Florentin Smarandache

This fourth edition of Representing Higher-Order Networks: A Survey of Graph-Based Frameworks presents a substantially expanded and conceptually enriched treatment of higher-order network models. While retaining the foundational survey character of earlier editions, the present version introduces not only editorial refinements but also significant conceptual and structural developments that deepen and broaden the scope of the work. Beyond corrections and improved exposition, this edition incorporates a range of newly developed frameworks, including advanced tensor-based representations, recursive and iterated graph constructions, and extended superhypergraph formalisms. These additions reflect a shift from a primarily descriptive catalog of models toward a more systematic and integrative perspective, emphasizing relationships, hierarchies, and interoperability among diverse higher-order structures. The book further strengthens its unifying framework by organizing higher-order graph models into coherent families—combinatorial, topological, tensorial, and semantic—while highlighting their underlying principles and modeling capabilities. In doing so, it advances a more comprehensive theoretical landscape that supports both comparative analysis and practical application across disciplines such as network science, data modeling, and knowledge representation. Overall, this edition represents a qualitative evolution of the work, transitioning from a survey of existing approaches to a more cohesive and forward-looking synthesis of higher-order network theory.

This book presents a systematic development of Refined Quadruple Neutrosophic (T, I, N, F) Logic, extending classical neutrosophic logic by distinguishing Neutrality (N) from Pure Indeterminacy (I) and by refining each of the four components—Truth, Indeterminacy, Neutrality, and Falsehood—into semantically labeled subcomponents. The resulting n-valued structure provides a multidimensional representation capable of distinguishing different sources, contexts, time intervals, criteria, or manifestations of truth, uncertainty, neutrality, and falsity. The mathematical framework includes quadruple neutrosophic numbers, refined n-valued representations, component-wise algebraic operations, normalization procedures, and a hierarchical total-ordering method based on Score, Accuracy, Extended Certainty, and Certainty functions. A domain-coverage analysis evaluates the state of refined neutrosophic applications across 51 fields of knowledge, distinguishing documented implementations from basic neutrosophic applications and theoretically promising but still unexplored areas. The interdisciplinary scope encompasses the social sciences and humanities, physical and natural sciences, formal sciences and engineering, decision sciences, economics and management, medicine, agriculture, education, and jurisprudence. Eight extended case studies illustrate the framework in evolutionary biology and genetics, infrastructure decision-making, medical diagnostics, autonomous navigation, algorithmic trading, judicial evidence evaluation, inheritance-pattern modeling, and oceanography. Particular emphasis is placed on the conceptual distinction between indeterminacy—representing missing, contradictory, insufficient, or unresolved information—and neutrality, representing a genuine non-aligned or intermediate state. By combining mathematical foundations, ordering and algebraic tools, cross-domain applications, performance considerations, a research roadmap for 2026–2030, and a Python reference implementation, the volume provides both a theoretical framework and a computational basis for further research in refined multidimensional uncertainty modeling.

This book develops a unified framework for refined and Over/Under/Off extensions of probability, measure, and statistics by systematically combining two generating operators: Refinement, which decomposes a component into finer, individually trackable subcomponents while preserving reduction consistency, and Over/Under/Off relaxation, which extends conventional bounded values beyond the standard interval ([0,1]). Their application produces nine interrelated families based on Classical Probability, Imprecise Probability, Neutrosophic Probability, Plithogenic Probability, Measure, and the corresponding statistical frameworks. For each family, formal definitions are accompanied by worked applications and analyses of fundamental probabilistic relations, including complementarity, independence and dependence, mutual exclusivity, conditional probability, De Morgan’s rules, and Bayes’ theorem. Particular attention is given to Refined and Over/Under/Off Neutrosophic and Plithogenic structures, including the quadruple ((T,I,N,F)) extension and contradiction-degree-based representations. The framework is situated within the broader landscape of uncertainty science through connections with imprecise probability, Dempster–Shafer and Dezert–Smarandache theories, Bayesian inference, multi-criteria decision-making, actuarial risk, machine-learning uncertainty quantification, and quantum probability. Numerical case studies and applications in medicine, engineering, finance, quality control, supply chains, risk analysis, and other domains demonstrate how refinement can localize heterogeneous sources of uncertainty while Over/Under/Off representations accommodate values expressing super-certainty, negative propensity, contradictory evidence, or anomalous contributions that conventional bounded models cannot represent directly. The resulting architecture provides a systematic vocabulary and mathematical framework for representing complex, granular, inconsistent, and off-interval uncertainty while explicitly identifying its limitations and directions for further research.

Neutrosophic Functional Analysis: Foundations, Fermatean n-Norms, and Approximation Theory presents a unified mathematical framework for the study of functional analysis under uncertainty, indeterminacy, and inconsistency by integrating neutrosophic theory with modern normed-space analysis. The book develops the fundamental concepts of neutrosophic intuitionistic algebra fuzzy normed spaces and Fermatean neutrosophic n-normed linear spaces, establishing their algebraic, topological, and geometric foundations. It systematically investigates convergence, Cauchy sequences, completeness, boundedness, compactness, neighbourhood structures, norm equivalence, and finite-dimensional properties, culminating in Heine–Borel type characterizations and compactness criteria. The theory is further extended through Banach-type fixed point theorems, contraction mappings, and applications to linear and nonlinear operator equations. A comprehensive treatment of weighted neutrosophic best approximation is also developed, including existence, Pareto efficiency, and uniqueness results under appropriate convexity assumptions. By combining classical functional analysis with neutrosophic logic and Fermatean uncertainty models, this volume provides a rigorous foundation for future research in operator theory, optimization, nonlinear analysis, approximation theory, and intelligent mathematical systems operating in uncertain environments.

Între Anzi și Viitor. Universități, oameni și idei. Fotojurnal instantaneu în Chile și Peru este un jurnal de călătorie care îmbină experiența academică cu reflecția personală și descoperirea culturală. Printr-o succesiune de impresii, întâlniri și fotografii, autorul străbate Chile și Peru, vizitând universități, participând la conferințe internaționale dedicate inteligenței artificiale și neutrosofiei, explorând orașe emblematice precum Santiago, Valparaíso și Lima și admirând unele dintre cele mai spectaculoase peisaje ale Anzilor. Dincolo de relatarea unei călătorii, volumul propune o reflecție asupra adevăratului scop al cercetării științifice, asupra responsabilității etice a cunoașterii și asupra rolului educației în construirea viitorului. Dimensiunea academică este completată de experiențele umane ale întâlnirilor cu profesori, studenți, prieteni și membri ai familiei, demonstrând că știința își găsește sensul deplin atunci când contribuie la binele generațiilor viitoare. Scris într-un stil accesibil și meditativ și însoțit de un bogat material fotografic, acest fotojurnal invită cititorul să descopere nu doar locurile vizitate, ci și oamenii, ideile și emoțiile care transformă o călătorie într-o experiență de creștere intelectuală și umană.

This book presents a comprehensive survey and unified treatment of HyperStructures, SuperHyperStructures, Hv-Structures, and SHv-Structures, emphasizing their role as general frameworks for modeling multivalued, weakly constrained, and hierarchically organized mathematical systems. Beginning with the fundamental concepts of hyperoperations and iterated powersets, the volume systematically develops Hv- and SHv-theories across a broad spectrum of algebraic structures, including groups, fields, rings, vector spaces, matrices, semigroups, and related algebraic systems. It further extends these frameworks to uncertainty-based environments through fuzzy, neutrosophic, plithogenic, and general uncertain Hv- and SHv-structures, providing unified definitions and hierarchical formulations that clarify their mutual relationships. The book also introduces generalized concepts such as ordered Hv-structures, meta-Hv-structures, multi-HyperStructures, multi-SHv-Structures, and chemically motivated Hv-models, demonstrating how weak identities and hierarchical hyperoperations naturally extend classical algebraic theories. By integrating established results with new conceptual developments, the volume offers a coherent mathematical foundation for studying hyperstructural systems and their applications in algebra, uncertainty modeling, artificial intelligence, complex systems, and the mathematical sciences. It serves both as a reference for researchers and graduate students and as a basis for future theoretical, computational, and interdisciplinary investigations.

This book develops several directions in generalized functional analysis within fuzzy, intuitionistic fuzzy, Fermatean intuitionistic fuzzy, and neutrosophic normed spaces. It investigates fundamental analytical concepts including convergence, Cauchy sequences, completeness, compactness, total boundedness, finite-dimensionality, equivalence of generalized norms, and fixed point principles. The first part studies compactness and finite-dimensional structures in intuitionistic algebra fuzzy normed spaces, establishing generalized versions of classical results such as completeness of finite-dimensional subspaces, equivalence of norms, and the Heine–Borel theorem. The second part introduces Fermatean intuitionistic fuzzy (n)-normed linear spaces and develops their convergence, completeness, and fixed point theory. The third part examines Tribonacci ideal and ideal-star convergence in non-Archimedean neutrosophic normed spaces, combining matrix transformations, ideal convergence, and neutrosophic analytical methods. Together, these contributions provide a unified framework for extending classical functional analysis to mathematical environments characterized by uncertainty, hesitation, and indeterminacy, with potential applications in applied mathematics, artificial intelligence, optimization, decision-making, and information sciences.

This book presents a comprehensive treatment of intersection graph theory within modern uncertainty-aware mathematical frameworks. Beginning with the foundations of classical graph theory and the major families of intersection graphs—including interval, disk, circle, circular-arc, ray, string, grid, and related geometric graph classes—it develops a unified perspective that integrates fuzzy, intuitionistic fuzzy, neutrosophic, plithogenic, and general uncertain graph models into a common theoretical framework. The volume surveys existing literature while introducing several original contributions, including a general definition of uncertain graphs based on arbitrary uncertainty models, a unified theory of uncertain intersection graphs, and corresponding geometric subclasses such as uncertain unit square graphs, uncertain circle graphs, uncertain ray graphs, uncertain string graphs, and uncertain grid intersection graphs. The book further proposes the novel concepts of Multi-Intersection Graphs, Quasi-Intersection Graphs, Metric Quasi-Intersection Graphs, and Multi-Quasi-Intersection Graphs, together with their fundamental properties and well-definedness results. By combining a broad survey with new theoretical developments, the work establishes a coherent foundation for modeling complex geometric and relational structures under uncertainty. It is intended as a reference for researchers and graduate students working in graph theory, uncertainty modeling, fuzzy systems, neutrosophic mathematics, artificial intelligence, computational geometry, and related interdisciplinary fields, while also providing a basis for future algorithmic, computational, and applied investigations.

Entre los Andes y el Futuro. Universidades, personas e ideas. Fotodiario instantáneo de Chile y Perú es un diario de viaje que entrelaza la experiencia académica con la reflexión personal y el descubrimiento cultural. A través de una serie de impresiones, encuentros y fotografías, el autor recorre Chile y Perú, visitando universidades, participando en conferencias internacionales sobre inteligencia artificial y neutrosofía, explorando ciudades emblemáticas como Santiago, Valparaíso y Lima, y contemplando algunos de los paisajes más representativos de los Andes. Más allá de la crónica de viaje, el libro propone una reflexión sobre el verdadero propósito de la investigación científica, la responsabilidad ética del conocimiento y el papel de la educación en la construcción del futuro. La dimensión académica se complementa con la experiencia humana del encuentro con colegas, estudiantes, amigos y familiares, mostrando que el conocimiento adquiere su sentido más profundo cuando contribuye al bienestar de las generaciones futuras. Con un estilo cercano, meditativo y acompañado por abundante material fotográfico, este fotodiario invita al lector a descubrir no solo los lugares visitados, sino también las personas, las ideas y las emociones que transforman un viaje en una experiencia de crecimiento intelectual y humano.

This book provides a systematic conceptual and numerical comparison between the α-Discounting Method for Multi-Criteria Decision Making (α-D MCDM) and approximately forty established MCDM techniques. Unlike conventional weighting approaches that require coherent or acceptably consistent judgments, α-Discounting transforms systems of potentially contradictory preference equations through a scalar discounting parameter, α. This parameter measures the degree of inconsistency, while the corresponding normalized solution generates a usable vector of criteria weights. The study positions α-Discounting primarily as a weight-derivation method and examines its relationships with AHP, ANP, entropy weighting, CRITIC, BWM, FUCOM, TOPSIS, VIKOR, PROMETHEE, ELECTRE, and numerous other weighting, outranking, utility-based, optimization, fuzzy, and uncertainty-oriented methods. The first part presents theoretical comparisons focusing on preference elicitation, inconsistency management, computational structure, and practical applicability. The second part applies the methods to a shared decision problem through computer-verified numerical examples, making it possible to identify areas of agreement and divergence among their rankings. Most ranking methods produce the same ordering when supplied with α-Discounting weights, while the exceptions reveal the effects of different compromise principles, distance measures, weighting philosophies, and treatments of individual regret. The volume demonstrates that α-Discounting complements rather than replaces established ranking procedures by providing a flexible mechanism for deriving weights from inconsistent, interval-valued, n-wise, or nonlinear expert judgments.

This book provides a comprehensive and updated survey of major mathematical frameworks developed for representing vagueness, incomplete information, partial truth, indeterminacy, inconsistency, and other forms of uncertainty. Beginning with fuzzy sets, intuitionistic fuzzy sets, neutrosophic sets, and plithogenic sets, it adopts the Uncertain Set framework as a unifying perspective in which uncertainty models are characterized through model-dependent degree-domains and their associated structural constraints. The study systematically examines the relationships, reductions, and generalizations connecting established uncertainty theories and develops an extensive taxonomy of Uncertain Set extensions, including multi-component, multidimensional, refined, interval-valued, linguistic, soft, rough, hierarchical, probabilistic, dynamic, credal, Markov, weighted, and uncertain-number constructions. By identifying common mathematical patterns across theories that have often developed independently, the book provides a coherent architecture for understanding their similarities and differences. The resulting framework accommodates fuzzy, intuitionistic fuzzy, neutrosophic, picture fuzzy, spherical fuzzy, plithogenic, and numerous other models through appropriate choices of degree-domain and model-specific constraints. In addition to consolidating existing theory, the survey seeks to reduce fragmentation within uncertainty mathematics and establish a flexible foundation for constructing, comparing, and extending uncertainty models and their applications across mathematics, computer science, engineering, decision science, machine learning, and related disciplines.

This book provides a comprehensive and updated survey of major mathematical frameworks developed for representing vagueness, incomplete information, partial truth, indeterminacy, inconsistency, and other forms of uncertainty. Beginning with fuzzy sets, intuitionistic fuzzy sets, neutrosophic sets, and plithogenic sets, it adopts the Uncertain Set framework as a unifying perspective in which uncertainty models are characterized through model-dependent degree-domains and their associated structural constraints. The study systematically examines the relationships, reductions, and generalizations connecting established uncertainty theories and develops an extensive taxonomy of Uncertain Set extensions, including multi-component, multidimensional, refined, interval-valued, linguistic, soft, rough, hierarchical, probabilistic, dynamic, credal, Markov, weighted, and uncertain-number constructions. By identifying common mathematical patterns across theories that have often developed independently, the book provides a coherent architecture for understanding their similarities and differences. The resulting framework accommodates fuzzy, intuitionistic fuzzy, neutrosophic, picture fuzzy, spherical fuzzy, plithogenic, and numerous other models through appropriate choices of degree-domain and model-specific constraints. In addition to consolidating existing theory, the survey seeks to reduce fragmentation within uncertainty mathematics and establish a flexible foundation for constructing, comparing, and extending uncertainty models and their applications across mathematics, computer science, engineering, decision science, machine learning, and related disciplines.

Graph theory provides a fundamental mathematical framework for representing and analyzing networks through vertices, edges, and their structural relationships. However, classical graph models are often insufficient for systems characterized by uncertainty, vagueness, incomplete information, or parameter-dependent relationships. This book presents a comprehensive and systematic survey of two major approaches for extending graph theory to such environments: Soft Graphs and Rough Graphs. Soft Graphs associate parameters with subgraphs of an ambient graph, allowing network structures to vary according to attributes, criteria, or contextual information, whereas Rough Graphs employ lower and upper approximations to distinguish structural components that are certainly present, possibly present, or situated within a boundary region. Beginning with the foundations of soft sets, rough sets, uncertain sets, graphs, HyperGraphs, and SuperHyperGraphs, the book develops an extensive taxonomy of Soft Graph and Rough Graph models. It examines numerous uncertainty-based, structural, and parameter-based extensions, including HyperSoft, SuperHyperSoft, fuzzy, intuitionistic fuzzy, neutrosophic, plithogenic, bipolar, multipolar, interval-valued, Pythagorean fuzzy, and q-rung orthopair formulations, as well as Soft HyperGraphs, Soft SuperHyperGraphs, Rough HyperGraphs, and Rough SuperHyperGraphs. The treatment also encompasses important special classes and structural properties, including soft and rough trees, regular and complete graphs, connectivity, Hamiltonian structures, paths, cycles, and covering-based models. By organizing these diverse frameworks within a unified graph-theoretic perspective, the book clarifies their mathematical foundations, structural relationships, and modeling capabilities, providing a reference for researchers working on uncertain networks, generalized graph structures, decision modeling, and related areas of applied mathematics.

This edited volume presents recent theoretical advances, methodological innovations, and interdisciplinary applications in the rapidly evolving field of neutrosophic theories and applications, with particular emphasis on uncertainty modelling, indeterminacy representation, and multi-criteria decision-making (MCDM). Building upon the foundational principles of neutrosophic set theory (NST) introduced by Florentin Smarandache, the volume explores the integration of neutrosophic approaches with Pythagorean fuzzy sets, Fermatean fuzzy sets, rough set theory, pentapartitioned neutrosophic structures, and continuum uncertainty frameworks to address complex real-world problems characterized by vagueness, incompleteness, inconsistency, and contradictory information. The book encompasses diverse theoretical contributions, including neutrosophic number–based machine learning, learning analytics, information geometry, graph theory, topological structures, self-reference stability, algebraic systems, and Gödel-type incompleteness extensions, alongside novel aggregation operators and advanced decision-making methodologies. Several chapters develop innovative MCDM and MADM frameworks in rough neutrosophic, interval rough neutrosophic, trapezoidal neutrosophic, Fermatean neutrosophic, and pentapartitioned neutrosophic environments, demonstrating applications in engineering, biomedical sciences, environmental sustainability, healthcare, artificial intelligence, education, manufacturing systems, and sustainable waste management. By integrating mathematical rigor with practical applicability, this volume advances the theoretical foundations of neutrosophic science while offering robust analytical tools for reasoning under uncertainty across multidisciplinary domains. The book serves as a valuable resource for researchers, academicians, practitioners, and graduate students working in fuzzy systems, artificial intelligence, computational intelligence, decision sciences, mathematical modelling, and uncertainty-aware systems.

This volume presents an Assamese translation of selected one-line poems by Florentin Smarandache, a Romanian-American writer, mathematician, and professor of science and mathematics at the University of New Mexico. The collection introduces Assamese readers to the literary form of one-line poetry (eshoriya kobita), a concise poetic genre that has gained recognition in Western literature but remains relatively unexplored in Assamese literary traditions. The poems, translated from English into Assamese by Dr. Ranjit Bhattacharyya, preserve the philosophical depth, irony, existential reflection, emotional resonance, and minimalist aesthetics characteristic of Smarandache’s poetic style. Through linguistic adaptation and cultural mediation, this work creates a bridge between global experimental poetry and Assamese literary expression, expanding the scope of contemporary Assamese poetic discourse. The translation also contributes to intercultural literary exchange by making internationally recognized minimalist poetry accessible to Assamese-speaking readers.

Este libro desarrolla un análisis comparativo integral de la plitogenia como un marco metafilosófico capaz de interpretar las relaciones entre diversas tradiciones filosóficas. La plitogenia, formulada originalmente en el ámbito de la lógica matemática, describe la formación y evolución de entidades mediante la interacción de múltiples elementos que pueden ser compatibles, contradictorios o neutrales. Al extender este concepto formal al campo de la filosofía, la obra examina cómo los sistemas filosóficos pueden entenderse no como doctrinas aisladas o mutuamente excluyentes, sino como estructuras conceptuales en interacción dentro de un campo más amplio de pluralidad intelectual. El estudio explora sistemáticamente las conexiones entre la plitogenia y los principales movimientos filosóficos, incluyendo el empirismo, el racionalismo, el idealismo, el realismo, la fenomenología y el epifenomenalismo, así como teorías éticas como la deontología, el utilitarismo y el hedonismo. Asimismo, aborda el posmodernismo, el pragmatismo y dominios contemporáneos como la ética digital, la bioética, la filosofía ambiental y la teoría de la justicia. Una contribución distintiva de esta edición ampliada reside en la integración de tradiciones filosóficas latinoamericanas e indígenas, incluyendo conceptos como yanantin, ch’ixi, pachakuti, ayni, Sumak Kawsay (Buen Vivir) y teotl, destacando sus afinidades estructurales con el razonamiento plitogénico. Mediante el empleo de una metodología comparativa consistente —centrada en similitudes, diferencias y conexiones conceptuales— el libro demuestra que la pluralidad filosófica puede modelarse como un sistema dinámico de perspectivas en interacción. La plitogenia se presenta así no como un reemplazo de las filosofías existentes, sino como una lente analítica unificadora que posibilita el diálogo entre tradiciones preservando al mismo tiempo sus diferencias. La obra contribuye a la filosofía contemporánea al ofrecer un marco adecuado para abordar la complejidad, la interdisciplinariedad y la coexistencia de epistemologías diversas tanto en contextos teóricos como aplicados.

The Honest Classroom explores how education must evolve in response to the widespread adoption of artificial intelligence and the emergence of highly fluent but epistemically unreliable AI systems. The book argues that traditional educational approaches are insufficient for preparing students to critically evaluate AI-generated content because they lack a formal framework for representing uncertainty, contradiction, and ignorance. Drawing on neutrosophic logic, the authors introduce the Truth–Indeterminacy–Falsity (T, I, F) framework as a practical and theoretical tool for navigating complex knowledge environments where confidence does not guarantee accuracy. The work examines the cognitive biases that influence human reasoning, demonstrates how AI systems can amplify these biases, and proposes pedagogical strategies that cultivate epistemic responsibility, critical thinking, and intellectual honesty. Through educational models, assessment instruments, AI tutoring frameworks, and classroom applications, the book develops the concept of the “Honest Classroom,” an educational environment that treats indeterminacy as a legitimate and valuable component of learning. By integrating neutrosophic logic into teaching, assessment, and AI-assisted learning, the book provides educators, researchers, and policymakers with a comprehensive framework for fostering informed judgment, responsible AI use, and resilient epistemic practices in the age of confident machines.

This book presents a comprehensive study of topological indices in the framework of single-valued neutrosophic graphs, extending classical graph-theoretical concepts to environments characterized by uncertainty, indeterminacy, and inconsistency. Building upon the foundations of graph theory and neutrosophic set theory, the authors define and analyze several important neutrosophic topological indices, including the connectivity index, Wiener index, and Sombor index. The work develops formal definitions, computational methods, theoretical properties, bounds, and proofs related to these indices within neutrosophic graph structures. Special attention is given to connected neutrosophic graphs, neutrosophic trees, cycles, stars, and super hyper power graphs. Through detailed examples and mathematical demonstrations, the book illustrates how neutrosophic representations provide a more realistic modeling framework for complex systems affected by ambiguous or conflicting information. The volume also explores applications of neutrosophic graph indices in network analysis, decision-making systems, and complex relational structures. Designed for researchers and graduate students in mathematics, computer science, engineering, and data science, this work contributes to the advancement of neutrosophic graph theory by establishing a rigorous foundation for future research on uncertainty-aware topological analysis and generalized graph invariants.

This book introduces Type-k Neutrosophic Sets, a recursive generalization of classical neutrosophic sets designed to represent nested and higher-order uncertainty. Unlike traditional neutrosophic frameworks, where truth, indeterminacy, and falsity are represented as independent scalar values, the Type-k model recursively assigns a complete neutrosophic triplet to each component, producing increasingly expressive structures capable of modeling meta-uncertainty and epistemic self-assessment. The work establishes the formal mathematical foundations of Type-k neutrosophic representations, proves their strict expressive hierarchy, and develops recursive aggregation operators that preserve compatibility with classical single-valued neutrosophic systems. The book further extends neutrosophic theory through tensorial frameworks, including single-valued neutrosophic tensors and plithogenic tensors, enabling advanced multi-criteria decision-making and multilinear analysis. A neutrosophic paraconsistent logic is introduced to address contradiction and indeterminacy within formal reasoning systems. The theoretical developments are applied to the epistemic auditing of large language models, where extensive empirical studies demonstrate the limitations of conventional probabilistic approaches and reveal the necessity of extended neutrosophic representations for capturing hallucinations, paradoxes, contradictions, and structural uncertainty. The volume concludes with open problems and future research directions connecting neutrosophic mathematics, artificial intelligence auditing, decision sciences, and paraconsistent reasoning.

This research monograph introduces the theory of Neutrosophic Super Hyper Sets (NSHS), a novel mathematical framework that integrates neutrosophic set theory with the hierarchical structure of iterated power sets to model uncertainty in multi-level and nested systems. Building upon the evolution from classical, fuzzy, intuitionistic fuzzy, and neutrosophic sets, the authors address situations in which information is associated not only with individual elements but also with collections, collections of collections, and higher-order hierarchical structures. The work develops the foundational theory of NSHS, including definitions, set-theoretic operations, algebraic properties, lattice structures, topology, relations, mappings, similarity measures, entropy, and distance metrics. Advanced extensions such as interval-valued, bipolar, plithogenic, and multi-polar neutrosophic super hyper sets are also formulated. The monograph establishes rigorous mathematical results concerning partial orders, distributive lattices, aggregation operators, score and accuracy functions, and topological structures, while demonstrating practical applications in decision-making, medical diagnosis, environmental risk assessment, data mining, machine learning, knowledge graphs, network science, and complex systems analysis. By combining hierarchical modeling with the expressive representation of truth, indeterminacy, and falsity, the NSHS framework provides a powerful tool for analyzing uncertainty in increasingly interconnected and multi-layered scientific, technological, and social systems.

This volume proposes a radical reconceptualization of reality, knowledge, and existence through the development of Meta-Garde theory, a structural framework designed to articulate systems governed not by binary exclusion, but by the coexistence of multiple, non-exclusive states. Moving beyond its origins in avant-garde aesthetics and paradoxism, Meta-Garde is extended here into the domains of epistemology, ontology, and decolonial philosophy to address the limitations of universalist, binary logic in capturing the complexity of Indigenous cosmologies and Latin American thought. The central thesis posits that many Indigenous worldviews—from Mesoamerican triadic ontologies and Andean yanantin (complementary duality) to Amazonian distributed agency and perspectivism—already operate through pluriversal structures where contradiction, indeterminacy, and relational multiplicity are constitutive rather than anomalous. To formally articulate these coexistential dynamics, the book integrates neutrosophic logic (the triadic system of Truth, Indeterminacy, and Falsity) and n-alectics (a logic of coexistence beyond dialectical synthesis) to model realities where affirmation and negation persist simultaneously without requiring resolution. Through a five-part interdisciplinary inquiry, the text traverses the foundations of pluriversal thought, analyzes specific Indigenous cosmological structures as proto-Meta-Garde systems, and explores the role of language as a site of epistemic resistance against colonial erasure. It further develops a Meta-Garde Complexity Model to address recursive contradictions and nonlinear epistemologies, culminating in the formulation of the Pluriversal Meta-Garde Space—a formal field where heterogeneous ontologies interact without collapsing into a singular universal order. The book concludes by outlining the ethical, political, and anthropological implications of this framework, proposing new models for environmental ethics (such as the Meta-Garde Sustainability Index) and a Meta-Garde Anthropology that studies contradiction as a structural principle of social life. By bridging formal logic, decolonial theory, and Indigenous epistemologies, Meta-Garde and the Pluriversal Condition offers a transformative philosophical orientation capable of engaging the dynamic, unresolved, and profoundly interconnected realities of the twenty-first century.

This book develops a formal, quantitative theory of anti-art by integrating multidimensional modeling, neutrosophic logic, and geometric analysis. Moving beyond traditional binary aesthetics, it reconceptualizes art as a structured configuration space in which affirmation, negation, and indeterminacy coexist. The core contribution is the construction of the Meta-Garde framework, which includes: (1) a high-dimensional state space representing artworks as vectors of structural variables, (2) the Meta-Garde Intensity Index (MGII) as a composite functional measuring anti-art intensity, and (3) a geometric and taxonomic system enabling clustering, comparison, and classification of artistic movements. Through this framework, diverse practices—ranging from Dada and Fluxus to Conceptual Art, Glitch Art, and AI Art—are mapped as distinct regions within a unified analytical space. Anti-art is thus shown to be multidimensional, structured, and measurable, rather than merely disruptive or anomalous. A central result of the study is the identification of Outer Art as a high-coherence meta-garde configuration, characterized by simultaneous elevation across key variables such as definitional destabilization, paradox density, reflexivity, and logical explicitness. While not maximizing intensity in any single dimension, Outer Art uniquely achieves multi-variable structural coherence, positioning it as a calibration case and representative of a distinct mathematical class within the state space. The framework culminates in a general theory of neutrosophic aesthetics, in which artworks are modeled through triadic logic ⟨T, I, F⟩ and embedded within a multidimensional geometric structure. This synthesis establishes the foundations for a formal science of art, capable of analyzing aesthetic complexity, contradiction, and evolution across both historical and emerging artistic practices.

This book develops a comprehensive theoretical framework for understanding pArAdOXisM as a central formation within contemporary avant-garde literature and introduces meta-garde as a broader structural condition of artistic and linguistic expression. Challenging traditional binary aesthetics—such as art versus non-art or meaning versus meaninglessness—the work argues that literary and artistic objects must be analyzed through the simultaneous coexistence of affirmation, negation, and indeterminacy. Through historical reconstruction, conceptual analysis, and comparative mapping, the study demonstrates that pArAdOXisM is not merely a marginal or derivative movement but a system in which contradiction functions as a generative principle rather than a stylistic device. The book formalizes paradoxist poetics by identifying key operations such as semantic inversion, anti-language, non-poems, and structural contradiction, and situates these within a multidimensional analytical model derived from neutrosophic logic. By comparing pArAdOXisM with other avant-garde movements—including Dada, Surrealism, Lettrism, Oulipo, and conceptual writing—the work establishes its distinctiveness as a meta-avant-garde system characterized by high paradox density, reflexivity, and definitional instability. Ultimately, the book proposes meta-garde as a generalizable framework extending beyond aesthetics into a broader philosophical logic of coexistence, where contradiction and indeterminacy are understood as fundamental structural conditions rather than anomalies.

This volume develops meta-garde as a structural condition of art in which aesthetic states are no longer governed by binary distinctions, but coexist in multivalent configurations. Moving beyond the traditional oppositions that have shaped aesthetic theory—such as art and non-art, meaning and non-meaning—the study argues that contemporary artistic practices reveal a condition in which affirmation, negation, and indeterminacy operate simultaneously within the same object. The book proceeds by first examining the historical development of aesthetics, from classical frameworks grounded in binary logic to the fragmentation associated with postmodern theory. While postmodernism recognizes the instability of aesthetic categories, it remains largely descriptive. In response, this work introduces a formal articulation of coexistence through a triadic structure that allows contradictory states to be represented without reduction. This framework is then applied to a range of aesthetic phenomena, including anti-art, paradoxism, conceptual practices, and hybrid and processual forms. Particular attention is given to the role of indeterminacy, reflexivity, and contradiction as constitutive features rather than anomalies. Through the development of a minimal axiomatic system and the concept of an aesthetic state space, the study provides tools for analyzing objects that resist classification within traditional models. Finally, the volume situates meta-garde within the history of aesthetic thought and argues for its extension beyond aesthetics. Rather than a movement or style, meta-garde is proposed as a general condition that may also be observed in language, culture, and non-binary philosophical frameworks. In this sense, the book advances a shift from exclusion-based models of understanding toward a logic of coexistence, offering a foundation for further theoretical and interdisciplinary development.

This book develops a comprehensive theoretical framework for the study of Hierarchical SuperHyperGraphs and Recursive SuperHyperGraphs, extending classical graph and hypergraph theory toward more expressive higher-order and multi-level relational structures. SuperHyperGraphs generalize hypergraphs through iterated powerset constructions, allowing vertices to be nested, set-valued, and hierarchically organized. The work introduces recursive SuperHyperGraphs by integrating n-level supervertices with recursive superhyperedges of bounded depth, where edges may contain both supervertices and lower-level recursive edges under typed incidence and well-foundedness constraints. Furthermore, the book formulates hierarchical SuperHyperGraphs as mixed-level incidence systems governed by coherence conditions to ensure structural consistency across powerset levels. Several important extensions are explored, including fuzzy, neutrosophic, uncertain, soft, rough, weighted, directed, and bidirected variants. Theoretical comparisons are also provided with related graph-theoretic models such as Meta-Graphs, Iterated Meta-Graphs, Filtrated Graphs, Iterated MultiGraphs, and Neural Graphs. Emphasizing definitions, formal properties, examples, and structural comparisons, the book establishes a foundational basis for future computational developments and interdisciplinary applications in complex systems, artificial intelligence, network science, and higher-order modeling.

This book develops a systematic and unified framework for the study of planar graph theory under uncertainty, integrating classical graph-theoretic concepts with fuzzy, intuitionistic fuzzy, neutrosophic, plithogenic, and uncertain graph models. The work revisits foundational graph classes—including planar, outerplanar, apex, quasi-planar, and related graph structures—and extends them into uncertainty-aware environments where vertices and edges may carry graded, indeterminate, or attribute-dependent information. Special attention is given to fuzzy planar graphs, intuitionistic fuzzy planar graphs, neutrosophic planar graphs, plithogenic planar graphs, and uncertain planar graph formulations, emphasizing both structural consistency and model-independent interpretations of planarity. In addition, the book explores related extensions to hypergraphs and SuperHyperGraphs, thereby broadening the conceptual landscape of planar structures in higher-order settings. By combining rigorous definitions, examples, comparative perspectives, and theoretical analysis, the book establishes a coherent foundation for future investigations in uncertain network modeling, graph theory, artificial intelligence, graph neural networks, computer vision, VLSI design, and complex systems analysis.

Artificial Intelligence (AI) is rapidly transforming teacher education through intelligent tutoring systems, automated assessment, learning analytics, virtual teaching simulations, and AI-assisted curriculum design. While these technologies offer substantial benefits in terms of efficiency, personalization, scalability, and data-informed decision-making, they also introduce significant challenges related to ethical responsibility, professional autonomy, algorithmic bias, privacy, and pedagogical integrity. Conventional evaluative approaches often rely on binary or probabilistic reasoning, limiting their ability to capture the complexity and uncertainty inherent in AI-supported educational environments. This book introduces a neutrosophic framework for analyzing AI-driven teacher education, grounded in the concepts of truth, indeterminacy, and falsity. Through this multidimensional perspective, AI technologies are examined as educational phenomena that simultaneously generate positive outcomes, unresolved uncertainties, and potential limitations. The work develops a comprehensive theoretical foundation, presents a neutrosophic assessment framework, explores empirical applications and case studies, and discusses ethical, methodological, and policy implications of AI integration in teacher preparation. By embracing uncertainty as an essential dimension of educational reality, the book advocates a balanced and human-centered approach to technological innovation. It argues that AI should function as an augmentative partner that enhances reflective practice, pedagogical creativity, and professional growth while preserving the relational and ethical foundations of teaching. The neutrosophic perspective offers researchers, educators, and policymakers a robust analytical tool for understanding and guiding the future development of AI-enabled teacher education systems.

This book presents a collection of recent advances in neutrosophic mathematics, highlighting the growing role of neutrosophic structures in addressing uncertainty, indeterminacy, and inconsistency within modern mathematical systems. The volume integrates developments from neutrosophic graph theory, algebra, metric spaces, and functional analysis, demonstrating both theoretical depth and practical applicability. The first part of the book investigates novel classes of neutrosophic graphs, including Min-Max neutrosophic graphs and neutrosophic bidirected graphs. New combinatorial constructions, structural properties, graph operations, and network modeling applications are developed, providing mathematical tools for the analysis of uncertain and complex networks. The second part focuses on neutrosophic algebraic structures through the study of neutrosophic Lie subalgebras and ideals. Fundamental results concerning homomorphisms, images, inverse images, quotient structures, and preservation properties are established, extending classical Lie algebra theory to neutrosophic environments. The third part explores analytical aspects of neutrosophic mathematics. New common fixed-point results in neutrosophic metric spaces are developed under integral-type contractive conditions and applied to dynamic programming problems. Furthermore, the theory of rough I-statistical convergence of order α′ in neutrosophic normed spaces is introduced and characterized, with applications to dynamical systems, iterative processes, and sensor-network data fusion. The book concludes with a comprehensive discussion of emerging research directions, including neutrosophic artificial intelligence, machine learning, quantum neutrosophic structures, network science, algebraic topology under uncertainty, and sustainability applications. By bringing together diverse mathematical perspectives, this volume provides a unified overview of contemporary developments in neutrosophic mathematics and serves as a valuable resource for researchers, graduate students, and professionals working in uncertainty modeling and advanced mathematical sciences.

his book introduces Type-k Neutrosophic Sets as a recursive extension of classical neutrosophic logic. While standard neutrosophic sets represent each proposition through three independent components—truth, indeterminacy, and falsity—Type-k Neutrosophic Sets allow each of these components to be recursively characterized by further neutrosophic triplets. This creates a hierarchical epistemic structure capable of representing nested uncertainty, meta-uncertainty, contradiction, ignorance, and paraconsistent states that cannot be adequately captured by classical probability, fuzzy logic, or intuitionistic fuzzy models. The book develops the formal foundations of this recursive framework, proves its expressive hierarchy, and extends it through single-valued neutrosophic tensors and plithogenic tensors for multi-criteria decision-making. It also explores connections with neutrosophic paraconsistent logic and applies the framework to the epistemic auditing of Large Language Models. Particular attention is given to hallucination detection, the limitations of softmax-based confidence, and the need for triadic and recursive models of truth, falsity, and indeterminacy in AI evaluation. By combining non-classical logic, tensorial structures, decision theory, and artificial intelligence auditing, the work proposes a new formal and computational framework for understanding uncertainty in complex epistemic systems. It is intended for researchers in neutrosophic logic, philosophy of logic, formal epistemology, artificial intelligence, paraconsistent reasoning, and decision sciences.