Superhypergraph Neural Networks And Plithogenic Graph Neural Networks Theoretical Foundations
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Superhypergraph Neural Networks And Plithogenic Graph Neural Networks Theoretical Foundations
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Author : Takaaki Fujita
language : en
Publisher: Infinite Study
Release Date : 2025-01-01
Superhypergraph Neural Networks And Plithogenic Graph Neural Networks Theoretical Foundations written by Takaaki Fujita and has been published by Infinite Study this book supported file pdf, txt, epub, kindle and other format this book has been release on 2025-01-01 with Mathematics categories.
Hypergraphs extend traditional graphs by allowing edges to connect multiple nodes, while superhypergraphs further generalize this concept to represent even more complex relationships. Neural networks, inspired by biological systems, are widely used for tasks such as pattern recognition, data classification, and prediction. Graph Neural Networks (GNNs), a well-established framework, have recently been extended to Hypergraph Neural Networks (HGNNs), with their properties and applications being actively studied. The Plithogenic Graph framework enhances graph representations by integrating multi-valued attributes, as well as membership and contradiction functions, enabling the detailed modeling of complex relationships. In the context of handling uncertainty, concepts such as Fuzzy Graphs and Neutrosophic Graphs have gained prominence. It is well established that Plithogenic Graphs serve as a generalization of both Fuzzy Graphs and Neutrosophic Graphs. Furthermore, the Fuzzy Graph Neural Network has been proposed and is an active area of research. This paper establishes the theoretical foundation for the development of SuperHyperGraph Neural Networks (SHGNNs) and Plithogenic Graph Neural Networks, expanding the applicability of neural networks to these advanced graph structures. While mathematical generalizations and proofs are presented, future computational experiments are anticipated.
Advancing Uncertain Combinatorics Through Graphization Hyperization And Uncertainization Fuzzy Neutrosophic Soft Rough And Beyond
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Author : Takaaki Fujita
language : en
Publisher: Infinite Study
Release Date : 2025-01-20
Advancing Uncertain Combinatorics Through Graphization Hyperization And Uncertainization Fuzzy Neutrosophic Soft Rough And Beyond written by Takaaki Fujita and has been published by Infinite Study this book supported file pdf, txt, epub, kindle and other format this book has been release on 2025-01-20 with Mathematics categories.
This book is the fifth volume in the series of Collected Papers on Advancing Uncertain Combinatorics through Graphization, Hyperization, and Uncertainization: Fuzzy, Neutrosophic, Soft, Rough, and Beyond. This volume specifically delves into the concept of Various SuperHyperConcepts, building on the foundational advancements introduced in previous volumes. The series aims to explore the ongoing evolution of uncertain combinatorics through innovative methodologies such as graphization, hyperization, and uncertainization. These approaches integrate and extend core concepts from fuzzy, neutrosophic, soft, and rough set theories, providing robust frameworks to model and analyze the inherent complexity of real-world uncertainties. At the heart of this series lies combinatorics and set theory—cornerstones of mathematics that address the study of counting, arrangements, and the relationships between collections under defined rules. Traditionally, combinatorics has excelled in solving problems involving uncertainty, while advancements in set theory have expanded its scope to include powerful constructs like fuzzy and neutrosophic sets. These advanced sets bring new dimensions to uncertainty modeling by capturing not just binary truth but also indeterminacy and falsity. In this fifth volume, the exploration of Various SuperHyperConcepts provides an innovative lens to address uncertainty, complexity, and hierarchical relationships. It synthesizes key methodologies introduced in earlier volumes, such as hyperization and neutrosophic extensions, while advancing new theories and applications. From pioneering hyperstructures to applications in advanced decision-making, language modeling, and neural networks, this book represents a significant leap forward in uncertain combinatorics and its practical implications across disciplines. The book is structured into 17 chapters, each contributing unique perspectives and advancements in the realm of Various SuperHyperConcepts and their related frameworks: Chapter 1 introduces the concept of Body-Mind-Soul-Spirit Fluidity within psychology and phenomenology, while examining established social science frameworks like PDCA and DMAIC. It extends these frameworks using Neutrosophic Sets, a flexible extension of Fuzzy Sets, to improve their adaptability for mathematical and programming applications. The chapter emphasizes the potential of Neutrosophic theory to address multi-dimensional challenges in social sciences. Chapter 2 delves into the theoretical foundation of Hyperfunctions and their generalizations, such as Hyperrandomness and Hyperdecision-Making. It explores higher-order frameworks like Weak Hyperstructures, Hypergraphs, and Cognitive Hypermaps, aiming to establish their versatility in addressing multi-layered problems and setting a foundation for further studies. Chapter 3 extends traditional decision-making methodologies into HyperDecision-Making and n-SuperHyperDecision-Making. By building on approaches like MCDM and TOPSIS, this chapter develops frameworks capable of addressing complex decision-making scenarios, emphasizing their applicability in dynamic, multi-objective contexts. Chapter 4 explores integrating uncertainty frameworks, including Fuzzy, Neutrosophic, and Plithogenic Sets, into Large Language Models (LLMs). It proposes innovative models like Large Uncertain Language Models and Natural Uncertain Language Processing, integrating hierarchical and generalized structures to advance the handling of uncertainty in linguistic representation and processing. Chapter 5 introduces the Natural n-Superhyper Plithogenic Language by synthesizing natural language, plithogenic frameworks, and superhyperstructures. This innovative construct seeks to address challenges in advanced linguistic and structural modeling, blending attributes of uncertainty, complexity, and hierarchical abstraction. Chapter 6 defines mathematical extensions such as NeutroHyperstructures and AntiHyperstructures using the Neutrosophic Triplet framework. It formalizes structures like neutro-superhyperstructures, advancing classical frameworks into higher-dimensional realms. Chapter 7 explores the extension of Binary Code, Gray Code, and Floorplans through hyperstructures and superhyperstructures. It highlights their iterative and hierarchical applications, demonstrating their adaptability for complex data encoding and geometric arrangement challenges. Chapter 8 investigates the Neutrosophic TwoFold SuperhyperAlgebra, combining classical algebraic operations with neutrosophic components. This chapter expands upon existing algebraic structures like Hyperalgebra and AntiAlgebra, exploring hybrid frameworks for advanced mathematical modeling. Chapter 9 introduces Hyper Z-Numbers and SuperHyper Z-Numbers by extending the traditional Z-Number framework with hyperstructures. These extensions aim to represent uncertain information in more complex and multidimensional contexts. Chapter 10 revisits category theory through the lens of hypercategories and superhypercategories. By incorporating hierarchical and iterative abstractions, this chapter extends the foundational principles of category theory to more complex and layered structures. Chapter 11 formalizes the concept of n-SuperHyperBranch-width and its theoretical properties. By extending hypergraphs into superhypergraphs, the chapter explores recursive structures and their potential for representing intricate hierarchical relationships. Chapter 12 examines superhyperstructures of partitions, integrals, and spaces, proposing a framework for advancing mathematical abstraction. It highlights the potential applications of these generalizations in addressing hierarchical and multi-layered problems. Chapter 13 revisits Rough, HyperRough, and SuperHyperRough Sets, introducing new concepts like Tree-HyperRough Sets. The chapter connects these frameworks to advanced approaches for modeling uncertainty and complex relationships. Chapter 14 explores Plithogenic SuperHyperStructures and their applications in decision-making, control, and neuro systems. By integrating these advanced frameworks, the chapter proposes innovative directions for extending existing systems to handle multi-attribute and contradictory properties. Chapter 15 focuses on superhypergraphs, expanding hypergraph concepts to model complex structural types like arboreal and molecular superhypergraphs. It introduces Generalized n-th Powersets as a unifying framework for broader mathematical applications, while also touching on hyperlanguage processing. Chapter 16 defines NeutroHypergeometry and AntiHypergeometry as extensions of classical geometric structures. Using the Geometric Neutrosophic Triplet, the chapter demonstrates the flexibility of these frameworks in representing multi-dimensional and uncertain relationships. Chapter 17 establishes the theoretical groundwork for SuperHyperGraph Neural Networks and Plithogenic Graph Neural Networks. By integrating advanced graph structures, this chapter opens pathways for applying neural networks to more intricate and uncertain data representations.
Exploring Concepts Of Hyperfuzzy Hyperneutrosophic And Hyperplithogenic Sets Ii
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Author : Takaaki Fujita
language : en
Publisher: Infinite Study
Release Date : 2025-01-01
Exploring Concepts Of Hyperfuzzy Hyperneutrosophic And Hyperplithogenic Sets Ii written by Takaaki Fujita and has been published by Infinite Study this book supported file pdf, txt, epub, kindle and other format this book has been release on 2025-01-01 with Mathematics categories.
This paper delves into the advancements of classical set theory to address the complexities and uncertainties inherent in real-world phenomena. It highlights three major extensions of traditional set theory - Fuzzy Sets [288], Neutrosophic Sets [237], and Plithogenic Sets [243] - and examines their further generalizations into Hyperfuzzy [106], HyperNeutrosophic [90], and Hyperplithogenic Sets [90]. Building on previous research [83], this study explores the potential applications of HyperNeutrosophic Sets and SuperHyperNeutrosophic Sets across various domains. Specifically, it extends f undamental c oncepts such as Neutrosophic Logic, Cognitive Maps, Graph Neural Networks, Classifiers, and Triplet Groups through these advanced set structures and briefly a nalyzes t heir m athematical properties.
Some Types Of Hyperneutrosophic Set 4 Cubic Trapozoidal Q Rung Orthopair Overset Underset And Offset
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Author : Takaaki Fujita
language : en
Publisher: Infinite Study
Release Date :
Some Types Of Hyperneutrosophic Set 4 Cubic Trapozoidal Q Rung Orthopair Overset Underset And Offset written by Takaaki Fujita and has been published by Infinite Study this book supported file pdf, txt, epub, kindle and other format this book has been release on with Mathematics categories.
This paper builds upon the foundational work presented in [38–40]. The Neutrosophic Set provides a comprehensive mathematical framework for managing uncertainty, defined by three membership functions: truth, indeterminacy, and falsity. Recent advancements have introduced extensions such as the Hyperneutrosophic Set and the SuperHyperneutrosophic Set, which are specifically designed to address increasingly complex and multidimensional problems. The formal definitions of these sets are available in [30]. In this paper, we extend the Neutrosophic Cubic Set, Trapezoidal Neutrosophic Set, q-Rung Orthopair Neutrosophic Set, Neutrosophic Overset, Neutrosophic Underset, and Neutrosophic Offset using the frameworks of the Hyperneutrosophic Set and the SuperHyperneutrosophic Set. Furthermore, we briefly examine their properties and potential applications.
Some Types Of Hyperneutrosophic Set 5 Support Paraconsistent Faillibilist And Others
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Author : Takaaki Fujita
language : en
Publisher: Infinite Study
Release Date : 2025-01-01
Some Types Of Hyperneutrosophic Set 5 Support Paraconsistent Faillibilist And Others written by Takaaki Fujita and has been published by Infinite Study this book supported file pdf, txt, epub, kindle and other format this book has been release on 2025-01-01 with Mathematics categories.
This paper builds upon the foundational advancements introduced in [14, 25–27]. The Neutrosophic Set offers a versatile mathematical framework for addressing uncertainty through its three membership functions: truth, indeterminacy, and falsity. Extensions such as the Hyperneutrosophic Set and the SuperHyperneutrosophic Set have been recently proposed to tackle increasingly sophisticated and multidimensional problems. Detailed formal definitions of these concepts can be found in [20]. In this paper, we extend various specialized classes of Neutrosophic Sets—namely, the Support Neutrosophic Set, Neutrosophic Intuitionistic Set (distinct from the Intuitionistic Fuzzy Set), Neutrosophic Paraconsistent Set, Neutrosophic Faillibilist Set, Neutrosophic Paradoxist Set, Neutrosophic Pseudo-Paradoxist Set, Neutrosophic Tautological Set, Neutrosophic Nihilist Set, Neutrosophic Dialetheist Set, and Neutrosophic Trivialist Set—by utilizing the frameworks of the Hyperneutrosophic Set and the SuperHyperneutrosophic Set.
Neutrosophic Twofold Superhyperalgebra And Anti Superhyperalgebra
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Author : Takaaki Fujita
language : en
Publisher: Infinite Study
Release Date : 2025-01-01
Neutrosophic Twofold Superhyperalgebra And Anti Superhyperalgebra written by Takaaki Fujita and has been published by Infinite Study this book supported file pdf, txt, epub, kindle and other format this book has been release on 2025-01-01 with Mathematics categories.
Neutrosophic Sets are conceptual frameworks designed to address uncertainty. A Neutrosophic TwoFold Algebra is a hybrid algebraic structure defined over a neutrosophic set, combining classical algebraic operations with neutrosophic components. Concepts such as Hyperalgebra and Superhyperalgebra extend classical Algebra using Power Sets and 𝑛-th powersets. Additionally, structures such as NeutroAlgebra and AntiAlgebra have been defined in recent y ears. This paper explores several related concepts, including TwoFold SuperhyperAlgebra and Anti SuperhyperAlgebra.
Some Types Of Hyperneutrosophic Set 2 Complex Single Valued Triangular Fermatean And Linguistic Sets
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Author : Takaaki Fujita
language : en
Publisher: Infinite Study
Release Date : 2025-01-01
Some Types Of Hyperneutrosophic Set 2 Complex Single Valued Triangular Fermatean And Linguistic Sets written by Takaaki Fujita and has been published by Infinite Study this book supported file pdf, txt, epub, kindle and other format this book has been release on 2025-01-01 with Mathematics categories.
This paper is a continuation of the work presented in [35]. The Neutrosophic Set provides a mathematical framework for managing uncertainty, characterized by three membership functions: truth, indeterminacy, and falsity. Recent advancements have introduced extensions such as the Hyperneutrosophic Set and SuperHyperneutrosophic Set to address more complex and multidimensional challenges. In this study, we extend the Complex Neutrosophic Set, Single-Valued Triangular Neutrosophic Set, Fermatean Neutrosophic Set, and Linguistic Neutrosophic Set within the frameworks of Hyperneutrosophic Sets and SuperHyperneutrosophic Sets. Furthermore, we investigate their mathematical structures and analyze their connections with other set-theoretic concepts.
Reconsideration Of Neutrosophic Social Science And Neutrosophic Phenomenology With Non Classical Logic
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Author : Takaaki Fujita
language : en
Publisher: Infinite Study
Release Date : 2025-01-01
Reconsideration Of Neutrosophic Social Science And Neutrosophic Phenomenology With Non Classical Logic written by Takaaki Fujita and has been published by Infinite Study this book supported file pdf, txt, epub, kindle and other format this book has been release on 2025-01-01 with Mathematics categories.
Body-Mind-Soul-Spirit Fluidity is a concept rooted in psychology and phenomenology, offering significant insights into human decision-making and well-being. Similarly, in social analysis and social sciences, frameworks such as PDCA, DMAIC, SWOT, and OODA have been established to enable structured evaluation and effective p roblem-solving. Furthermore, in phenomenology and social sciences, various logical systems have been developed to address specific objectives and practical applications. This paper extends these concepts using the Neutrosophic theory, revisiting their mathematical definitions and exploring their properties. The Neutrosophic Set, an extension of the Fuzzy Set, is a highly flexible framework that has been widely studied in fields such as social s ciences. By incorporating Neutrosophic Sets, we aim to improve their suitability for programming and mathematical analysis, providing advanced methods to tackle complex, multi-dimensional problems. We hope that this research will inspire further studies and foster the development of practical applications across various related disciplines.
Some Types Of Hyperneutrosophic Set 3 Dynamic Quadripartitioned Pentapartitioned Heptapartitioned M Polar
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Author :
language : en
Publisher: Infinite Study
Release Date : 2025-01-01
Some Types Of Hyperneutrosophic Set 3 Dynamic Quadripartitioned Pentapartitioned Heptapartitioned M Polar written by and has been published by Infinite Study this book supported file pdf, txt, epub, kindle and other format this book has been release on 2025-01-01 with Mathematics categories.
This paper builds upon the foundation established in [50, 51]. The Neutrosophic Set provides a robust mathematical framework for handling uncertainty, defined by three membership functions: truth, indeterminacy, and falsity. Recent developments have introduced extensions such as the Hyperneutrosophic Set and SuperHyperneutrosophic Set to tackle increasingly complex and multidimensional problems. In this study, we explore further extensions, including the Dynamic Neutrosophic Set, Quadripartitioned Neutrosophic Set, Pentapartitioned Neutrosophic Set, Heptapartitioned Neutrosophic Set, and m-Polar Neutrosophic Set, to address advanced challenges and applications.
Plithogenic Superhypersoft Set And Plithogenic Forest Superhypersoft Set
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Author : Takaaki Fujita
language : en
Publisher: Infinite Study
Release Date : 2025-01-01
Plithogenic Superhypersoft Set And Plithogenic Forest Superhypersoft Set written by Takaaki Fujita and has been published by Infinite Study this book supported file pdf, txt, epub, kindle and other format this book has been release on 2025-01-01 with Mathematics categories.
The Plithogenic Set is renowned for generalizing concepts such as Fuzzy Sets and Neutrosophic Sets. The Extended Plithogenic Set represents an advanced concept of the Plithogenic Set, as recently defined in [62]. A hypersoft set is a mathematical structure that maps distinct attributes with non-overlapping values to subsets of a universal set, facilitating multi-criteria decision analysis. Recent studies have explored the combination of Plithogenic Sets and Hypersoft Sets, leading to the development of the Plithogenic Hypersoft Set. This paper further extends these concepts to introduce and examine the Plithogenic SuperHypersoft Set and the Extended Plithogenic SuperHypersoft Set.