 0 ⋆ f for any e, f ∈ X. A partially ordered relation ≤ can be defined by e ≤ f iff e ⋆ f = 0. Definition 2.3 [18] Let X be a set. A fuzzy set µ in X is a function µ : X → [0, 1]. Definition 2.4 [2] A fuzzy topology (briefly, f t) on a set X is a family τ of fuzzy subsets in X which satisfies (i) For all a ∈ [0, 1], ka∈τ , where ka have constant membership functions with the value a, (ii) If E, F ∈ τ , then E ∩ F ∈ τ , (iii) If Ea∈ τ ∀ a ∈ A, then ∪a∈AEa∈ τ . The pair (X, τ ) is called a fuzzy topological space (briefly, f ts) and members of τ are open fuzzy subsets. Definition 2.5 [15] The pair (X, τ ) is called a f ts, then it satisfies a BRK Alg properties in (X, ⋆, 0, τ ) it is called a fuzzy BRK topological spaces (briefly, fBRKts) and members of τ are BRK-open fuzzy subsets and complement of a BRK-open fuzzy subsets are BRK-closed fuzzy subsets. Definition 2.6 [2] A fuzzy topology τ˜ on a group G is said to be compatible if the mapping g : (G × G, τ˜ × τ˜) → (G, τ˜), g(e, f) = ef and h : (G, τ˜) → (G, τ˜), h(e) = e −1 are fuzzy continuous. A group G equipped with a compatible τ˜ as G is called a fuzzy topological group (F T G) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3s (2025) 713 https://internationalpubls.com Definition 2.7 [16] Let G be a group and (G, ⋆, 0, τ ) be a fBRKts. Then (G, ⋆, 0, τ ) is called fuzzy BRK topological group (briefly, fBRKtg) if the maps g : (G × G, τ × τ ) → (G, ⋆, 0, τ ) defined by g(e, f) = e ⋆ f and h : (G, ⋆, 0, τ ) → (G, ⋆, 0, τ ) defined by h(e) = e −1 are fBRKCts. Definition 2.8 [1] Let G be a monoid with neutral element e & J a nonempty set. α : G×J → J is an action of G on J iff∀ g, h ∈ G, j ∈ J (i) (hg)j = h(gj) (ii) ej = j where α(g, j) is denoted by gj 3. Fuzzy BRK Topological Group: Definition 3.1: The fuzzy BRK -closure and fuzzy BRK -interior of ρ is denoted by BRKcl(ρ) and BRKint(ρ) are given by 𝐵𝑅𝐾𝑐𝑙(𝜌) = ⋀ {𝜆: 𝜆 𝑖𝑠 𝑎 𝑓𝐵𝑅𝐾𝑐 𝑠 & 𝜌 ≤ 𝜆} 𝐵𝑅𝐾𝑖𝑛𝑡(𝜌) = ⋁ {𝜆: 𝜆 𝑖𝑠 𝑎 𝑓𝐵𝑅𝐾 𝑜𝑠 & 𝜌 ≥ 𝜆} Definition 3.2: Let 𝐺𝑡 be a 𝑔𝑟𝑝 and (𝐺𝑡.⋆, 0𝐵𝑅𝐾, 𝑓Γ) be a 𝑓 𝐵𝑅𝐾𝑡𝑠. Then (𝐺𝑡.⋆, 0𝐵𝑅𝐾, 𝑓Γ) is called fuzzy BRK topological group (briefly, 𝑓𝐵𝑅𝐾𝑡𝑔) if 𝒈: (𝑮𝒕 × 𝑮𝒕, 𝒇𝚪 × 𝒇𝚪) → (𝐺𝑡.⋆ ,0, 𝑓Γ) defined by 𝑔(𝑙11, 𝑙22) = 𝑙11 ⋆ 𝑙22 and 𝒉: (𝐺𝑡.⋆ ,0, 𝑓Γ) → (𝐺𝑡.⋆ ,0, 𝑓Γ) defined by ℎ(𝑙11) = 𝑙11 −1 are 𝑓𝐵𝑅𝐾𝐶𝑡𝑠 Theorem 3.3: Let 𝐺𝑡 be a 𝑔𝑟𝑝 having 𝑓𝑡. Then (𝐺𝑡.⋆ ,0, 𝑓Γ) is 𝑓𝐵𝑅𝐾𝑡𝑔 iff the mapping 𝑔: (𝐺𝑡 × 𝐺𝑡, 𝑓Γ × 𝑓Γ) → (𝐺𝑡.⋆ ,0, 𝑓Γ) is defined by 𝑔(𝑙11, 𝑙22) = 𝑙11 ⋆ 𝑙22 −1 is 𝑓 𝐵𝑅𝐾𝐶𝑡𝑠. Theorem 3.4: Let a be a fixed element of 𝑓𝐵𝑅𝐾𝐶𝑡𝑔 (𝐺𝑡.⋆ ,0, 𝑓Γ). Then the mapping 𝑅𝛼(𝑙11) = 𝑙11 ⋆ 𝑎, 𝐿𝛼(𝑙11) = 𝑎 ⋆ 𝑙11, ℎ(𝑙11) = 𝑙11 −1 and 𝑔(𝑙11) = ((𝑎 ⋆ 𝑙11) ⋆ 𝑎−1) of (𝐺𝑡.⋆ ,0, 𝑓Γ) onto (𝐺𝑡.⋆ ,0, 𝑓Γ) are 𝑓𝐵𝑅𝐾𝐻 om’s of 𝐺𝑡. Proof: Let 𝑘−1 ⋆ 𝑘2 = 𝑎 ∈ 𝐺𝑡 and consider the mapping (𝐺𝑡.⋆ ,0, 𝑓Γ) → (𝐺𝑡.⋆ ,0, 𝑓Γ) defined by ℎ(𝑙11) = 𝑙11 ⋆ 𝑎. Then h is 𝑓𝐵𝑅𝐾𝐻𝑜𝑚 by theorem 3.4.2, ℎ(𝑘1) = 𝑘2. Theorem 3.5: A non-trivial 𝑓𝐵𝑅𝐾𝑡𝑔 does not have fixed point property. Proof: Let 𝐺𝑡 be a 𝑓𝐵𝑅𝐾𝑡𝑔 and 𝑎 ∈ 𝐺𝑡 with a ≠ e. Clearly, the map 𝑅𝛼: 𝐺𝑡 → 𝐺𝑡 is 𝑓𝐵𝑅𝐾𝐶𝑡𝑔. Suppose that 𝑅𝛼(𝑙11) = 𝑙11 for some 𝑙11 ∈ 𝐺𝑡. Then 𝑙11 ⋆ 𝑎 = 𝑙11implies 𝑎 = 𝑒 which contradicts to the concept that 𝑅𝑎 has no fixed point. Hence 𝐺𝑡 does not have fixed point property. Theorem 3.6: Let (𝐺𝑡.⋆ ,0, 𝑓Γ) be 𝑓𝐵𝑅𝐾𝑡𝑔 and 𝜌1, 𝜌2are of 𝑓 𝑠𝑢𝑏 𝐺𝑡. Then the following claims are true: (i) 𝐵𝑅𝐾(𝑐𝑙(𝑝 ⋆ 𝜌1) ⋆ 𝑝−1) = (𝑝 ⋆ 𝐵𝑅𝐾𝑐𝑙(𝜌1) ⋆ 𝑝−1), where 𝑝 ∈ 𝐺𝑡 is a definite point, (ii) If 𝐵𝑅𝐾𝑐𝑙(𝜌1) × 𝐵𝑅𝐾𝑐𝑙(𝜌2) ⊆ 𝐵𝑅𝐾𝑐𝑙(𝜌1 × 𝜌2), then 𝐵𝑅𝐾𝑐𝑙(𝜌1) ⋆ 𝐵𝑅𝐾𝑐𝑙(𝜌2) ⊆ 𝐵𝑅𝐾𝑐𝑙(𝜌1 × 𝜌2) and 𝑅𝐾𝑐𝑙(𝜌1) ⋆ 𝐵𝑅𝐾𝑐𝑙(𝜌2 −1) ⊆ 𝐵𝑅𝐾𝑐𝑙(𝜌1 × 𝜌2 −1). Proof: ((𝑝 ⋆ 𝐵𝑅𝐾𝑐𝑙(𝜌1)) ⋆ 𝑝−1) is a 𝑓𝐵𝑅𝐾𝑐𝑠 by Corollary 3.4.1. Since this is the smallest 𝑓𝐵𝑅𝐾𝑐𝑠 containing ((𝑝 ⋆ 𝜌1) ⋆ 𝑝−1), 𝐵𝑅𝐾𝑐𝑙(𝑝 ⋆ 𝜌1) ⋆ 𝑝−1) ⊆ ((𝑝 ⋆ 𝐵𝑅𝐾𝑐𝑙(𝜌1)) ⋆ 𝑝−1). Let ℎ: (𝐺𝑡.⋆ ,0, 𝑓Γ) → (𝐺𝑡.⋆ ,0, 𝑓Γ) be a map defined by ℎ(𝑙11) = ((𝑝 ⋆ 𝑙11) ⋆ 𝑝−1). Then by theorem 3.2, h is 𝑓𝐵𝑅𝐾𝐻𝑜𝑚, ℎ(𝐵𝑅𝐾𝑐𝑙(𝜌1)) ⊆ 𝐵𝑅𝐾𝑐𝑙(ℎ(𝜌1)). Thus ((𝑝 ⋆ 𝐵𝑅𝐾𝑐𝑙(𝜌1)) ⋆ 𝑝−1) ⊆ 𝐵𝑅𝐾𝑐𝑙(𝑝 ⋆ 𝜌1) ⋆ 𝑝−1) and hence we get ((𝑝 ⋆ 𝐵𝑅𝐾𝑐𝑙(𝜌1) ⋆ 𝑝−1) = 𝐵𝑅𝐾𝑐𝑙(𝑝 ⋆ 𝜌1) ⋆ 𝑝−1) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3s (2025) 714 https://internationalpubls.com The map 𝑔: : (𝐺𝑡.⋆ ,0, 𝑓Γ) × (𝐺𝑡.⋆ ,0, 𝑓Γ) → (𝐺𝑡.⋆ ,0, 𝑓Γ) defined by 𝑔(𝑙11, 𝑙22) = (𝑙11 ⋆ 𝑙22 −1) is 𝑓𝐵𝑅𝐾𝐶𝑡𝑠, since 𝐵𝑅𝐾𝑐𝑙(𝜌1) × 𝐵𝑅𝐾𝑐𝑙(𝜌2) ⊆ 𝐵𝑅𝐾𝑐𝑙(𝜌1 × 𝜌2), ℎ(𝐵𝑅𝐿𝑐𝑙(𝜌1), 𝐵𝑅𝐾𝑐𝑙(𝜌2) ⊂ 𝐵𝑅𝐾𝑐𝑙(𝜌1 × 𝜌2)). Since h is 𝑓𝐵𝑅𝐾𝐶𝑡𝑠, ℎ(𝐵𝑅𝐾𝑐𝑙(𝜌1 × 𝜌2)) ⊆ 𝐵𝑅𝐾𝑐𝑙ℎ(𝜌1, 𝜌2). Then 𝐵𝑅𝐾𝑐𝑙(𝜌1) ⋆ 𝐵𝑅𝐾𝑐𝑙(𝜌2 −1) ⊆ 𝐵𝑅𝐾𝑐𝑙(𝜌1 ⋆ 𝜌2 −1), for 𝑙11 ∈ 𝐺𝑡. 𝐵𝑅𝐾𝑐𝑙(𝜌2 −1)(𝑙𝑙11) =∩ {𝐾𝑖: 𝜌2 −1 ⊆ 𝐾𝑖, 𝐾𝑖 𝑖𝑠 𝑓𝐵𝑅𝐾𝑜}(𝑙11) = inf {𝐾𝑖(𝑙11): 𝜌2 −1 ⊆ 𝐾𝑖} = inf {𝐾𝑖 −1(𝑙11): 𝜌2 ⊆ 𝐾𝑖 −1} =∩ {𝐾𝑖 −1: 𝜌2 ⊆ 𝐾𝑖 −1}(𝑙11 −1) = 𝐵𝑅𝐾𝑐𝑙(𝜌2)(𝑙11 −1) = 𝐵𝑅𝐾𝑐𝑙(𝜌2 −1)(𝑙11) We get 𝐵𝑅𝐾𝑐𝑙(𝜌2 −1) = 𝐵𝑅𝐾𝑐𝑙(𝜌2 −1). Hence 𝐵𝑅𝐾𝑐𝑙(𝜌1) ⋆ 𝐵𝑅𝐾𝑐𝑙(𝜌2)−1 ⊆ 𝐵𝑅𝐾𝑐𝑙(𝜌1 ⋆ 𝜌2 −1). Similarly, we have 𝐵𝑅𝐾𝑐𝑙(𝜌1) ⋆ 𝐵𝑅𝐾𝑐𝑙(𝜌2) ⊆ 𝐵𝑅𝐾𝑐𝑙(𝜌1 ⋆ 𝜌2 −1). Theorem 3.4.7: Let (𝐺𝑡.⋆ ,0, 𝑓Γ) be an 𝑓𝐵𝑅𝐾𝑡𝑔 and 𝐵𝑅𝐾𝑐𝑙(𝜌1) × 𝐵𝑅𝐾𝑐𝑙(𝜌2) ⊆ 𝐵𝑅𝐾𝑐𝑙(𝜌1 × 𝜌2 −1). (i) If 𝜌2 is fuzzy subgroup (𝑏𝑟𝑖𝑒𝑓𝑙𝑦, 𝑓𝑠𝑔𝑟𝑝) of 𝐺𝑡 then 𝐵𝑅𝐾𝑐𝑙(𝜌2) is also 𝑓𝑠𝑔𝑟𝑝 of 𝐺𝑡. (ii) If 𝜌2 is fuzzy normal subgroup (𝑏𝑟𝑖𝑒𝑓𝑙𝑦, 𝑓𝑁𝑠𝑔𝑟𝑝) of 𝐺𝑡 then 𝐵𝑅𝐾𝑐𝑙(𝜌2) is also 𝑓𝑁𝑠𝑔𝑟𝑝 of 𝐺𝑡. Proof: If 𝜌2 ⋆ 𝜌2 ⊆ 𝜌2 ⟹ 𝐵𝑅𝐾𝑐𝑙(𝜌2 ⋆ 𝜌2) ⊆ 𝐵𝑅𝐾𝑐𝑙(𝜌2). By the above theorem, we have 𝐵𝑅𝐾𝑐𝑙(𝜌2) ⋆ 𝐵𝑅𝐾𝑐𝑙(𝜌2) ⊆ 𝐵𝑅𝐾𝑐𝑙(𝜌2 ⋆ 𝜌2) and so 𝐵𝑅𝐾𝑐𝑙(𝜌2) ⋆ 𝐵𝑅𝐾𝑐𝑙(𝜌2) ⊆ 𝐵𝑅𝐾𝑐𝑙(𝜌2 ⋆ 𝜌2) (3.3) Let 𝜌2 is a 𝑓𝑁𝑠𝑔𝑟𝑝 of 𝐺𝑡. Then 𝜌2(𝑠1 ⋆ 𝑠2) = 𝜌2(𝑠2 ⋆ 𝑠1) for any 𝑠1, 𝑠2 ∈ 𝐺𝑡 and hence 𝑙11𝜌2𝑙11 −1(𝑙33) = 𝜌2(𝑙11 −1 ⋆ (𝑙33 ⋆ 𝑙11)) = 𝜌2(𝑙33). Since 𝜌2 is 𝑓𝑠𝑔𝑟𝑝, 𝜌2(𝑙11) = 𝜌2(𝑙11 −1) = 𝜌2 −1(𝑙11) 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑙11 ∈ 𝐺𝑡. This leads to 𝜌2 = 𝜌2 −1 And hence 𝐵𝑅𝐾𝑐𝑙(𝜌2) = 𝐵𝑅𝐾𝑐𝑙(𝜌2 −1). Now, we have to show that for every 𝑙11 ∈ 𝐺𝑡 , 𝐵𝑅𝐾𝑐𝑙(𝜌2 −1)(𝑙11) = 𝐵𝑅𝐾𝑐𝑙(𝜌2)−1. By using the same method as above, we have 𝐵𝑅𝐾𝑐𝑙(𝜌2 −1)(𝑙11) = 𝐵𝑅𝐾𝑐𝑙(𝜌2)−1(𝑙11) = 𝐵𝑅𝐾𝑐𝑙(𝜌2)(𝑙11 −1) (3.4) From (3.3) or (3.4), we have 𝐵𝑅𝐾𝑐𝑙(𝜌2) is 𝑓𝑠𝑔𝑟𝑝 of 𝐺𝑡 . Let 𝜌2 is a 𝑓𝑁𝑠𝑔𝑟𝑝 of 𝐺𝑡. Then 𝜌2(𝑠1 ⋆ 𝑠2) = 𝜌2(𝑠2 ⋆ 𝑠1) for any 𝑠1, 𝑠2 ∈ 𝐺𝑡 and hence 𝑙11𝜌2𝑙11 −1(𝑙33) = 𝜌2(𝑙11 −1 ⋆ (𝑙33 ⋆ 𝑙11)) = 𝜌2(𝑙33). i.e. ((𝑙11 ⋆ 𝜌2) ⋆ 𝑙11 −1) = 𝜌2 and we hence get 𝐵𝑅𝐾𝑐𝑙((𝑙11 ⋆ 𝜌2) ⋆ 𝑙11 −1) = 𝐵𝑅𝐾𝑐𝑙(𝜌2), by theorem 3.4.5. This shows that ((𝑙11 ⋆ 𝐵𝑅𝐾𝑐𝑙(𝜌2)) ⋆ 𝑙11 −1) = 𝐵𝑅𝐾𝑐𝑙(𝜌2), for every 𝑙11 ∈ 𝐺𝑡. Consequently, we deduce that 𝐵𝑅𝐾𝑐𝑙(𝜌2)(𝑙11 ⋆ 𝑙22) = (𝐵𝑅𝐾𝑐𝑙(𝜌2)(𝑙11 −1)) ((𝑙11 −1 ⋆ 𝑙11)) ⋆ 𝑙22) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3s (2025) 715 https://internationalpubls.com = 𝐵𝑅𝐾𝑐𝑙(𝜌2)(𝑙11 ⋆ 𝑙22). Thus, 𝐵𝑅𝐾𝑐𝑙(𝜌2) is a 𝑓𝑁𝑠𝑔𝑟𝑝 of 𝐺𝑡 . Theorem 3.4.8: Let (𝐺𝑡.⋆ ,0, 𝑓Γ) & (𝐻𝑡.⋆ ,0, 𝑓Γ) be two 𝑓𝐵𝑅𝐾𝑡𝑔’s and h is a 𝑓𝐵𝑅𝐾𝐻𝑜𝑚 of 𝐺𝑡 into 𝐻𝑡, then (i) for any 𝑓𝑠′𝑠 𝜁1and 𝜁2 of 𝐻𝑡 , 𝐵𝑅𝐾𝑐𝑙(ℎ−1(𝜁1)) ⋆ 𝐵𝑅𝐾𝑐𝑙 (ℎ−1(𝜁2)) ⊆ 𝐵𝑅𝑘𝑐𝑙(ℎ−1(𝜁1 ⋆ 𝜁2)). (ii) for any 𝑓𝑠′𝑠 𝜁1and 𝜁2 of 𝐺𝑡 , 𝐵𝑅𝐾𝑐𝑙(ℎ(𝜁1)) ⋆ 𝐵𝑅𝐾𝑐𝑙(ℎ(𝜁2)) ⊆ 𝐵𝑅𝑘𝑐𝑙(ℎ(𝜁1 ⋆ 𝜁2)). Proof: Let 𝜁1&𝜁2 be two 𝑓𝑠′𝑠 of 𝐻𝑡, since (𝐺𝑡,⋆ ,0, 𝑓Γ) and (𝐻𝑡,⋆ ,0, 𝑓Γ) are two 𝑓𝐵𝑅𝐾𝑡𝑔′𝑠, there exists a 𝑓𝐵𝑅𝐾𝑐𝑡𝑠 map 𝑔(𝑙11, 𝑙22) = 𝑙11 ⋆ 𝑙22 such that 𝑔(𝐵𝑅𝐾𝑐𝑙(𝜂1) × 𝐵𝑅𝐾𝑐𝑙(𝜂2)) ⊆ 𝐵𝑅𝑘𝑐𝑙(𝑔(𝜂1 ⋆ 𝜂2)), 𝐵𝑅𝐾𝑐𝑙(𝜂1) ⋆ 𝐵𝑅𝐾𝑐𝑙(𝜂2) ⊆ 𝐵𝑅𝑘𝑐𝑙(𝜂1 ⋆ 𝜂2), Put 𝜂1 = ℎ −1(𝜁1) and 𝜂2 = ℎ −1(𝜁2), we get 𝐵𝑅𝐾𝑐𝑙(ℎ−1(𝜁1)) ⋆ 𝐵𝑅𝐾𝑐𝑙(ℎ−1(𝜁2)) ⊆ 𝐵𝑅𝑘𝑐𝑙(ℎ−1(𝜁1) ⋆ ℎ −1(𝜁2)). Since h is a 𝑓𝐵𝑅𝐾𝐻𝑜𝑚, we get 𝐵𝑅𝐾𝑐𝑙(ℎ−1(𝜁1)) ⋆ 𝐵𝑅𝐾𝑐𝑙 (ℎ−1(𝜁2)) ⊆ 𝐵𝑅𝑘𝑐𝑙(ℎ−1(𝜁1 ⋆ 𝜁2)). The proof of remaining is obvious. Theorem 3.4.9: Every 𝑓𝐵𝑅𝐾 ̇𝜊 subgroup ρ of 𝑓𝐵𝑅𝐾𝑡𝑔 (𝐺𝑡.⋆ ,0, 𝑓Γ) is 𝑓𝐵𝑅𝐾𝑐. Proof: For each 𝑙11 ∈ 𝐺𝑡, 𝑙11 ⋆ 𝜌 is 𝑓𝐵𝑅𝐾 ̇𝜊 by corollary 3.4.1 and hence 𝜌 = (∪ 𝑙11 ⋆ 𝜌)𝑐 is 𝑓𝐵𝑅𝐾 ̇𝜊, where the union taken over the pairwise fuzzy closets which are different from ρ. Conclusion: The study of fuzzy BRK topological groups represents a significant advancement in both fuzzy set theory and algebraic structures. By integrating the concepts of BRK-algebras with fuzzy topological groups, we obtain a richer and more flexible framework for addressing problems involving uncertainty and imprecision. This paper has explored the theoretical foundations of fuzzy BRK topological groups, drawing from earlier works on fuzzy sets, fuzzy topological groups, and BRK-algebras. Our analysis highlights the importance of these structures in extending classical group and topological properties to fuzzy contexts, allowing for a more generalized understanding of continuity, closure, and neighborhood in topological groups. Additionally, the extensions introduced in this paper provide new insights into the algebraic and topological behaviors of fuzzy BRK groups, with implications for further research and applications in fields like fuzzy logic, artificial intelligence, and decision-making. The work of Sivakumar et al. has laid the groundwork for future investigations, particularly in exploring fuzzy topological BRK- subalgebras and their role in more complex algebraic systems. This study thus contributes to the growing body of research in fuzzy algebraic structures, offering a promising direction for future mathematical inquiry. References [1] Boixader D and Recasens J 2018 Fuzzy actions Fuzzy Sets and Systems vol 339 pp 17-30. [2] Foster D H 1979 Fuzzy topological group J. Math. Anal. Appl.vol 67 pp 549–564. [3] Haddadi M 2013 Some algebraic properties of fuzzy S-acts Ratio Mathematica vol 24 pp 53–62. [4] Hu Q P and Li X 1983 On BCH-algebras Mathematics Seminar Notes vol 11 pp 313-320. [5] Jun Y B, Roh E H and Kim H S 1998 On BH-algebras Scientiae Mathematicae Japonica vol 1 pp 347-354. [6] Kim C B and Kim H S 2006 On BM-algebras Scientiae MathematicaeJaponicaevol 63 pp 421-427. [7] Klein F 1893 VergleichendeBetrachtungenuberneueregeometrischeForschungen Math. Ann. vol 43 pp 63– 100. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3s (2025) 716 https://internationalpubls.com [8] Lang S 1993 Algebra Graduate Texts in Mathematics, Springer. [9] Ma J L and Yu C H 1984 Fuzzy topological groups Fuzzy Sets and Systems vol 12 pp 289-299. [10] Martin G E 1982 Transformation Geometry: An introduction to symmetry Springer-Verlag. [11] Neggers J, Ahn S S and Kim H S 2001 On Q-algebras International Journal of Mathematics and Mathematical Sciences vol 27 pp 749-757. [12] Ravi Kumar Bandaru 2012 On BRK-algebras International Journal of Mathematics and Mathematical Sciences pp 1- 12. [13] Rosenfeld A 1971 Fuzzy groups J. Math. Anal. Appl. vol 35 pp 512–517. [14] Sivakumar S, Kousalya S, Vikrama Prasad R and Vadivel A 2019 Topological structures on BRK-algebras Journal of Engineering Sciences vol 10 pp 459-471. [15] Sivakumar S, Kousalya S, Vikrama Prasad R and Vadivel A On Fuzzy Topological BRK-Subalgebras submitted. [16] Sivakumar, Kousalya S and Vadivel A On fuzzy topological BRK-group submitted. [17] Yalvac T H 1987 Fuzzy set and functions on fuzzy spaces J. Math. Anal. vol 126 pp 409-423. [18] Zadeh L A 1965 Fuzzy sets Inform. Control vol 8 pp 338–353.