Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 500 https://internationalpubls.com Theoretical Foundation, Topological Technique, and Decision-Making Application of Intuitionistic Fuzzy D-Algebra M. Siva1, A.Jeeva2, S. Kathirvizhi3, N. Kumaran4 1Assistant Professor, Department of Mathematics, St. Joseph's Institute of Technology, Chennai, Tamil Nadu 600119, India. sivam@stjosephstechnology.ac.in 2Assistant Professor , Vel Tech Rangarajan Dr.sagunthala R&D Institute of Science and Technology, Avadi, Chennai 600062. drjeevaa@veltech.edu.in 3Assistant Professor Jeppiaar University, OMR Road Semmanchery, Chennai-600119,Tamil Nadu.India. kathirvizhi02@gmail.com 4Assistant Professor, Department of Mathematics, Vel Tech Rangarajan Dr. Sagunthala R&D Institute of Science and Technology, Avadi, Chennai - 600062. nkumaran@veltech.edu.in Article History: Received: 20-04-2024 Revised: 10-06-2024 Accepted: 28-06-2024 Abstract: Intuitionistic fuzzy D-algebra introduces a novel framework extending classical fuzzy algebra by incorporating degrees of membership and non-membership. This approach addresses the inherent uncertainty and imprecision in real-world systems. Topological techniques facilitate the analysis of convergence and continuity properties, ensuring the robustness of mathematical models. The need for such a framework arises from the limitations of classical fuzzy algebra in capturing nuanced degrees of uncertainty. Real- world decision-making processes often involve complex, ambiguous information that cannot be adequately represented by binary membership functions alone. Intuitionistic fuzzy D-algebra offers a more nuanced representation, expressing hesitation and uncertainty inherent in decision-making contexts. The proposed work comprehensively explores intuitionistic fuzzy D-algebra, including the formal definition of core structures, mathematical modelling, validation strategies through examples and counterexamples, and the development of interactive visualizations. By integrating computational tools and theoretical insights, this framework provides a versatile platform for addressing uncertainty in various domains, from decision-making systems to artificial intelligence, thus paving the way for innovative solutions and improved decision outcomes. The results provide an immersive exploration into the intricacies of intuitionistic fuzzy D-algebra. From the transformation of fuzzy sets into topological spaces to the dynamic manipulation of algebraic operations, each visualization offers an intense dive into understanding uncertainty and imprecision. The visuals serve as powerful educational tools, enabling a profound grasp of complex mathematical concepts and their practical implications in decision-making systems and artificial intelligence. Keywords: Intuitionistic fuzzy D-algebra, Topological techniques, Classical fuzzy algebra, Mathematical modelling, Validation strategies, Decision-making systems, Artificial intelligence. 1. Introduction The exploration of abstract algebraic structures has led to the introduction of various algebras and their generalizations, which play a crucial role in mathematical logic and theoretical computer science. Among these, BCK-algebras and BCI-algebras form fundamental classes with intriguing properties. BCK-algebras are a subset of the more general BCI-algebras, distinguished by certain Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 501 https://internationalpubls.com additional axioms and properties, focusing on a binary operation and an inequality relation [1]. This distinction allows BCK-algebras to provide a rich structure for logical deduction and reasoning while maintaining a specific set of constraints that differentiate them from the broader category of BCI- algebras. In the realm of generalizations, d-algebra represents another important advancement [2]. This structure, characterized by a binary operation that adheres to a set of axioms, extends the applicability of BCK-algebraic concepts. The flexibility introduced by d-algebra allows for a more comprehensive framework in which the properties of BCK-algebras can be studied and applied in broader contexts, facilitating new lines of inquiry and application. Further enhancing the study of d-algebras is the concept of d-filters, which are specific subsets of d- algebras satisfying certain filtering properties. These filters are crucial for understanding the internal structure of d-algebras and for studying their substructures in detail [3]. The notion of d-filters provides a powerful tool for analyzing and decomposing d-algebras into more manageable components, thereby facilitating deeper insights into their algebraic properties and enhancing the overall comprehension of these complex structures [4]. The introduction of fuzzy sets in 1965 revolutionized the way uncertainty and imprecision are handled in mathematics [5]. By allowing elements to have degrees of membership, fuzzy sets enable more nuanced modeling of real-world phenomena, offering a flexible approach to dealing with ambiguity [6]. This innovation paved the way for further advancements, such as the introduction of intuitionistic fuzzy sets in 1986. Intuitionistic fuzzy sets extend the concept of fuzzy sets by incorporating a degree of non- membership alongside the degree of membership [7]. This dual-degree system provides a richer framework for capturing uncertainty and ambiguity, making it a valuable tool for a wide range of applications. The application of intuitionistic fuzzy sets to d-algebras has led to the concept of intuitionistic fuzzy d-algebras [8]. This approach integrates the flexibility of intuitionistic fuzzy sets with the algebraic structure of d-algebras, creating a hybrid system that benefits from the strengths of both frameworks. Intuitionistic fuzzy d-filters, introduced as an extension of d-filters within this context, exhibit unique properties that blend the characteristics of intuitionistic fuzzy sets with the filtering criteria of d- algebras. These filters play a significant role in the analysis and understanding of intuitionistic fuzzy d-algebras, offering new perspectives and methodologies for studying these complex systems. Studying relations on intuitionistic fuzzy d-algebras involves examining how these structures interact under various operations and mappings. This examination uncovers new properties and relationships, enriching the theoretical landscape of fuzzy algebraic systems and providing deeper insights into their behavior and characteristics. The development and study of these abstract algebraic structures, from BCK-algebras and BCI-algebras to d-algebras and their fuzzy extensions, provide a versatile and robust framework for exploring complex algebraic structures. This framework not only enhances our understanding of abstract algebra but also offers powerful tools for addressing problems in logic, computation, and applied mathematics, thereby contributing significantly to the advancement of these fields [9]. 2. Preliminaries of fuzzy D-algebra Fuzzy D-algebra, an interdisciplinary extension of classical algebraic structures, emerges as a pivotal discipline in handling uncertain or imprecise data across diverse domains such as decision theory, Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 502 https://internationalpubls.com pattern recognition, and artificial intelligence. Rooted in foundational definitions, this field provides a sophisticated framework to capture and manipulate fuzzy information effectively. A fuzzy set A within the universe X is characterized by its membership function μA(x), assigning each element x a degree of membership in the range [0,1] [10]. This nuanced representation allows for the modeling of imprecise or uncertain elements, crucial for real-world applications. By virtue of its membership function μR(x, y), a fuzzy relation R between sets X and Y quantifies the degree of association between elements x and y, providing a flexible framework to capture vague or indeterminate relationships [11]. Generalizing traditional algebraic structures, a fuzzy algebraic structure defined on set A, equipped with fuzzy operations ∘, encompasses imprecise elements and operations while adhering to specific algebraic properties, thus accommodating the inherent uncertainty [12]. Rigorously formalized, a fuzzy D-algebra A comprises a non-empty set endowed with fuzzy binary operations ⨁ and ⨀, ensuring commutativity, associativity, identity, and distributive properties. This framework enables seamless manipulation of fuzzy data, crucial for decision-making under uncertainty. Within the context of a fuzzy D-algebra A, a fuzzy ideal I emerges as a key concept, constituting a fuzzy subset where fuzzy operations maintain elements within the ideal [13]. This notion facilitates the exploration of fuzzy algebraic structures under specific constraints. Building upon the foundation of fuzzy D-algebra, the concept of a fuzzy quotient D-algebra A/I arises, representing equivalence classes of elements in A modulo the fuzzy ideal I. This construction provides a powerful tool for analyzing fuzzy algebraic structures in a quotient space setting. Given fuzzy relations R and S, their composition R ∘ S is defined by the membership function μR∘S (x, y), capturing the degree of association between elements x and z through a series of intermediary elements y [14]. A fuzzy relation R on set A is termed as a fuzzy equivalence relation if it is reflexive, symmetric, and transitive, allowing for the classification of elements into equivalence classes based on their fuzzy relationships [15]. Extending the concept of classical lattices, a fuzzy lattice L is defined on a partially ordered set P, equipped with fuzzy operations ⋁(join) and ⋀(meet), enabling the modeling of imprecise ordering relationships among elements [16]. In the context of a fuzzy algebraic structure, a fuzzy congruence relation Θ is defined as an equivalence relation that preserves fuzzy operations, providing a foundation for quotient structures in fuzzy algebraic contexts. 3. Proposed Work 3.1 Mathematical modeling of intuitionistic fuzzy 𝑫 -algebra structures Advancing the theoretical foundations of intuitionistic fuzzy D -algebra begins with defining the core structures of this innovative field. Intuitionistic fuzzy D -algebra extends classical fuzzy algebra by incorporating degrees of membership and non-membership, providing a richer framework for dealing with uncertainty and imprecision. Formally, an intuitionistic fuzzy set A in a universe X is characterized by a membership function μA: X → [0.1] and a non-membership function VA: X → [0.1] such that 0 ≤ μA(x) + VA(x) ≤ 1 for all x ∈ X. This dual-function approach allows for the expression of hesitation, represented by πA(x) = 1 − μA(x) − VA(x). Building upon this foundation, an intuitionistic fuzzy D algebra structure is defined as a tuple (X, μA, VA, D), where D denotes a Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 503 https://internationalpubls.com binary operation that satisfies the properties of a D -algebra. The primary operation in an intuitionistic fuzzy D-algebra can be expressed as: DA(x, y) = (μA(x)⋀μA(y), VA(x)⋁VA(y)) (1) Where ⋀ and ⋁ denote the minimum and maximum operations, respectively. This definition encapsulates the interplay between membership and non-membership degrees under the operation D. Building upon these definitions, mathematical models are developed to represent these structures, leveraging advanced techniques from lattice theory and topology. These models are constructed to faithfully represent the interactions and relationships inherent in intuitionistic fuzzy D-algebra, allowing for the exploration of their properties through rigorous analysis and simulation. By integrating these models with computational tools such as MATLAB and mathematica, a versatile platform is created for both theoretical exploration and practical application, paving the way for innovative solutions in areas ranging from decision-making systems to artificial intelligence. Consider two elements x, y ∈ Xwith their respective membership and non-membership values μA(x), μA(y), VA(x) and VA(y). Applying the operation DA , we obtain: μDA(x, y) = μA(x)⋀μA(y) (2) VDA(x, y) = VA(x)⋁VA(y) (3) To illustrate, choose specific values μA(x) = 0.7, μA(y) = 0.4, VA(x) = 0.3 and VA(y) = 0.2 then, μDA(x, y) = min(0.7,0.4) = 0.4 (4) VDA(x, y) = max(0.2,0.3) = 0.3 (5) The hesitation degree πDA(x, y) is computed as: πDA(x, y) = 1 − μDA(x, y) − VDA(x, y) = 1 − 0.4 − 0.3 = 0.3 (6) Fig.1 Operation 𝐃𝐀 for intuitionistic fuzzy D-algorithm Fig.1 visualizes the operation DA within an intuitionistic fuzzy DDD-algebra, showcasing how it combines the membership and non-membership degrees of two elements x and y from the universe of discourse. The blue curve represents the resulting membership degree μDA(x, y),showing the degree to which the pair of elements belong to the resulting intuitionistic fuzzy set. Conversely, the red curve illustrates the non-membership degree VDA(x, y), indicating the extent to which the pair of elements do not belong to the resulting set. The graph demonstrates how DA integrates the degrees Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 504 https://internationalpubls.com of membership and non-membership to define the characteristics of the intuitionistic fuzzy set formed by the operation. 3.2 Topological techniques in intuitionistic fuzzy 𝑫-algebra Topological techniques, intuitionistic fuzzy sets are mapped onto topological spaces, which are mathematical structures equipped with a notion of closeness or proximity. Let X be the universe of discourse, and F(X) be the set of all intuitionistic fuzzy subsets of X. The mapping Φ: F(X) → T. where T is a topological space, assigns each intuitionistic fuzzy set A in X a corresponding subset Φ(A) in T, preserving certain topological properties. Topological techniques allows to investigate the convergence behavior of sequences of intuitionistic fuzzy sets and the compactness of sets within intuitionistic fuzzy D -algebra structures. A sequence {An} of intuitionistic fuzzy sets is said to converge to a limit set A if, for every point x in the universe of discourse XXX, the membership and non-membership degrees of An approach the membership and non-membership degrees of A, respectively, as n approaches infinity. The compactness of sets in intuitionistic fuzzy D-algebra captures the notion of boundedness and finite subcoverings, providing insights into the structure and behavior of intuitionistic fuzzy sets under algebraic operations. Topological techniques enable to analyze the continuity properties of operations within intuitionistic fuzzy D -algebra. An operation D in intuitionistic fuzzy D -algebra is said to be continuous if, for every pair of intuitionistic fuzzy sets A and B, the operation DDD preserves limits of convergent sequences. Mathematically, this can be expressed as: lim n→∞ → D(An, Bn) = D( lim n→∞ An, lim n→∞ Bn) (7) Where An and Bn are convergent sequences of intuitionistic fuzzy sets. To illustrate the convergence behaviour and continuity properties. Consider an example of an intuitionistic fuzzy D-algebraic operation D defined as follows: D(A, B) = (μA⋀ μB, VA⋁ VB) (8) where μAand VA are the membership and non-membership functions of intuitionistic fuzzy set A, and μB and VB are the membership and non-membership functions of intuitionistic fuzzy set B. We aim to analyze the convergence behaviour of sequences An and Bn prove the continuity of operation D. 3.3 Validation strategies Examples serve as concrete instances that illustrate the application and validity of mathematical models. In the context of intuitionistic fuzzy D-algebra, examples can be constructed to demonstrate the behavior of algebraic operations, the convergence of sequences, or the satisfaction of specific properties. Consider the following example illustrating the application of an intuitionistic fuzzy D- algebraic operation. Let A and B be two intuitionistic fuzzy sets defined over a universe X. Suppose A = (0.7,0.3) and B = (0.5,0.6). The result of the D-algebraic operation D(A, B) can be computed of the as follows: D(A, B) = (μA⋀ μB, VA⋁ VB) = (0.5,0.6) (9) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 505 https://internationalpubls.com This example demonstrates the application of the D-algebraic operation to two intuitionistic fuzzy sets, showcasing how the operation combines their membership and non-membership degrees. Counterexamples, on the other hand, are instances that challenge the validity or applicability of a mathematical model by demonstrating scenarios where the model's predictions or assumptions fail to hold. In the context of intuitionistic fuzzy D-algebra, counterexamples can reveal limitations or unexpected behaviors of the models, prompting to refine or reconsider their approaches. Consider the following counterexample illustrating a scenario where an algebraic property does not hold. Let A and B be two intuitionistic fuzzy sets defined over a universe X. Suppose A = (0.9,0.2) and A = (0.3,0.8). Now, consider the associative property of the D-algebraic operation. D(A, D(B, C)) ≠ D(D(A, B), C) (10) This counterexample demonstrates a violation of the associative property, highlighting a scenario where the DDD-algebraic operation does not exhibit the expected behaviour. (a) (b) Fig.2 Intuitionistic fuzzy sets - decision making Fig.2 enabling users to explore and understand the concept of intuitionistic fuzzy sets in decision- making scenarios. Through this visualization, users can dynamically adjust the membership and non- membership degrees of two candidate sets, visualized as heatmaps, and observe the resulting optimal candidate in real-time. By interacting with sliders that control the degrees of membership and non- membership for each candidate, users gain hands-on experience in understanding how different degrees of qualification and experience affect the selection of the optimal candidate. This interactive approach facilitates intuitive comprehension of the complex mathematical concepts underlying intuitionistic fuzzy sets and their application in decision-making processes. Additionally, the visualization enhances user engagement by providing immediate feedback on how changes in the degrees of membership and non-membership influence the selection of the optimal candidate. Users can experiment with various scenarios, gaining insights into the trade-offs between different criteria and the decision outcomes based on intuitionistic fuzzy logic principles. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 506 https://internationalpubls.com 4. Results To effectively perform the visualizations and interactive plots for intuitionistic fuzzy D-algebra operations, the system should adhere to specific requirements. It should run on Windows, macOS, or Linux operating systems and be equipped with a multi-core processor, ideally Intel i5 or AMD Ryzen 5 and above, to handle computational tasks efficiently. A minimum of 8 GB of RAM is essential to support the processing demands of the plotting algorithms and ensure smooth performance. Adequate storage space of at least 1 GB should be available for storing Python scripts, libraries, and generated plots. Python 3.6 or higher is necessary, along with essential libraries like NumPy, Matplotlib, and mpl_toolkits, which can be installed using pip package manager. While having a dedicated graphics card would enhance the rendering performance, it is not mandatory. Furthermore, a stable internet connection is recommended for downloading libraries and updates to maintain compatibility and functionality. Fig.3 Mapping intuitionistic fuzzy set A to topological space Fig.3 achieves its result by visualizing the mapping of an intuitionistic fuzzy set A onto a topological space through the application of a mapping function Φ. Initially, the membership and non- membership functions of the intuitionistic fuzzy set A are plotted over the universe of discourse X, providing insights into the degrees of membership and non-membership across different elements of X. Subsequently, the mapping function Φ is applied to transform these membership and non- membership degrees into a new set of values, which represent the mapped set Φ(A) in the topological space. This mapping function serves to preserve certain topological properties while translating the intuitionistic fuzzy set A into the topological realm. The resulting graph illustrates how the original fuzzy set A is transformed into a new representation within the topological space, thereby demonstrating the application of topological techniques in intuitionistic fuzzy D-algebra. Through this visualization, the relationship between intuitionistic fuzzy sets and their corresponding topological representations could be understanding, facilitating the analysis of convergence behaviour, continuity properties, and other key aspects of intuitionistic fuzzy D-algebra structures. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 507 https://internationalpubls.com Fig.4 Convergence behaviour and operation D in intuitionistic fuzzy D-algebra Fig.5 Convergence behaviour of intuitionistic fuzzy sets Fig.5 illustrates the convergence behaviour of sequences of intuitionistic fuzzy sets towards a limit set A, providing valuable insights into the dynamics of convergence within intuitionistic fuzzy D- algebra structures. Each line on the graph represents a sequence of fuzzy sets, with their membership and non-membership degrees gradually approaching those of the limit set A as the sequence progresses. The dashed lines represent the membership and non-membership functions of the limit set A, serving as reference points for convergence. As the sequences approach the limit, their curves converge towards these dashed lines, indicating the increasing similarity of their membership and non-membership degrees to those of the limit set A. The graph thus visually demonstrates the convergence process, allowing to analyse the convergence rate and understand how intuitionistic fuzzy sets evolve towards a specific limit under defined sequences. Through this visualization, a deeper understanding of the convergence behaviour within intuitionistic fuzzy D-algebra is achieved, aiding in the exploration and analysis of uncertainty and imprecision in various applications. Fig.6 Membership degree surface plot Fig.6 visualizes the membership degree μDA(x, y)resulting from the intuitionistic fuzzy D-algebra operation DA. In this plot, the x-axis and y-axis represent the membership degrees μA(x)and μA(y)of two elements, while the z-axis shows the resulting membership degreeμDA(x, y). The surface plot is colored using the 'viridis' colormap to highlight the variations in membership degrees. This visualization clearly demonstrates how the membership degrees are combined using the minimum operation ⋀, indicating the degree to which the resulting pair of elements belong to the intuitionistic fuzzy set. By examining this plot, one can understand the behavior and interaction of membership values under the intuitionistic fuzzy D-algebra operation. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 508 https://internationalpubls.com Fig.7 Non-membership degree surface plot Fig.7 depicts the non-membership degree VDA(x, y)resulting from the same intuitionistic fuzzy D- algebra operation DA. Here, the x-axis and y-axis represent the non-membership degrees VA(x) and VA(y)of the two elements, while the z-axis shows the resulting non-membership degree VDA(x, y). This plot uses the 'plasma' colormap to illustrate the variations in non-membership degrees. The visualization highlights how the non-membership degrees are combined using the maximum operation ⋁, indicating the extent to which the resulting pair of elements does not belong to the intuitionistic fuzzy set. This plot provides insights into the interplay between non-membership values and how they aggregate under the D-algebra operation, complementing the understanding gained from the membership degree surface. Fig.8 Intuitionistic fuzzy D-algebra operation Fig.8 with sliders allows for a dynamic exploration of the intuitionistic fuzzy D-algebra operation. Users can adjust the membership (μA(x), μA(y)) and non-membership (VA(x), VA(y)) degrees of two elements using sliders, and the resulting membership (μDA), non-membership (VDA), and hesitation (πDA) degrees are displayed in a bar chart. This real-time updating of the bar chart as the sliders are adjusted provides an intuitive understanding of how different input values affect the outcome of the D-algebra operation. The plot enhances engagement and comprehension by allowing users to experiment with various scenarios, seeing first-hand how changes in membership and non- membership degrees influence the resulting degrees. This interactive tool is particularly useful for demonstrating the complex relationships within intuitionistic fuzzy sets and for educational purposes, helping users to grasp the principles of intuitionistic fuzzy D-algebra through direct manipulation and observation. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 509 https://internationalpubls.com 5. Conclusion and Future Work The advancement of intuitionistic fuzzy D-algebra involves defining core structures, utilizing mathematical models, and validating through examples. It extends classical fuzzy algebra by incorporating degrees of membership and non-membership, enriching frameworks for handling uncertainty. Convergence behaviour and continuity property is analysed through topological techniques, providing deeper insights. Validation strategies elucidate the applicability and limitations of theoretical models. Interactive visualizations aid in understanding intuitionistic fuzzy sets in decision-making contexts. This comprehensive approach bridges theory with practical applications, fostering innovative solutions. Visualization representations help map sets onto topological spaces to illustrate convergence dynamics. The results offer insights into transformation processes and operational characteristics, enhancing comprehension of complex concepts. Future research may refine visualization techniques and explore machine learning applications to further enhance decision-making processes based on intuitionistic fuzzy logic. Integrating computational tools and advanced methodologies holds promise for innovative solutions across domains, from AI to decision support systems. References [1] van Oostrom, V. Commutative Residual Algebra motivation, decision, and applications. a/a, 1, 2. [2] Öztürk, M. A. (2021). Nearness d-algebras. Journal of Algebraic Hyperstructures and Logical Algebras, 2(3), 73-84. [3] Eyssidieux, P., Hwang, J. M., Kebekus, S., & Păun, M. (2023). Komplexe Analysis–Differential and Algebraic methods in Kähler spaces. Oberwolfach Reports, 20(2), 965-1030. [4] Singh, H. K., & Azman, H. H. (2023). Visualizing Algebra: Exploring a Gifted 3rd Grader's Problem-Solving Techniques for Enhanced Conceptual Understanding in Algebra Education. International STEM Journal, 4(2), 1-11. [5] Wu, H. C. (2023). Mathematical Foundations of Fuzzy Sets. John Wiley & Sons. [6] Božić, V. Fuzzy Approach to Risk Management: Enhancing Decision-Making Under Uncertainty. [7] Kahraman, C., Cebi, S., Oztaysi, B., & Cevik Onar, S. (2023). Intuitionistic fuzzy sets with ordered pairs and their usage in multi-attribute decision making: A novel intuitionistic fuzzy TOPSIS method with ordered pairs. Mathematics, 11(18), 3867. [8] Muhiuddin, G., Mahboob, A. H. S. A. N., & Balamurugan, M. (2020). Hesitant anti-intuitionistic fuzzy soft commutative ideals of BCK-algebras. Annals of Communications in Mathematics, 3(2), 158-170. [9] Shrivastava, H., Gopalakrishnan, S., Al-Fatlawy, M. H., Sivasubramanian, S., Sheela, M. S., & Zhu, K. (2023, August). Construction of Evaluation Model of Regional Economic Development Level Based on Fuzzy Clustering Algorithm. In 2023 Second International Conference On Smart Technologies For Smart Nation (SmartTechCon) (pp. 678-682). IEEE. [10] Bayro-Corrochano, E. (2021). A survey on quaternion algebra and geometric algebra applications in engineering and computer science 1995–2020. IEEE Access, 9, 104326-104355. [11] Baghini, A. Z., Babaei, H., & Mirhosseini, R. T. (2024). Determining the membership degree by fuzzy hypergraphs. Mathematical Foundations of Computing, 0-0. [12] Rawbone, T. (2024). Fuzzy relational music perception: Resemblance and implication in the individuation and assembly of concepts. Journal of Creative Music Systems, 8(1), 1-54. [13] Saeed, M., Saeed, M. H., Khalid, M., & Mekawy, I. (2023). Development of hamming and hausdorff distance metrics for cubic intuitionistic fuzzy hypersoft set in cement storage quality control: Development and evaluation. Plos one, 18(9), e0291817. [14] Fan, X. (2022). Access control on provenance (Doctoral dissertation, Macquarie University). [15] Ulieru, M. Electrical and Computer Engineering Department The University of Calgary 2500 University Dr. NW Calgary, Alberta, T1N 1N4 Canada. [16] Syropoulos, A., & Grammenos, T. (2020). A modern introduction to fuzzy mathematics. John Wiley & Sons.