Since the world is full of indeterminacy, the neutrosophics found their place into contemporary research. We now introduce for the first time the notions of neutrosophic measure and neutrosophic integral. Neutrosophic Science means development and applications of neutrosophic logic/set/measure/integral/ probability etc. and their applications in any field. It is possible to define the neutrosophic measure and consequently the neutrosophic integral and neutrosophic probability in many ways, because there are various types (...) of indeterminacies, depending on the problem we need to solve. Indeterminacy is different from randomness. Indeterminacy can be caused by physical space materials and type of construction, by items involved in the space, or by other factors. Neutrosophic measure is a generalization of the classical measure for the case when the space contains some indeterminacy. Neutrosophic Integral is defined on neutrosophic measure. Simple examples of neutrosophic integrals are given. (shrink)

Since the world is full of indeterminacy, the Neutrosophics found their place into contemporary research. We now introduce for the first time the notions of Neutrosophic Crisp Sets and Neutrosophic Topology on Crisp Sets. We develop the 2012 notion of Neutrosophic Topological Spaces and give many practical examples. Neutrosophic Science means development and applications of Neutrosophic Logic, Set, Measure, Integral, Probability etc., and their applications in any field. It is possible to define the (...) class='Hi'>neutrosophic measure and consequently the neutrosophic integral and neutrosophic probability in many ways, because there are various types of indeterminacies, depending on the problem we need to solve. Indeterminacy is different from randomness. Indeterminacy can be caused by physical space, materials and type of construction, by items involved in the space, or by other factors. In 1965 [51], Zadeh generalized the concept of crisp set by introducing the concept of fuzzy set, corresponding to the situation in which there is no precisely defined set;there are increasing applications in various fields, including probability, artificial intelligence, control systems, biology and economics. Thus, developments in abstract mathematics using the idea of fuzzy sets possess sound footing. In accordance, fuzzy topological spaces were introduced by Chang [12] and Lowen [33]. After the development of fuzzy sets, much attention has been paid to the generalization of basic concepts of classical topology to fuzzy sets and accordingly developing a theory of fuzzy topology [1-58]. In 1983, the intuitionistic fuzzy set was introduced by K. Atanassov [55, 56, 57] as a generalization of the fuzzy set, beyond the degree of membership and the degree of non-membership of each element. In 1999 and 2002, Smarandache [71, 72, 73, 74] defined the notion of Neutrosophic Sets, which is a generalization of Zadeh’s fuzzy set and Atanassov's intuitionistic fuzzy set. Some neutrosophic concepts have been investigated by Salama et al. [61-70]. Forwarding the study of neutrosophic sets, this book consists of seven chapters, targeting to:  generalize the previous studies in [1-59], and[91-94] so to define the neutrosopic crisp set and neutrosophic set concepts;  discuss their main properties; A. A. Salama & Florentin Smarandache  introduce and study some concepts of neutrosophic crisp and neutrosophic topological spaces and deduce their properties;  deduce many types of functions and give the relationships between different neutrosophic topological spaces, which helps to build new properties of neutrosophic topological spaces;  stress once more the importance of Neutrosophic Ideal as a nontrivial extension of neutrosophic set and neutrosophic logic [71, 72, 73, 74];  propose applications on computer sciences by using neutrosophic sets. (shrink)

The topics approached in the 52 papers included in this book are: neutrosophic sets; neutrosophic logic; generalized neutrosophic set; neutrosophic rough set; multigranulation neutrosophic rough set (MNRS); neutrosophic cubic sets; triangular fuzzy neutrosophic sets (TFNSs); probabilistic single-valued (interval) neutrosophic hesitant fuzzy set; neutro-homomorphism; neutrosophic computation; quantum computation; neutrosophic association rule; data mining; big data; oracle Turing machines; recursive enumerability; oracle computation; interval number; dependent degree; possibility degree; power aggregation operators; multi-criteria (...) group decision-making (MCGDM); expert set; soft sets; LA-semihypergroups; single valued trapezoidal neutrosophic number; inclusion relation; Q-linguistic neutrosophic variable set; vector similarity measure; fundamental neutro-homomorphism theorem; neutro-isomorphism theorem; quasi neutrosophic triplet loop; quasi neutrosophic triplet group; BE-algebra; cloud model; Maclaurin symmetric mean; pseudo-BCI algebra; hesitant fuzzy set; photovoltaic plan; decision-making trial and evaluation laboratory (DEMATEL); Choquet integral; fuzzy measure; clustering algorithm; and many more. (shrink)

This work introduces the framework for selecting architecture in 5G networks, considering various technological, performance, economic, and operational factors. With the emergence of 5G technology, the architecture selection process has become pivotal in meeting diverse requirements for ultra-high-speed connectivity, low latency, scalability, and diverse service demands. The evaluation comprehensively analyses different architecture options, including centralized, distributed, cloud-based, and virtualized architectures. Factors such as network performance, scalability, cost-effectiveness, security, and compatibility are considered within a multi-criteria decision-making framework. Findings reveal each architecture (...) option's diverse strengths and limitations, emphasizing the importance of aligning architectural choices with specific use cases, deployment scenarios, and business objectives. We used the concept of multi-criteria decision-making (MCDM) methodology to control the various criteria. We used the single-valued neutrosophic set (SVNS) to deal with vague and inconsistent data. The SVNS is integrated with the hyperSoft set. The proposed methodology uses a single-valued neutrosophic hyperSoft set (SVNHSS) to select the best 5G architecture. We used the VIKOR method as an MCDM method to rank the alternatives. The proposed framework provides stakeholders with a structured methodology to evaluate and prioritize architecture options, facilitating informed decision-making in the complex landscape of 5G network deployments. (shrink)

This book is the sixth volume in the series of Collected Papers on Advancing Uncertain Combinatorics through Graphization, Hyperization, and Uncertainization: Fuzzy, Neutrosophic, Soft, Rough, and Beyond. Building upon the foundational contributions of previous volumes, this edition focuses on the exploration and development of Various New Uncertain Concepts, further enriching the study of uncertainty and complexity through innovative theoretical advancements and practical applications. The volume is meticulously organized into 15 chapters, each presenting unique perspectives and contributions to the field. (...) From theoretical explorations to real-world applications, these chapters provide a cohesive and comprehensive overview of the state of the art in uncertain combinatorics, emphasizing the versatility and power of the newly introduced concepts and methodologies. The first chapter (SuperHypertree-depth – Structural Analysis in SuperHyperGraphs) explores the concept of SuperHypertree-depth, an extension of the classical graph parameter Tree-depth and its hypergraph counterpart Hypertree-depth. By introducing hierarchical nesting within SuperHyperGraphs, where both vertices and edges can represent recursive subsets, this study investigates the mathematical properties and structural implications of these extended parameters. The findings highlight the relationships between SuperHypertree-depth and its traditional graph-theoretic equivalents, providing a deeper understanding of their applicability to hierarchical and complex systems. The second chapter (Obstructions for Hypertree-width and SuperHypertree-width) examines the role of ultrafilters as obstructions in determining Hypertree-width and extends the concept to SuperHypertree-width. Building on hypergraph theory, which abstracts traditional graph frameworks into more complex domains, the study investigates how recursive structures within SuperHyperGraphs redefine the computational and structural properties of these parameters. Ultrafilters, with their broad mathematical significance, serve as critical tools for understanding the limitations and potentials of these advanced graph metrics. The third chapter (SuperHypertree-Length and SuperHypertree-Breadth in SuperHyperGraphs) investigates the extension of the graph-theoretic parameters Tree-length and Tree-breadth to the realms of hypergraphs and SuperHyperGraphs. By leveraging the hierarchical nesting of SuperHyperGraphs, the study explores how these parameters adapt to increasingly complex and multi-level structures. Comparative analyses between these extended parameters and their classical counterparts reveal new insights into their relevance and utility in advanced graph and hypergraph theory. Plithogenic Sets, which generalize Fuzzy and Neutrosophic Sets, are extended in the fourth chapter (Extended HyperPlithogenic Sets and Generalized Plithogenic Graphs) to Extended Plithogenic Sets, HyperPlithogenic Sets, and SuperHyperPlithogenic Sets. This study further investigates their application to graph theory through the concepts of Extended Plithogenic Graphs and Generalized Extended Plithogenic Graphs. The chapter provides a concise exploration of these frameworks, offering insights into their potential for addressing uncertainty and complexity in graph structures. Soft Sets provide an effective framework for decision-making by mapping parameters to subsets of a universal set, addressing uncertainty and vagueness. The fifth chapter (Double-Framed Superhypersoft Set and Double-Framed Treesoft Set) introduces the Double-Framed SuperHypersoft Set and the Double-Framed Treesoft Set as extensions of traditional and advanced soft set frameworks, such as Hypersoft and SuperHypersoft Sets. The chapter explores their relationships with existing concepts, offering new tools to handle complex decision-making scenarios with enhanced structural flexibility. The sixth paper (HyperPlithogenic Cubic Set and SuperHyperPlithogenic Cubic Set) introduces the concepts of the HyperPlithogenic Cubic Set and SuperHyperPlithogenic Cubic Set, which extend the Plithogenic Cubic Set by integrating both interval-valued and single-valued fuzzy memberships. These sets leverage multi-attribute aggregation techniques inherent to plithogenic structures, allowing for nuanced representations of uncertainty. Additionally, related constructs such as the HyperPlithogenic Fuzzy Cubic Set, HyperPlithogenic Intuitionistic Fuzzy Cubic Set, and HyperPlithogenic Neutrosophic Cubic Set are explored, further enriching the theoretical and practical applications of this framework. The seventh chapter (L-Neutrosophic Sets and Nonstationary Neutrosophic Sets) extends the foundational concepts of fuzzy sets by integrating Neutrosophic and Plithogenic frameworks. By introducing L-Neutrosophic Sets and Nonstationary Neutrosophic Sets, the study enhances the representation of uncertainty through independent membership components: truth, indeterminacy, and falsity. These advanced constructs also incorporate multi-dimensional and contradictory attributes, providing a robust means of modeling complex decision-making and uncertain data. Plithogenic and Rough Sets, known for generalizing uncertainty modeling and classification, are extended in the eight chapter (Forest HyperPlithogenic and Forest HyperRough Sets) to Forest HyperPlithogenic Sets, Forest SuperHyperPlithogenic Sets, Forest HyperRough Sets, and Forest SuperHyperRough Sets. These frameworks incorporate hierarchical and recursive structures to advance existing set-theoretic paradigms. The chapter explores their applications in multi-level data analysis and uncertainty classification, demonstrating their adaptability to complex systems. Building on Fuzzy, Neutrosophic, and Plithogenic Sets, the tenth chapter (Symbolic HyperPlithogenic Sets) introduces Symbolic HyperPlithogenic Sets and Symbolic n-SuperHyperPlithogenic Sets. These sets incorporate symbolic components and algebraic coefficients, enabling flexible operations within a defined prevalence order. By extending symbolic representation into hyperplithogenic and superhyperplithogenic domains, the chapter opens new pathways for addressing uncertainty and hierarchical complexity in mathematical modeling. Soft Sets, designed to manage uncertainty and imprecision, have evolved through various extensions like Hypersoft Sets and SuperHypersoft Sets. The eleventh chapter (N-SuperHypersoft and Bijective SuperHypersoft Sets) introduces N-SuperHypersoft Sets, N-Treesoft Sets, Bijective SuperHypersoft Sets, and Bijective Treesoft Sets. These new constructs enhance decision-making frameworks by incorporating advanced hierarchical and bijective relationships, building on existing theories and expanding their applications. Plithogenic Sets, known for integrating multi-valued attributes and contradictions, and Rough Sets, which partition data into definable approximations, are combined in the twelfth chapter (Plithogenic Rough Sets) to form Plithogenic Rough Sets. This fusion provides a powerful framework for addressing uncertainty in dynamic and complex decision-making scenarios, offering a novel approach to uncertainty modeling. Expanding on Neutrosophic Sets, which represent truth, indeterminacy, and falsehood, this chapter introduces Plithogenic Duplets and Plithogenic Triplets. These constructs leverage the Plithogenic framework to incorporate attributes, values, and contradiction measures. The thirteenth chapter (Plithogenic Duplets and Triplets) examines their relationships with Neutrosophic Duplets and Triplets, offering new tools for multi-dimensional data representation and decision-making. Building on foundational concepts like Rough Sets and Vague Sets, the fourteenth chapter (SuperRough and SuperVague Sets) introduces SuperRough Sets and SuperVague Sets. These generalized frameworks extend uncertainty modeling by incorporating hierarchical structures. The study also demonstrates that SuperRough Sets can evolve into SuperHyperRough Sets, providing further generalizations for advanced data classification and analysis. The fifteenth chapter (Neutrosophic TreeSoft Expert and ForestSoft Sets) revisits the Neutrosophic TreeSoft Set, which combines the hierarchical structure of TreeSoft Sets with the Neutrosophic framework for uncertainty representation. Additionally, it introduces the Neutrosophic TreeSoft Expert Set, incorporating expert knowledge into the model. The chapter also explores the ForestSoft Set and its extension, the Neutrosophic ForestSoft Set, to provide multi-level, tree-structured approaches for complex data representation and analysis. (shrink)

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. (shrink)

The second volume of “Advancing Uncertain Combinatorics through Graphization, Hyperization, and Uncertainization: Fuzzy, Neutrosophic, Soft, Rough, and Beyond” presents a deep exploration of the progress in uncertain combinatorics through innovative methodologies like graphization, hyperization, and uncertainization. This volume integrates foundational concepts from fuzzy, neutrosophic, soft, and rough set theory, among others, to further advance the field. Combinatorics and set theory, two central pillars of mathematics, focus on counting, arrangement, and the study of collections under defined rules. Combinatorics excels (...) in handling uncertainty, while set theory has evolved with concepts such as fuzzy and neutrosophic sets, which enable the modeling of complex real-world uncertainties by addressing truth, indeterminacy, and falsehood. These advancements, when combined with graph theory, give rise to novel forms of uncertain sets in "graphized" structures, including hypergraphs and superhypergraphs. Innovations such as Neutrosophic Oversets, Undersets, and Offsets, as well as the Nonstandard Real Set, build upon traditional graph concepts, pushing both theoretical and practical boundaries. The synthesis of combinatorics, set theory, and graph theory in this volume provides a robust framework for addressing the complexities and uncertainties inherent in both mathematical and real-world systems, paving the way for future research and application. In the first chapter, “A Review of the Hierarchy of Plithogenic, Neutrosophic, and Fuzzy Graphs: Survey and Applications”, the authors investigate the interrelationships among various graph classes, including Plithogenic graphs, and explore other related structures. Graph theory, a fundamental branch of mathematics, focuses on networks of nodes and edges, studying their paths, structures, and properties. A Fuzzy Graph extends this concept by assigning a membership degree between 0 and 1 to each edge and vertex, representing the level of uncertainty. The Turiyam Neutrosophic Graph is introduced as an extension of both Neutrosophic and Fuzzy Graphs, while Plithogenic graphs offer a potent method for managing uncertainty. The second chapter, “Review of Some Superhypergraph Classes: Directed, Bidirected, Soft, and Rough”, examines advanced graph structures such as directed superhypergraphs, bidirected hypergraphs, soft superhypergraphs, and rough superhypergraphs. Classical graph classes include undirected graphs, where edges lack orientation, and directed graphs, where edges have specific directions. Recent innovations, including bidirected graphs, have sparked ongoing research and significant advancements in the field. Soft Sets and their extension to Soft Graphs provide a flexible framework for managing uncertainty, while Rough Sets and Rough Graphs address uncertainty by using lower and upper approximations to handle imprecise data. Hypergraphs generalize traditional graphs by allowing edges, or hyperedges, to connect more than two vertices. Superhypergraphs further extend this by allowing both vertices and edges to represent subsets, facilitating the modeling of hierarchical and group-based relationships. The third chapter, “Survey of Intersection Graphs, Fuzzy Graphs, and Neutrosophic Graphs”, explores the intersection graph models within the realms of Fuzzy Graphs, Intuitionistic Fuzzy Graphs, Neutrosophic Graphs, Turiyam Neutrosophic Graphs, and Plithogenic Graphs. The chapter highlights their mathematical properties and interrelationships, reflecting the growing number of graph classes being developed in these areas. Intersection graphs, such as Unit Square Graphs, Circle Graphs, and Ray Intersection Graphs, are crucial for understanding complex graph structures in uncertain environments. The fourth chapter, “Fundamental Computational Problems and Algorithms for SuperHyperGraphs”, addresses optimization problems within the SuperHypergraph framework, such as the SuperHypergraph Partition Problem, Reachability, and Minimum Spanning SuperHypertree. The chapter also adapts classical problems like the Traveling Salesman Problem and the Chinese Postman Problem to the SuperHypergraph context, exploring how hypergraphs, which allow hyperedges to connect more than two vertices, can be used to solve complex hierarchical and relational problems. The fifth chapter, “A Short Note on the Basic Graph Construction Algorithm for Plithogenic Graphs”, delves into algorithms designed for Plithogenic Graphs and Intuitionistic Plithogenic Graphs, analyzing their complexity and validity. Plithogenic Graphs model multi-valued attributes by incorporating membership and contradiction functions, offering a nuanced representation of complex relationships. The sixth chapter, “Short Note of Bunch Graph in Fuzzy, Neutrosophic, and Plithogenic Graphs”, generalizes traditional graph theory by representing nodes as groups (bunches) rather than individual entities. This approach enables the modeling of both competition and collaboration within a network. The chapter explores various uncertain models of bunch graphs, including Fuzzy Graphs, Neutrosophic Graphs, Turiyam Neutrosophic Graphs, and Plithogenic Graphs. In the seventh chapter, “A Reconsideration of Advanced Concepts in Neutrosophic Graphs: Smart, Zero Divisor, Layered, Weak, Semi, and Chemical Graphs”, the authors extend several fuzzy graph classes to Neutrosophic graphs and analyze their properties. Neutrosophic Graphs, a generalization of fuzzy graphs, incorporate degrees of truth, indeterminacy, and falsity to model uncertainty more effectively. The eighth chapter, “Short Note of Even-Hole-Graph for Uncertain Graph”, focuses on Even-Hole-Free and Meyniel Graphs analyzed within the frameworks of Fuzzy, Neutrosophic, Turiyam Neutrosophic, and Plithogenic Graphs. The study investigates the structure of these graphs, with an emphasis on their implications for uncertainty modeling. The ninth chapter, “Survey of Planar and Outerplanar Graphs in Fuzzy and Neutrosophic Graphs”, explores planar and outerplanar graphs, as well as apex graphs, within the contexts of fuzzy, neutrosophic, Turiyam Neutrosophic, and plithogenic graphs. The chapter examines how these types of graphs are used to model uncertain parameters and relationships in mathematical and real-world systems. The tenth chapter, “General Plithogenic Soft Rough Graphs and Some Related Graph Classes”, introduces and explores new concepts such as Turiyam Neutrosophic Soft Graphs and General Plithogenic Soft Graphs. The chapter also examines models of uncertain graphs, including Fuzzy, Intuitionistic Fuzzy, Neutrosophic, and Plithogenic Graphs, all designed to handle uncertainty in diverse contexts. The eleventh chapter, “Survey of Trees, Forests, and Paths in Fuzzy and Neutrosophic Graphs”, provides a comprehensive study of Trees, Forests, and Paths within the framework of Fuzzy and Neutrosophic Graphs. This chapter focuses on classifying and analyzing graph structures like trees and paths in uncertain environments, contributing to the ongoing development of graph theory in the context of uncertainty. (shrink)

The third volume of “Advancing Uncertain Combinatorics through Graphization, Hyperization, and Uncertainization: Fuzzy, Neutrosophic, Soft, Rough, and Beyond” presents an in-depth exploration of the cutting-edge developments in uncertain combinatorics and set theory. This comprehensive collection highlights innovative methodologies such as graphization, hyperization, and uncertainization, which enhance combinatorics by incorporating foundational concepts from fuzzy, neutrosophic, soft, and rough set theories. These advancements open new mathematical horizons, offering novel approaches to managing uncertainty within complex systems. Combinatorics, a discipline focused on (...) counting, arrangement, and structure, often faces challenges when uncertainty is present. Set theory, which underpins combinatorial problems, has evolved to tackle these challenges. The introduction of fuzzy and neutrosophic sets has expanded the toolkit for modeling uncertainty by incorporating elements of truth, indeterminacy, and falsehood into decision-making processes. These innovations seamlessly intersect with graph theory, providing new ways to represent uncertain structures through "graphized" forms such as hypergraphs and superhypergraphs. This volume also introduces advanced concepts like Neutrosophic Oversets, Undersets, and Offsets, which push the boundaries of classical graph theory and offer deeper insights into the mathematical and practical challenges posed by real-world systems. By blending combinatorics, set theory, and graph theory, the authors have created a robust framework for addressing uncertainty in both mathematical systems and their real-world applications. This foundation sets the stage for future breakthroughs in combinatorics, set theory, and related fields. Each chapter in this volume contributes both theoretical foundations and practical applications, demonstrating the power of integrating graph theory, set theory, and uncertainty models. The new ideas, algorithms, and mathematical tools presented here will drive the future of combinatorial research and its applications in uncertain environments. In the first chapter, “Introduction to Upside-Down Logic: Its Deep Relation to Neutrosophic Logic and Applications”, the authors present Upside-Down Logic, a novel logical framework that systematically transforms truths into falsehoods and vice versa, based on contextual shifts. Introduced by F. Smarandache, this paper provides a mathematical definition of Upside-Down Logic, including applications related to the Japanese language. The chapter also introduces Contextual Upside-Down Logic, an extension that adjusts logical connectives alongside flipped truth values, as well as Indeterm-Upside-Down Logic and Certain Upside-Down Logic to address indeterminacy. A simple algorithm is also proposed to demonstrate the computational aspects of this logic. In the second chapter, “Local-Neutrosophic Logic and Local-Neutrosophic Sets: Incorporating Locality with Applications”, the authors introduce Local-Neutrosophic Logic and Local-Neutrosophic Sets, which integrate the concept of locality into Neutrosophic Logic. By defining locality as the influence of immediate surroundings on an object or system, this chapter explores how it affects indeterminacy in real-world problems. The paper also examines potential applications and provides mathematical definitions for these new concepts. The third chapter, “A Review of Fuzzy and Neutrosophic Offsets: Connections to Some Set Concepts and Normalization Function”, extends the concept of offsets in uncertain set-theoretic frameworks, such as Fuzzy Sets, Neutrosophic Sets, and Plithogenic Sets. This chapter introduces several advanced types of offsets, including Nonstationary Fuzzy Offset, Multi-valued Plithogenic Offset, and Subset-valued Neutrosophic Offset, offering deeper insights into handling uncertainty in mathematical models. In the fourth chapter, “Review of Plithogenic Directed, Mixed, Bidirected, and Pangene OffGraph”, the authors build upon Plithogenic Graphs to propose extensions such as Plithogenic Directed OffGraph, Plithogenic BiDirected OffGraph, and Plithogenic Mixed OffGraph. These new concepts, including the Plithogenic Pangene OffGraph, are explored in detail, with a focus on their mathematical properties and potential applications in uncertain graph theory. The fifth chapter, “Short Note on Neutrosophic Closure Matroids”, explores the extension of matroid concepts into Neutrosophic and Turiyam Neutrosophic set theories, introducing Neutrosophic closure matroids. This concept integrates uncertainty, indeterminacy, and liberal states into matroid theory, enhancing its applicability in optimization and combinatorial problems. In the sixth chapter, “Some Graph Parameters for Superhypertree-width and Neutrosophic Tree-width”, the authors discuss graph parameters such as Superhypertree-width and Neutrosophic tree-width. These parameters play a crucial role in the study of graph characteristics, particularly in algorithms and real-world applications. The chapter explores the generalization of hypergraphs to SuperHyperGraphs and examines how these concepts extend tree-width parameters within the context of Neutrosophic logic. (shrink)

The present article delves into the extension of Knuth’s fundamental Boolean logic table to accommodate the complexities of indeterminate truth values through the integration of neutrosophic logic (Smarandache & Christianto, 2008). Neutrosophic logic, rooted in Florentin Smarandache’s groundbreaking work on Neutrosophic Logic (cf. Smarandache, 2005, and his other works), introduces an additional truth value, ‘indeterminate,’ enabling a more comprehensive framework to analyze uncertainties inherent in computational systems. By bridging the gap between traditional boolean operations and the indeterminacy (...) present in various real-world scenarios, this extension redefines logic tables, introducing neutrosophic operators that capture nuances beyond the binary realm. Through a thorough exploration of neutrosophic logic's principles and its implications in computational paradigms, this study proposes a novel approach to logic design that accommodates uncertain, imprecise, and incomplete information. This paradigm shift in logic tables not only broadens the spectrum of computing methodologies but also holds promise in fields such as decision-making systems and data analytics. This article amalgamates insights from over twelve key references encompassing seminal works in boolean logic, neutrosophic logic, and their applications in diverse scientific and computational domains, aiming to pave the way for a more robust and adaptable logic framework in computation. (shrink)

Study represents an application of the neutrosophic method, for solving the contradiction between communication and information. In addition, it recourse to an appropriate method of approaching the contradictions: Extensics, as the method and the science of solving the contradictions. The research core is the reality that the scientific research of communication-information relationship has reached a dead end. The bivalent relationship communicationinformation, information-communication has come to be contradictory, and the two concepts to block each other. After the critical examination of (...) conflicting positions expressed by many experts in the field, the extensic and inclusive hypothesis is issued that information is a form of communication. The object of communication is the sending of a message. The message may consist of thoughts, ideas, opinions, feelings, beliefs, facts, information, intelligence or other significational elements. When the message content is primarily informational, communication will become information or intelligence. The arguments of supporting the hypothesis are: a) linguistic (the most important being that there is "communication of information" but not "information of communication"; also, it is clarified and reinforced the over situated referent, that of the communication as a process), b) systemic-procedural (in the communication system is developing an information system; the informing actant is a type of communicator, the information process is a communication process), c) practical (the delimitation eliminates the efforts of disparate and inconsistent understanding of the two concepts), d) epistemological arguments (the possibility of intersubjective thinking of reality is created), linguistic arguments, e) logical and realistic arguments (it is noted the situation that allows to think coherently in a system of concepts - derivative series or integrative groups) f) and arguments from historical experience (the concept of communication has temporal priority, it appears 13 times in Julius Caesar’s writings ). In an axiomatic conclusion, the main arguments are summarized in four axioms: three are based on the pertinent observations of specialists, and the fourth is a relevant application of Florentin Smarandache’s neutrosophic theory. (shrink)

Beyond the predominant paradigm of an essentially rational human cognition, based on the classical binary logic, we want to propose some reflections that are organized around the intuition that the representations we have of the world are weighted with appreciations, for example affective ones. resulting from our integration into a social environment. We see these connotations as essentially ternary in nature, depending on the concepts underlying neutrosophy: either positive, negative or neutral. This form of representation would then influence the very (...) nature of the cognitive process, which in complex real-world situations, has to deal with problems of a combinatorial nature leading to a number of cases too large for our abilities. Forced to proceed by shortcuts on the basis of heuristics, cognition would use these assessments of the representations it manipulates to decide whether partial solutions are attractive for solving the problem or on the contrary are judged negative and are then quickly rejected. There is still the case of a neutral weighting that allows processing to continue. Thus a neutrosophical conception of our representations of the world explains how our cognition functions in its treatment of combinatorial problems in the form of producing processing accelerating heuristics, both in terms of partial solutions selection and processing optimization. (shrink)

La presente investigación tiene como objetivo desarrollar un Método Neutrosófico para evaluar la eficacia de los enjuagues bucales en el control de la placa dental y la gingivitis. Se parte del entendimiento de que el biofilm dental, una acumulación de microorganismos sobre la superficie dentaria, puede provocar la inflamación reversible de los tejidos periodontales, dando lugar a gingivitis. Los enjuagues bucales con propiedades antimicrobianas han demostrado ser efectivos para reducir estos problemas, aunque su uso prolongado puede conducir a la resistencia (...) bacteriana. Para determinar la eficacia de los diferentes tipos de enjuagues bucales, se ha observado que los enjuagues herbales, a base de aceites y de clorhexidina son particularmente eficaces en la reducción de placa y gingivitis, presentando diferencias estadísticamente significativas en sus efectos. A pesar de que cada tipo de enjuague contiene componentes diversos, todos contribuyen al control de la placa y la gingivitis de manera complementaria. Es fundamental destacar que el uso de enjuagues bucales debe ser parte de un enfoque integral que incluya tratamientos profesionales, así como técnicas complementarias como el cepillado adecuado y el uso de hilo dental. Además, este método preventivo se refuerza con la educación sobre la higiene bucal, lo que potencia su eficacia a largo plazo. (shrink)

La presente investigación tiene como objetivo desarrollar un Modelo de Agregación para evaluar los factores que influyen en la correcta aplicación del Proceso de Atención de Enfermería (PAE) en el Hospital Marco Vinicio Iza, Ecuador. El PAE es una herramienta esencial que garantiza un cuidado integral, fundamentado en conocimientos, habilidades y valores adquiridos durante la formación profesional de los enfermeros. Sin embargo, diversos factores limitan su correcta implementación, como la falta de capacitación, el desconocimiento de la metodología del proceso, (...) la ausencia de formatos estandarizados, y la sobrecarga laboral. Para abordar estas limitaciones, se propondrá un modelo neutrosófico que permitirá evaluar y sintetizar los factores identificados a través de un estudio de caso. Este modelo se diseñará para facilitar la toma de decisiones informadas, promoviendo la actualización continua en el uso del PAE y la adopción de estrategias que fomenten su correcta aplicación. Los resultados de este estudio no solo contribuyen a la mejora del PAE en el contexto específico del hospital, sino que también ofrece un marco de referencia potencial para la implementación de prácticas similares en otras instituciones de salud. De esta manera, se espera elevar la calidad del cuidado brindado y garantizar una atención más adecuada y personalizada a los pacientes. (shrink)

Este proyecto de investigación se centra en la evaluación del incremento de la inseguridad ciudadana en la parroquia urbana 10 de noviembre del cantón Ventanas, Ecuador. Se aborda el fenómeno de la delincuencia desde diversas perspectivas, analizando las estrategias y métodos utilizados en los actos delictivos más relevantes que afectan a la comunidad. La investigaci ón destaca la importancia de identificar cuáles de estos actos delincuenciales tienen un mayor impacto en la percepción de la inseguridad. Con ese fin, se llevaron (...) a cabo encuestas cualitativas y cuantitativas, que proporcionaron una visión integral sobre la realidad cotidiana de los habitantes de la parroquia, permitiendo así identificar el aumento real de la inseguridad. La evaluación del incremento de la delincuencia se realiza mediante la implementación del Proceso Analítico Jerárquico Neutrosófico, un método que permite estructurar y priorizar la información recopilada, facilitando la toma de decisiones informadas. Este enfoque metodológico no solo ayuda a categorizar los actos delictivos según su gravedad y frecuencia, sino que también permite comprender mejor la percepción social sobre la seguridad en la parroquia. Esta investigación busca generar un diagnóstico claro de la situación de la inseguridad ciudadana, brindando herramientas que puedan contribuir a la formulación de estrategias y políticas efectivas para mejorar la seguridad en la comunidad. (shrink)

Big Data is a large-sized and complex dataset, which cannot be managed using traditional data processing tools. Mining process of big data is the ability to extract valuable information from these large datasets. Association rule mining is a type of data mining process, which is indented to determine interesting associations between items and to establish a set of association rules whose support is greater than a specific threshold. The classical association rules can only be extracted from binary data where an (...) item exists in a transaction, but it fails to deal effectively with quantitative attributes, through decreasing the quality of generated association rules due to sharp boundary problems. In order to overcome the drawbacks of classical association rule mining, we propose in this research a new neutrosophic association rule algorithm. The algorithm uses a new approach for generating association rules by dealing with membership, indeterminacy, and non-membership functions of items, conducting to an efficient decision-making system by considering all vague association rules. To prove the validity of the method, we compare the fuzzy mining and the neutrosophic mining. The results show that the proposed approach increases the number of generated association rules. (shrink)

In this paper, we prove that Neutrosophic Statistics is more general than Interval Statistics, since it may deal with all types of indeterminacies (with respect to the data, inferential procedures, probability distributions, graphical representations, etc.), it allows the reduction of indeterminacy, and it uses the neutrosophic probability that is more general than imprecise and classical probabilities and has more detailed corresponding probability density functions. While Interval Statistics only deals with indeterminacy that can be represented by intervals. And we (...) respond to the arguments by Woodall et al. [1]. We show that not all indeterminacies (uncertainties) may be represented by intervals. Also, in some cases, we should better use hesitant sets (that have less indeterminacy) instead of intervals. We redirect the authors to the Plithogenic Probability and Plithogenic Statistics which are the most general forms of MultiVariate Probability and Multivariate Statistics respectively (including, of course, the Imprecise Probability and Interval Statistics as subclasses). (shrink)

“Neutrosophics Knowledge” has been created for publications on advanced studies in neutrosophy, neutrosophic set, neutrosophic logic, neutrosophic probability, neutrosophic statistics that started in 1995 and their applications in any field, such as the neutrosophic structures developed in algebra, geometry, topology, etc. The submitted papers should be professional, in good English and Arabic, containing a brief review of a problem and obtained results. Neutrosophy is a new branch of philosophy that studies the origin, nature, and scope (...) of neutralities, as well as their interactions with different ideational spectra. (shrink)

Neutrosophic LA-semigroup is a midway structure between a neutrosophic groupoid and a commutative neutrosophic semigroup. Rings are the old concept in algebraic structures. We combine the neutrosophic LA-semigroup and ring together to form the notion of neutrosophic LA-semigroup ring. Neutrosophic LAsemigroup ring is defined analogously to neutrosophic group ring and neutrosophic semigroup ring.

Neutrosophic Over-/Under-/Off-Set and -Logic were defined for the first time by Smarandache in 1995 and published in 2007. They are totally different from other sets/logics/probabilities. He extended the neutrosophic set respectively to Neutrosophic Overset {when some neutrosophic component is > 1}, Neutrosophic Underset {when some neutrosophic component is < 0}, and to Neutrosophic Offset {when some neutrosophic components are off the interval [0, 1], i.e. some neutrosophic component > 1 and other (...) neutrosophic component < 0}. This is no surprise with respect to the classical fuzzy set/logic, intuitionistic fuzzy set/logic, or classical/imprecise probability, where the values are not allowed outside the interval [0, 1], since our real-world has numerous examples and applications of over-/under-/off-neutrosophic components. (shrink)

Neutrosophic set has been derived from a new branch of philosophy, namely Neutrosophy. Neutrosophic set is capable of dealing with uncertainty, indeterminacy and inconsistent information. Neutrosophic set approaches are suitable to modeling problems with uncertainty, indeterminacy and inconsistent information in which human knowledge is necessary, and human evaluation is needed. Neutrosophic set theory was firstly proposed in 1998 by Florentin Smarandache, who also developed the concept of single valued neutrosophic set, oriented towards real world scientific (...) and engineering applications. Since then, the single valued neutrosophic set theory has been extensively studied in books and monographs, the properties of neutrosophic sets and their applications, by many authors around the world. Also, an international journal - Neutrosophic Sets and Systems started its journey in 2013. Neutrosophic triplet was first defined in 2016 by Florentin Smarandache and Mumtaz Ali and they also introduced the neutrosophic triplet groups in the same year. For every element “x” in a neutrosophic triplet set A, there exist a neutral of “x” and an opposite of “x”. Also, neutral of “x” must be different from the classical neutral element. Therefore, the NT set is different from the classical set. Furthermore, a NT of “x” is showed by . This first volume collects original research and applications from different perspectives covering different areas of neutrosophic studies, such as decision making, Triplet, topology, and some theoretical papers. This volume contains three sections: NEUTROSOPHIC TRIPLET, DECISION MAKING, AND OTHER PAPERS. (shrink)

Neutrosophy has been introduced by Smarandache [7, 8] as a new branch of philosophy. The purpose of this paper is to construct a new set theory called the neutrosophic set. After given the fundamental definitions of neutrosophic set operations, we obtain several properties, and discussed the relationship between neutrosophic sets and others. Finally, we extend the concepts of fuzzy topological space [4], and intuitionistic fuzzy topological space [5, 6] to the case of neutrosophic sets. Possible application (...) to superstrings and space–time are touched upon. (shrink)

In this book we define some new notions of soft neutrosophic algebraic structures over neutrosophic algebraic structures. We define some different soft neutrosophic algebraic structures but the main motivation is two-fold. Firstly the classes of soft neutrosophic group ring and soft neutrosophic semigroup ring defined in this book is basically the generalization of two classes of rings: neutrosophic group rings and neutrosophic semigroup rings. These soft neutrosophic group rings and soft neutrosophic (...) semigroup rings are defined over neutrosophic group rings and neutrosophic semigroup rings respectively. This is basically the collection of parameterized subneutrosophic group ring and subneutrosophic semigroup ring of a neutrosophic group ring and neutrosophic semigroup ring respectively. These soft neutrosophic algebraic structures are even bigger than the corresponding classical algebraic structures. It is interesting to see that it wider the area of research for the researcher of algebraic structures. (shrink)

Misinformation and biased information significantly impact public perception and political decisions, especially on critical issues such as climate change and environmental conservation. This study aims to understand how indeterminacy and contradiction influence public perception and policy formulation by applying neutrosophic theory to model the complexity and multi-dimensionality of ignorance. Using neutrosophic Likert scales, we capture a nuanced spectrum of opinions on the scientific certainty of human impact on climate change. The results are analyzed through a k-means clustering algorithm (...) to identify patterns and segment participants into groups based on their levels of truth, indeterminacy, and falsehood. This approach reveals deeper insights into public perceptions and aids in evaluating their implications for effective communication and policy-making. (shrink)

This study provides an innovative approach to neutrosophic algebraic structures by introducing a new structure called Neutrosophic Soft Cubic T-ideal (NSCTID), which combines T-ideal (TID) and neutrosophic Soft Cubic Sets (NSCSs) within the framework of PS-Algebra. Within the already-existing neutrosophic cubic structures, the addition of soft sets with the characteristics of TID makes this structure more desirable. The theoretical development of the proposed structure includes the application of fundamental ideas as union, intersection, the Cartesian product, and (...) homomorphism. We also introduce the notions of NSCTID-translation and NSCTID-multiplication to further enhance the structure of NSCTID. By applying the idea of translation and multiplication, we offer improved algorithm for neutrosophic cubic sets to deal with different parameters that are satisying the TID’s distinctive characteristics. Through this thorough research, we offer an elementary understanding of NSCTID and its capabilities, providing the way to new algebraic structures. (shrink)

Both neutrosophic sets theory and rough sets theory are emerging as powerful tool for managing uncertainty, indeterminate, incomplete and imprecise information .In this paper we develop an hybrid structure called “ rough neutrosophic sets” and studied their properties.

Soft set theory is a general mathematical tool for dealing with uncertain, fuzzy, not clearly defined objects. In this paper we introduced soft neutrosophic biLA-semigroup,soft neutosophic sub bi-LA-semigroup, soft neutrosophic N -LA-semigroup with the discuission of some of their characteristics. We also introduced a new type of soft neutrophic bi-LAsemigroup, the so called soft strong neutrosophic bi-LAsemigoup which is of pure neutrosophic character. This is also extend to soft neutrosophic strong N-LA-semigroup. We also given some (...) of their properties of this newly born soft structure related to the strong part of neutrosophic theory. (shrink)

Bipolar neutrosophic sets are the extension of neutrosophic sets and are based on the idea of positive and negative preferences of information. Projection measure is a useful apparatus for modelling real life decision making problems. In the paper, we define projection, bidirectional projection and hybrid projection measures between bipolar neutrosophic sets. Three new methods based on the proposed projection measures are developed for solving multi-attribute decision making problems. In the solution process, the ratings of performance values of (...) the alternatives with respect to the attributes are expressed in terms of bipolar neutrosophic values. We calculate projection, bidirectional projection, and hybrid projection measures between each alternative and ideal alternative with bipolar neutrosophic information. All the alternatives are ranked to identify the best alternative. Finally, a numerical example is provided to demonstrate the applicability and effectiveness of the developed methods. Comparison analysis with the existing methods in the literature in bipolar neutrosophic environment is also performed. (shrink)

The notion of neutrosophic cubic set is originated from the hybridization of the concept of neutrosophic set and interval valued neutrosophic set. We define similarity measure for neutrosophic cubic sets and prove some of its basic properties. We present a new multi criteria group decision making method with linguistic variables in neutrosophic cubic set environment. Finally, we present a numerical example to demonstrate the usefulness and applicability of the proposed method.

In this paper we introduce the notion of filter on the neutrosophic crisp set, then we consider a generalization of the filter’s studies. Afterwards, we present the important neutrosophic crisp filters. We also study several relations between different neutrosophic crisp filters and neutrosophic topologies. Possible applications to database systems are touched upon.

In this paper, the cosine similarity measure of neutrosophic refined (multi-) sets is proposed and its properties are studied. The concept of this cosine similarity measure of neutrosophic refined sets is the extension of improved cosine similarity measure of single valued neutrosophic. Finally, using this cosine similarity measure of neutrosophic refined set, the application of medical diagnosis is presented.

The purpose of this paper is to introduce new types of neutrosophic crisp sets with three types 1, 2, 3. After given the fundamental definitions and operations, we obtain several properties, and discussed the relationship between neutrosophic crisp sets and others. Also, we introduce and study the neutrosophic crisp point and neutrosophic crisp relations. Possible applications to database are touched upon.

Neutrosophy considers a proposition, theory, event, concept, or entity, "A" in relation to its opposite, "Anti A" and that which is not A, "Non-A", and that which is neither "A" nor "Anti-A", denoted by "Neut-A". Neutrosophy is the basis of neutrosophic logic, neutrosophic probability, neutrosophic set, and neutrosophic statistics.

In this paper, we introduce the concept of ""neutrosophic crisp neighborhoods system for the neutrosophic crisp point ". Added to, we introduce and study the concept of neutrosophic crisp local function, and construct a new type of neutrosophic crisp topological space via neutrosophic crisp ideals. Possible application to GIS topology rules are touched upon.

This Paper combines interval- valued neutrouphic sets and rough sets. It studies roughness in interval- valued neutrosophic sets and some of its properties. Finally we propose a Hamming distance between lower and upper approximations of interval valued neutrosophic sets.

This paper is devoted to present Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) method for multi-attribute group decision making under rough neutrosophic environment. The concept of rough neutrosophic set is a powerful mathematical tool to deal with uncertainty, indeterminacy and inconsistency. In this paper, a new approach for multi-attribute group decision making problems is proposed by extending the TOPSIS method under rough neutrosophic environment. Rough neutrosophic set is characterized by the upper and lower (...) approximation operators and the pair of neutrosophic sets that are characterized by truth-membership degree, indeterminacy membership degree, and falsity membership degree. In the decision situation, ratings of alternatives with respect to each attribute are characterized by rough neutrosophic sets that reflect the decision makers’ opinion. Rough neutrosophic weighted averaging operator has been used to aggregate the individual decision maker’s opinion into group opinion for rating the importance of attributes and alternatives. Finally, a numerical example has been provided to demonstrate the applicability and effectiveness of the proposed approach. (shrink)

In this paper, we introduce and study the concept of" neutrosophic closed set" and" neutrosophic continuous function". Possible application to GIS topology rules are touched upon.

The Neutrosophic Precalculus and the Neutrosophic Calculus can be developed in many ways, depending on the types of indeterminacy one has and on the method used to deal with such indeterminacy. This article is innovative since the form of neutrosophic binomial factorial theorem was constructed in addition to its refrains. Two other important theorems were proven with their corollaries, and numerical examples as well. As a conjecture, we use ten (indeterminate) forms in neutrosophic calculus taking an (...) important role in limits. To serve article's aim, some important questions had been answered. (shrink)

Neutrosophic set is a new mathematical tool for handling problems involving imprecise, indetermi nacy and inconsistent data. Pseudo-BCI algebra is a kind of non-classical logic algebra in close connection with various non-commutative fuzzy logics. Recently, we applied neutrosophic set theory to pseudo-BCI al gebras. In this paper, we study neutrosophic filters in pseudo-BCI algebras. The concepts of neutrosophic regular filter, neutrosophic closed filter and fuzzy regular filter in pseudo-BCI algebras are introduced, and some basic properties (...) are discussed. Moreover, the relationships among neutrosophic regular filter, fuzzy filters and anti-grouped neutrosophic filters are prese nted, and the results are proved: a neutrosophic filter (fuzzy filter) is a neutrosophic regular filter (fuzzy regular filter), if and only if it is both a neutrosophic closed filter (fuzzy closed filter) and an anti-grouped neutrosophic filter (fuzzy anti-grouped filter). (shrink)

Soft neutrosophic group and soft neutrosophic subgroup are generalized to soft neutrosophic bigroup and soft neutrosophic N-group respectively in this paper. Different kinds of soft neutrosophic bigroup and soft neutrosophic N-group are given. The structural properties and theorems have been discussed with a lot of examples to disclose many aspects of this beautiful man made structure.

In this paper, we propose a new hybrid neutrosophic multi criteria decision making (MCDM) framework that employs a collection of neutrosophic analytical network process (ANP), and order preference by similarity to ideal solution (TOPSIS) under bipolar neutrosophic numbers. The MCDM framework is applied for chief executive officer (CEO) selection in a case study at the Elsewedy Electric Group, Egypt. The proposed approach allows us to assemble individual evaluations of the decision makers and therefore perform accurate personnel selection. (...) The outcomes of the proposed method are compared with those of the related works such as weight sum model (WSM), weight product model (WPM), analytical hierarchy process (AHP), multiobjective optimization based on simple ratio analysis (MOORA) and ANP methods to prove and validate the results. (shrink)

We present in this paper the neutrosophic randomized variables, which are a generalization of the classical random variables obtained from the application of the neutrosophic logic (a new nonclassical logic which was founded by the American philosopher and mathematical Florentin Smarandache, which he introduced as a generalization of fuzzy logic especially the intuitionistic fuzzy logic ) on classical random variables. The neutrosophic random variable is changed because of the randomization, the indeterminacy and the values it takes, which (...) represent the possible results and the possible indeterminacy. Then we classify the neutrosophic randomized variables into two types of discrete and continuous neutrosophic random variables and we define the expected value and variance of the neutrosophic random variable then offer some illustrative examples. -/- . (shrink)

In this paper, we present an expansion of one of the well-known classical inventory management models, which is the static model of inventory management without a deficit and for a single substance, based on the neutrosophic logic, where we provide through this study a basis for dealing with all data, whether specific or undefined in the field of inventory management, as it provides safe environment to manage inventory without running into deficit , and give us an approximate ideal volume (...) of inventory. Since the ideal size is affected by the rate of demand for inventory, we present in this paper a study of the rate of demand for inventory when it is not precisely refined, and we find that the indeterminate values of the demand rate cannot be ignored, because they actually affect determining the ideal size of inventory and calculating its costs, thus affecting on the efficiency of the facility and achieve great profits for it. (shrink)

The objective of this paper is to develop neutrosophic quadruple algebraic hyperstructures. Specifically, we develop neutrosophic quadruple semihypergroups, neutrosophic quadruple canonical hypergroups and neutrosophic quadruple hyperrings and we present elementary properties which characterize them.

The field of neutrosophic set theory and its applications has been rapidly expanding, particularly since the introduction of the journal "Neutrosophic Sets and Systems." New theories, techniques, and algorithms are being developed at a very high rate. One of the most notable trends in neutrosophic theory is its hybridization with other set theories such as rough set theory, bipolar set theory, soft set theory, hesitant fuzzy set theory, and more. Various hybrid structures like rough neutrosophic sets, (...) neutrosophic soft set, single valued neutrosophic hesitant fuzzy sets, among others, have been proposed in a short period. Neutrosophic sets have proven to be crucial tools across a wide array of fields including data mining, decision making, e-learning, engineering, medical diagnosis, social sciences, and beyond. -/- The third volume in the series “New Trends in Neutrosophic Theories and Applications” focuses on theories, methods, and algorithms for decision making, as well as applications involving neutrosophic information. Some topics introduce new sets such as the Pythagorean neutrosophic vague soft set, the triangular fuzzy penta-partitioned neutrosophic set, interval-valued neutrosophic b-open sets, and interval-valued neutrosophic b-closed sets. Other topics present applications in medical diagnosis, non-preemptive neutrosophic priority queues with uneven services (labeled as NM/NM/1), AHP in an interval neutrosophic set environment, MAGDM in a triangular fuzzy neutrosophic number environment, MAGDM in a pentapartitioned neutrosophic environment, the entropy-ARAS strategy in a single-valued neutrosophic number environment, and the MABAC strategy in a rough neutrosophic set environment. (shrink)

Neutrosophic theory and its applications have been expanding in all directions at an astonishing rate especially after of the introduction the journal entitled “Neutrosophic Sets and Systems”. New theories, techniques, algorithms have been rapidly developed. One of the most striking trends in the neutrosophic theory is the hybridization of neutrosophic set with other potential sets such as rough set, bipolar set, soft set, hesitant fuzzy set, etc. The different hybrid structures such as rough neutrosophic set, (...) single valued neutrosophic rough set, bipolar neutrosophic set, single valued neutrosophic hesitant fuzzy set, etc. are proposed in the literature in a short period of time. Neutrosophic set has been an important tool in the application of various areas such as data mining, decision making, e-learning, engineering, medicine, social science, and some more. (shrink)

Science is the basis for managing the affairs of life and human activities, and living without knowledge is a form of wandering and a kind of loss. Using scientific methods helps us understand the foundations of choice, decision-making, and adopting the right solutions when solutions abound and options are numerous. Operational research is considered the best that scientific development has provided because its methods depend on the application of scientific methods in solving complex issues and the optimal use of available (...) resources in various fields, private and governmental work in peace and war, in politics and economics, in planning and implementation, and in various aspects of life. Its basic essence is to use the data provided for the issue under study to build a mathematical model that is the optimal solution. It is the basis on which decision makers rely in managing institutions and companies, and when operations research methods meet with the neutrosophic teacher, we get ideal solutions that take into account all the circumstances and fluctuations that may occur in the work environment over time. One of the most important operations research methods is the linear programming method. Which prompted us to reformulate the linear models, the graphical method, and the simplex method, which are used to obtain the optimal solution for linear models using the concepts of neutrosophic science. In this research, and as a continuation of what we presented previously, we will reformulate the modified simplex algorithm that was presented to address the difficulty that we were facing when applying the direct simplex algorithm. It is the large number of calculations required to be performed in each step of the solution, which requires a lot of time and effort. (shrink)

Impostor syndrome or Impostor phenomenon is a belief that a person thinks their success is due to luck or external factors, not their abilities. This psychological trait is present in certain groups like women. In this paper, we propose a neutrosophic trait measure to represent the psychological concept of the trait-anti trait using refined neutrosophic sets. This study analysed a group of 200 undergraduate students for impostor syndrome, perfectionism, introversion and self-esteem: after the COVID pandemic break in 2021. (...) Data labelling was carried out using these neutrosophic trait measures. Machine learning models like Support Vector Machine(SVM), K-nearest neighbour (K-NN), and random forest were used to model the data; SVM provided the best accuracy of 92.15%. (shrink)

Neutrosophic set has been derived from a new branch of philosophy, namely Neutrosophy. Neutrosophic set is capable of dealing with uncertainty, indeterminacy and inconsistent information. Neutrosophic set approaches are suitable to modeling problems with uncertainty, indeterminacy and inconsistent information in which human knowledge is necessary, and human evaluation is needed. Neutrosophic set theory firstly proposed in 1998 by Florentin Smarandache, who also developed the concept of single valued neutrosophic set, oriented towards real world scientific and (...) engineering applications. Since then, the single valued neutrosophic set theory has been extensively studied in books and monographs introducing neutrosophic sets and its applications, by many authors around the world. (shrink)

A huge amount of data is being generated everyday through different transactions in industries, social networking, communication systems etc. Big data is a term that represents vast volumes of high speed, complex and variable data that require advanced procedures and technologies to enable the capture, storage, management, and analysis of the data. Big data analysis is the capacity of representing useful information from these large datasets. Due to characteristics like volume, veracity, and velocity, big data analysis is becoming one of (...) the most challenging research problems. Semantic analysis is method to better understand the implied or practical meaning of the input dataset. It is mostly applied with ontology to analyze content mainly in web resources. This field of research combines text analysis and Semantic Web technologies. The use semantic knowledge is to aid sentiment analysis of queries like emotion mining, popularity analysis, recommendation systems, user profiling, etc. A new method has been proposed to extract semantic relationships between different data attributes of big data which can be applied to a decision system. (shrink)

One of the most important theories in linear programming is the dualistic theory and its basic idea is that for every linear model has dual linear model, so that solving the original linear model gives a solution to the dual model. Therefore, when we solving the linear programming model, we actually obtain solutions for two linear models. In this research, we present a study of the models. The neutrosophic dual and the binary simplex algorithm, which works to find the (...) optimal solution for the original and dual models at the same time. The importance of this algorithm is evident in that it is relied upon in several operations research topics, such as integer programming algorithms, some nonlinear programming algorithms, and sensitivity analysis in linear programming. (shrink)