Florentin Smarandache

For more than a century, singularities and ultraviolet divergences have stood at the frontiers of modern theoretical physics, marking points where our most successful theories cease to be mathematically well defined. Infinitesimal Punctures proposes a structural shift in perspective: instead of inserting point-like sources into smooth manifolds, matter and physical attributes are interpreted as intrinsic geometric defects—measure-zero punctures—within spacetime itself. In this framework, curvature, charge, and quantum behavior arise not as external additions but as distributionally supported features of geometry. The series develops this idea systematically, moving from foundational geometry, through dynamical physical laws, to a unified S-MultiSpace and S-MultiStructure structural formalism. This volume establishes the geometric and analytic foundations of infinitesimally punctured manifolds. It introduces the weak–strong decomposition of geometry, in which smooth differential structure coexists with defect-supported distributional curvature on measure-zero sets. Classical singularities are reinterpreted as punctures carrying finite, integrable curvature, replacing divergent invariants with well-defined distributional objects. The book develops codimension-based classification of defects, holonomy effects, regularisation via smooth metric families, and the structural Neutrosophic triad as a geometric—not yet dynamical—decomposition. A substantial portion of the volume is devoted to operator theory on punctured manifolds, including self-adjoint extensions of Laplace-type operators, spectral stability, defect-induced bound states, and scattering without external potentials. Explicit low-dimensional toy models—punctured circles, conical planes, and point defects in three dimensions—demonstrate how geometry alone generates observable analytic consequences. This book contains no field equations and no cosmological models; its purpose is to provide a rigorous, self-contained mathematical groundwork upon which the subsequent volumes build.

Soft set theory provides a direct framework for parameterized decision modeling by assigning to each attribute (parameter) a subset of a given universe, thereby representing uncertainty in a structured way [1, 2]. Over the past decades, the theory has expanded into numerous variants— including hypersoft sets, superhypersoft sets, TreeSoft sets, bipolar soft sets, and dynamic soft sets—and has been connected to diverse areas such as topology and matroid theory. In this book, we present a survey-style overview of soft sets and their major extensions, highlighting core definitions, representative constructions, and key directions of current development.

This volume presents some original research on SuperHyperStructures and SuperHyperTopologies, new branches of mathematics with applications in applied sciences. These frameworks extend classical notions of sets, graphs and topologies by incorporating uncertainty, indeterminacy and multivalued membership, rooted in neutrosophic and Plithogenic logic. They provide non-rigorous methods for the analysis of complex systems in which ambiguity and dynamic interactions are fundamental. The book is largely deals with 3 main subject-matters. The first one introduces mathematical foundations, including SuperHyperTopologies, Neutrosophic and Plithogenic extensions, fuzzy and soft set frameworks, and links with Łukasiewicz algebras. The second one develops structural perspectives through SuperHyperGraphs, weighted hyperedges, fuzzy neural networks, and neutrosophic automata. The third subject-matter illustrates interdisciplinary applications in medicine, finance, agriculture, tourism, economics, and data science. An important feature of the volume is its interdisciplinary orientation. Mathematical theories are linked to practical problems, showing how SuperHyper concepts can support decision-making, optimize resources and model uncertainty in different fields. By combining fuzzy, neutrosophic and plithogenic logics within hyperstructural frameworks, the book highlights innovative approaches to indeterminate and multilayered phenomena. The subject-matters and content of this volume is aimed not only at researchers but also graduate students and professionals in mathematics, computer science and related disciplines. It offers a comprehensive reference and several practical research avenues, paving the way for further interdisciplinary research and applications. In this volume, there are 25 research studies with different approaches, results and findings.

Collected Papers, Vol. XVII – A Multi-Disciplinary Approach to Uncertainty, Indeterminacy, and Complex Systems is a substantial and wide-ranging volume, bringing together 74 articles across nearly 1,000 pages. The contributions, originally published in a variety of international scientific journals, reflect the breadth and maturity of contemporary research in neutrosophy and its related domains. The volume explores a rich spectrum of interdisciplinary fields, including logic, philosophy, physics, mathematics, statistics, information fusion, artificial intelligence, decision theory, complex systems, and robotics, offering both foundational developments and applied methodologies for modeling uncertainty, indeterminacy, and contradiction. All papers are authored or co-authored by Florentin Smarandache, either individually or in collaboration with 69 distinguished co-authors from 18 countries (Ahmed Abdelhafeez, Mohamed Abdel Basset, Hamiden Abd El-Wahed Khalifa, Ijaz Ahmad, Muhammad Ahsan, Alhanouf Alburaikana, Ahmed M. Ali, Anum Ali, Mubashir Ali, Wajid Ali, Sulaiman Al Amro, Rafif Alhabib, Saleh I. Alzahrani, Farrukh Arslan, Muhammad Arshad, Büşra Aydoğan, Robert N. Boyd, Victor Christianto, Ajoy Kanti Das, Rakhal Das, Suman Das, Rama Debbarma, Srila Dey, Dinh Van Dzung, Alaa Elmor, Takaaki Fujita, Daniela Gifu, Mohammad Hamidi, Hoang Anh Tuan, A. Kalavathi, Khuram Ali Khan, S. Krishnaprakash, Duraisamy Kumar, Luong Thi Hong Lan, Le Nhu Dieu Huon, Maikel Yelandi Leyva-Vázquez, Priyanka Majumder, Ahmed A. Metwaly, Mona Mohamed, Noreen Mushtaq, Ion Nălbitoru, Nguyen Long Giang, Nguyen Thi Hien, Nguyen Thi Lan Nhi, Nguyen Thien Luong, Nguyen Tho Thong, Nguyen Thu Huong, Murat Olgun, Gabrijela Popovic, Michael Angelo B. Promenti, Afshan Qayyum, Atiqe Ur Rahman, Shazia Rana, Muhammad Saeed, Alaa Salem, A. Saranya, Alexandra Şandru, Ovidiu Ilie Şandru, Tanzeela Shaheen, Dragisa Stanujkic, Maja Stanujkic, Mohadeseh Taghinezhad, John Frederick D. Tapia, Mehmet Ünver, Vo Si Nam, Michael Voskoglou, Muhammad Waqas, Muhammad Yasir, Badria Almaz Ali Yousif), underscoring the global reach and collaborative nature of neutrosophic research, and revealing a coherent yet diverse intellectual landscape, ranging from theoretical frameworks—such as neutrosophic logic, hypersoft sets, superhyperstructures, and revolutionary topologies—to practical applications in healthcare analytics, artificial intelligence, sustainability, engineering, and decision-making under uncertainty. Together, the works collected in this seventeenth volume provide both a comprehensive reference and a forward-looking perspective on the evolving role of neutrosophic and multi-valued approaches in addressing complex real-world problems.

This sixteenth volume of Collected Papers is an extensive work, comprising 78 articles that span almost 1000 pages. The papers, originally published in various scientific journals, cover the broad and interdisciplinary fields of neutrosophics and other areas of study, including logic, philosophy, physics, mathematics, statistics, information fusion, and robotics. The volume features articles authored by Florentin Smarandache, either alone or in collaboration with 81 co-authors (from 21 different countries): Shumaila Abbas, Mohammed Abdel-Sattar, M. Modather M. Abdoug, Usama Afzal, Daud Ahmad, Abdulrahman AlAita, Shahbaz Ali, Fahad Alsharari, Suriana Alias, Saima Anis, Muhammad Aslam, Hamdi Ayed, M.B. Bera, Noel Batista-Hernández, Robert N. Boyd, Said Broumi, Claudia Camacho-Zuñiga, T. Daniel Chandra, Ebenezer Chifu, Victor Christianto, Mihaela Colhon, Shrouk El-Amir, Ibrahim El-Henawy, Takaaki Fujita, Nguyen Long Giang, Ana González-Marcos, Phung The Huan, Sadia Iqbal, Azhar Ali Janjua, Maissam Jdid, Madad Khan, Waris Khan, Muhammad Shahid Khan, Anamika Kumari, Luong Thi Hong Lan, M. Lathamaheswari, B.S. Mahapatra, G.S. Mahapatra, Muhammad Khalid Mahmood, Nivetha Martin, Pierre Millette, Mona Mohamed, M.K. Mondal, Bahnishikha Roy Muhuri, Norzieha Mustapha, Fermín Navaridas-Nalda, Tran Thi Ngan, Antonios Paraskevas, P. Prabakaran, Madiha Qayyum, Shazia Rana, Alina Reșceanu, Jesus Estupiñan Ricardo, Andreas Ries, Yaser Saber, Muhammad Saeed, Taoufik Saidani, P. Sathya, Noorazliyana Shafii, Muhammad Ahmed Shehzad, Rehan Ahmad Khan Sherwani, Rajesh Singh, Le Hoang Son, S. Sudha, Vu Duc Thai, Hugo Terashima-Marin, Nguyen Tho Thong, Pham Huy Thong, Monica Tilea, Tran Manh Tuan, Yunita Umniyati, Maikel Leyva Vázquez, Michael Gr. Voskoglou, Mohammad Kaif Wajid, Mohammad Saif Wajid, Mohd Anas Wajid, Roliza Md Yasin, Shouzhen Zeng, Muhammad Zeeshan, Alia Nur Izzah Zulkifli.

This fifteenth volume of Collected Papers is an extensive work, comprising 77 articles that span almost 1000 pages. The papers, originally published in various scientific journals, cover the broad and interdisciplinary fields of neutrosophics and other areas of study, including mathematics, fuzzy sets, intuitionistic fuzzy sets, picture fuzzy sets, information fusion, robotics, statistics, and extenics. The volume features articles authored by Florentin Smarandache, either alone or in collaboration with 123 co-authors. A complete list of these co-authors is provided below, arranged alphabetically: Shumaila Abbas, Raja Abdullah Abdulfatah, Sabah S. Abdulrahman, Mohammad Abobala, Mohamed Abosalih, Sunday Adesina Adebisi, Lorenzo Affè, Abuobida Ahal, Ashik Ahirwar, Abuobida Alfahal, Yaser Ahmad Alhasan, Hanan Alohali, B. Ananda Priya, S. Angammal, N. Angel, T. Ashika, Faiza Asif, Muhammad Aslam, Aws A. Azzidan, Abdelaziz Babiker, Esam Bader, Assia Bakali, Ranulfo Paiva Barbosa, Noel Batista-Hernández, Sania Batool, Romualdas Bausys, Mahmut Can Bozyıgt, Said Broumi, Erick González Caballero, T. Chalapathi, Victor Christianto, Bing Cui, Tugrul Daim, Bijan Davvaz, Muhammet Deveci, Meghana Dirisala, Seyyed Ahmad Edalatpanah, Ahmed El-Massry, Tawfik Elmasry, A. Elrawy, Takaaki Fujita, N. Ramila Gandhi, Harish Garg, Mona Gharib, P. Gnanachandra, İlgin Gökasar, Hasan Gökbaş, Hannah Grace, S. Karunya Helen Gunaseeli, Mohammad Hamidi, Iftikhar Ul Haq, D. Harish Babu, Masooma Raza Hashmi, Farnaz Sheikh Hassani, Jesús Rafael Hechavarría-Hernández, K. Hemabala, A. Ibrahim, Sadia Iqbal, Mehtap Isik, Saeid Jafari, Maissam Jdid, Ilanthenral Kandasamy, Vasantha Kandasamy, Muthusamy Karthika, Huda E. Khalid, Neha Andaleeb Khalid, Rehan Ahmad Khan Sherwani, B. Srinivasa Kumar, K. Kumaraswamy Naidu, Anamika Kumari, Mukesh Kushwaha, Yingcang Ma, Madelaine Al Tahan, Nivetha Martin, Mohamed Talea, Mona Mohamed, Swaminathan Mohanaselvi, Alaa A. Mohammed, D. Nagarajan, Muhammad Ali Najjar, Giorgio Nordo, Murat Olgun, Dragan Pamucar, P. Pandiammal, Antonios Paraskevas, Mani Parimala, Aurimas Petrovas, S. Rajeswari, Muhammad Riaz, Rafael Rojas-Gualdron, Yaser Saber, Muhammad Saeed, Rajesh Kumar Saini, A. A. Salama, Muhammad Saleem, S. Sandhiya, Tanzeela Shaheen, Riya Eliza Shaju, Rehan Ahmad Khan Sherwani, Rajesh Singh, Ali Sorourkhah, B. Srinivasa Kumar, C. Sugapriya, Ayat A. Temraz, Hugo Terashima-Marin, Mahima Thakur, S. S. Thakur, A. Theeba, Shobh Nath Tiwari, Hamza Ghazanfar Toor, Selçuk Topal, Mehmet Ünver, Maikel Leyva Vázquez, Satya Bhushan Verma, Mohammad Kaif Wajid, Mohammad Saif Wajid, Mohd Anas Wajid, Tomasz Witczak, Subhash Kumar Yadav, Aws A. Zaidan, Kazimieras Edmundas Zavadskas, Dan Zhang, Hengdong Zhu.

The twentieth-century edifice of physics rests on a handful of foundational pillars: Newtonian mechanics, Maxwellian electrodynamics, quantum theory, and Einsteinian relativity. Within this framework, the vast majority of research proceeds by refining, extending, or experimentally testing the consequences of these established laws. Over the past three decades, however, a persistent current of inquiry has sought to probe the limits of this canonical paradigm, asking whether the logical and ontological foundations of the discipline might admit richer, more flexible structures. Central to this current is Neutrosophy, a meta-theory that offers a systematic way to treat truth (T), falsity (F), and indeterminacy (I) as mutually independent components of any physical proposition. This volume brings together, for the first time, a curated collection of abstracts submitted by the author (alone or in collaboration) to the American Physical Society (APS) meetings between 2008 and 2018. These entries, originally intended as invitations to oral and poster presentations, collectively map an ambitious agenda: to reinterpret familiar phenomena through the lens of Neutrosophic Physics. By gathering these dispersed records, we provide scholars and historians of science with a panoramic view of a research program that challenges the very "constancy" of universal constants and the locality of physical laws. In addition to the formal APS records, this collection includes a chapter of unsubmitted abstracts and exploratory drafts (2018–present). Unlike the preceding chapters, these entries were selected from the author’s private research archives to illustrate the continuing evolution of his theories. They explore the frontiers of Unmatter—matter-antimatter states motivated by neutrosophic neutrality—and the Neutrosophic Expansion Constant, which addresses emerging tensions in modern cosmology.

This book explores the theoretical and philosophical implications of transcending the speed of light limit, building upon Florentin Smarandache's "no speed barrier" hypothesis. This collection of works develops the hypothesis that no physical entity is fundamentally constrained by a speed barrier, suggesting a framework for extending classical and relativistic physics into the realms of Superluminal Physics and Instantaneous Physics. The book’s core technical argument centers on the phenomenon of quantum tunneling and the Hartman effect, where the tunneling time appears independent of barrier width, leading to seemingly superluminal results. The authors propose a novel interpretation of the Hartman effect, suggesting that it is a manifestation of the multivaluedness of solutions to the Schrödinger equation, implying a quantum entity can exist in multiple locations simultaneously and thus appear to tunnel instantaneously. To provide a plausible mechanism for this superluminal effect, the book considers soliton tunnelling, specifically the Falaco soliton, which is presented as a known exact solution to the three-dimensional Navier-Stokes equations and is proposed as a realistic mechanism for quantum tunneling. Ultimately, the research seeks to find a general theory, such as the S-Multispace Theory, that unites physics at subluminal, relativistic, superluminal, and instantaneous speeds. This theoretical path is explored as an alternative for deep space and interstellar travel, moving beyond conventional concepts like the Alcubierre warp drive or wormholes in General Relativity.

Education is the foundation of any progressive society and understanding the challenges within educational systems is crucial for their improvement. However, educational problems are often characterised by uncertainty, indeterminacy and incompleteness—factors that traditional methods sometimes struggle to address adequately. This book introduces neutrosophic theory as a powerful framework for investigating educational challenges. Neutrosophy, which deals with neutralities and indeterminacies, offers a unique perspective that goes beyond classical logic and fuzzy logic. It provides researchers and educators with tools to handle the ambiguities inherent in educational settings. Our motivation for writing this book stems from the recognition that educational research often encounters situations where data is incomplete, opinions are conflicting, or outcomes are uncertain. Traditional binary or fuzzy approaches may not fully capture these complexities. Neutrosophic methods, with their ability to represent truth, indeterminacy and falsity simultaneously, offer a more comprehensive approach. This book is designed for educational researchers, postgraduate students, policymakers and practitioners who wish to explore innovative methodologies for addressing educational problems. We have made conscious efforts to present the material in simple Indian English, making it accessible to a wide audience while maintaining academic rigour. The book is organised into ten chapters, beginning with an introduction to educational challenges and progressing through the theoretical foundations, mathematical frameworks and practical applications of neutrosophic approaches. We include numerous examples from the Indian educational context to illustrate concepts and demonstrate applicability.

Hypergraphs generalize this framework by allowing hyperedges that connect more than two vertices. Superhypergraphs further enrich the model through iterated powerset constructions, capturing hierarchical and self-referential structures among hyperedges. An (m,n)-SuperHyperGraph is a mathematical structure in which each vertex corresponds to an (m,n)-superhyperfunction defined on a base set, while the hyperedges group such functions together to represent higher-order relationships and contextual connections. Systematic research on SuperHyperGraphs is still relatively limited compared with the extensive literature on graphs and hypergraphs. To help bridge this gap, this book presents a survey of fundamental and advanced concepts related to SuperHy-perGraphs. Our aim is twofold: (i) to increase the visibility and accessibility of SuperHyperGraph theory and thereby stimulate further research, and (ii) to deepen the mathematical understanding of their structures among researchers and practitioners who work with graph- and hypergraph-based models.

En un mundo moldeado por contradicciones, incertidumbres y paradigmas en evolución, la Filosofía Neutrosófica emerge como un marco innovador que trasciende el pensamiento binario. Arraigada en el estudio de las neutralidades, las contradicciones y su interacción dinámica, esta filosofía redefine la lógica clásica, la epistemología y la ontología, ofreciendo un enfoque integral para comprender la realidad. A través de la lente de la Neutrosofía, este libro recopila artículos que exploran conceptos fundamentales como el continuo de neutralidades, el equilibrio de ideas y la tesis-antítesis-neutrotesis, desafiando las estructuras dialécticas tradicionales. Expande las fronteras de la filosofía al integrar la matematización, las lógicas multivaluadas y los enfoques transdisciplinarios al conocimiento. Desde la mecánica cuántica y la inteligencia artificial hasta la ética, la sociología y la literatura, las aplicaciones de la Neutrosofía son vastas y transformadoras. Ya sea reinterpretando paradojas, remodelando fundamentos filosóficos o explorando la naturaleza infinita de la verdad, esta obra allana el camino hacia una nueva forma de pensar: una que abraza la ambigüedad, la indeterminación y la coexistencia de opuestos. Al tender puentes entre disciplinas e introducir principios innovadores como la Lógica Neutrosófica, la Evolución Social Neutrosófica y el Materialismo Neutrosófico, este libro sirve como una base teórica y una guía práctica para académicos, investigadores y pensadores que buscan una comprensión más profunda de la complejidad en el mundo moderno. La Filosofía Neutrosófica no es solo una exploración del conocimiento, es una invitación a repensar la esencia misma de la verdad, la realidad y la comprensión humana.

Dans un monde façonné par les contradictions, les incertitudes et des paradigmes en constante mutation, la philosophie neutrosophique s'impose comme un cadre révolutionnaire qui transcende la pensée binaire. Enracinée dans l'étude des neutralités, des contradictions et de leur interaction dynamique, cette philosophie redéfinit la logique classique, l'épistémologie et l'ontologie, offrant une approche globale pour la compréhension de la réalité. Ce volume rassemble des articles explorant des concepts fondamentaux tels que le continuum des neutralités, l'équilibre des idées et la triade thèse-antithèse-neutrothèse, remettant en question les structures dialectiques traditionnelles. Les frontières de la philosophie sont élargies par l'intégration de la mathématisation, des logiques multivaleurs et des approches transdisciplinaires de la connaissance. De la mécanique quantique et l'intelligence artificielle jusqu'à l'éthique, la sociologie et la littérature, les applications de la neutrosophie sont vastes et transformatrices. Qu'il s'agisse de réinterpréter des paradoxes, de reconfigurer les fondements philosophiques ou d'explorer la nature infinie de la vérité, cet ouvrage ouvre la voie à un nouveau mode de pensée — un mode qui intègre l'ambiguïté, l'indétermination et la coexistence des contraires. En reliant les disciplines et en introduisant des principes innovants tels que la logique neutrosophique, l'évolution sociale neutrosophique et le matérialisme neutrosophique, ce livre sert à la fois de fondement théorique et de guide pratique pour les chercheurs, les penseurs et les spécialistes désireux d'approfondir leur compréhension de la complexité du monde contemporain. La philosophie neutrosophique n'est pas seulement une exploration du savoir — c'est une invitation à repenser l'essence même de la vérité, de la réalité et de la compréhension humaine.

Într-o lume modelată de contradicții, incertitudini și paradigme în continuă schimbare, filosofia neutrosofică se conturează ca un cadru revoluționar care transcende gândirea binară. Înrădăcinată în studiul neutralităților, al contradicțiilor și al interacțiunii lor dinamice, această filosofie redefinește logica clasică, epistemologia, ontologia, oferind o abordare cuprinzătoare a înțelegerii realității. Acest volum reunește articole care explorează concepte fundamentale precum continuumul neutralităților, echilibrul ideilor și triada teză–antiteză–neutroteză, provocând structurile dialectice tradiționale. Se extind granițele filosofiei prin integrarea matematizării, a logicilor multi-valorice și a abordărilor transdisciplinare asupra cunoașterii. De la mecanica cuantică și inteligența artificială, până la etică, sociologie și literatură, aplicațiile neutrosofiei sunt vaste și transformatoare. Fie că reinterpretăm paradoxuri, reconfigurăm fundamentele filosofice sau explorăm natura infinită a adevărului, această lucrare deschide drumul către un mod nou de a gândi—unul care integrează ambiguitatea, indeterminarea și coexistența contrariilor. Prin conectarea disciplinelor și introducerea unor principii inovatoare precum logica neutrosofică, evoluția socială neutrosofică și materialismul neutrosofic, această carte funcționează atât ca fundament teoretic, cât și ca ghid practic pentru cercetători, gânditori și specialiști interesați să aprofundeze înțelegerea complexității lumii contemporane. Filosofia neutrosofică nu este doar o explorare a cunoașterii—este o invitație de a regândi însăși esența adevărului, a realității și a înțelegerii umane.

Las plataformas tradicionales de mapeo de opinión emplean análisis de componentes principales (PCA) y agrupamiento para visualizar paisajes de opinión pública. Sin embargo, estos enfoques se basan en marcos lógicos binarios o difusos que pueden no capturar adecuadamente la ambigüedad inherente y la naturaleza multidimensional de las creencias humanas. Este artículo presenta una aplicación novedosa de la lógica neutrosófica al análisis de percepciones sobre el cambio climático, demostrando ventajas significativas sobre los métodos tradicionales de análisis de consenso. Utilizando una encuesta de 84 residentes en Guayaquil, Ecuador, empleamos una escala Likert neutrosófica que captura tres dimensiones independientes: verdad (T), indeterminación (I) y falsedad (F) para cada elemento de la encuesta. Nuestro análisis revela dos grupos de respondientes distintos: un grupo "Altamente Comprometido pero Conflictivo" (34.5% de los respondientes) caracterizado por puntuaciones elevadas en las tres dimensiones, y un grupo "Moderadamente Aceptante" (65.5%) con perfiles dominados por la verdad. Los hallazgos clave demuestran que el análisis neutrosófico descubre complejidad oculta en datos de opinión que los enfoques binarios o difusos obscurecerían, permitiendo estrategias de comunicación más dirigidas y efectivas.

Uncertainty permeates most real-world contexts, motivating mathematical frameworks that can faithfully represent vagueness, inconsistency, and incomplete information. Classical approaches include fuzzy sets and intuitionistic fuzzy sets. Extending these ideas, the neutrosophic framework introduces neutrosophic sets in which each element x is characterized by a triplet of independent degrees (T (x), I (x), F (x)) ∈ [0, 1]3 , representing, respectively, truth, indeterminacy, and falsity, typically subject to T (x) + I (x) + F (x) ≤ 3. This book concentrates on a selection of concepts developed and discussed in the Nidus idearum series, with particular emphasis on Neutrosophy, Plithogenic Sets, Physics, and related scientific domains. We focus on ideas that, despite their conceptual richness and potential applicability, have so far received relatively limited systematic treatment. Our aim is to present these notions in a coherent and accessible manner, in order to support further theoretical development and to encourage new applications in Neutrosophy, Plithogenic Set theory, Physics, and allied fields.

Real-world phenomena frequently involve vagueness, partial truth, and incomplete information. To capture such uncertainty in a mathematically rigorous manner, numerous generalized set-theoretic frameworks have been introduced, including Fuzzy Sets [1], Intuitionistic Fuzzy Sets [2], Neutrosophic Sets [3, 4], Vague Sets [5], Hesitant Fuzzy Sets [6], Picture Fuzzy Sets [7], Quadripartitioned Neutro-sophic Sets [8], PentaPartitioned Neutrosophic Sets [9], Plithogenic Sets [10], HyperFuzzy Sets [11], and HyperNeutrosophic Sets [12]. Within these frameworks, a vast number of concepts have been proposed and studied, especially in the contexts of Fuzzy, Intuitionistic Fuzzy, Neutrosophic, and Plithogenic Sets. This extensive body of work highlights both the importance of these theories and the breadth of their application domains. Consequently, many ideas, constructions, and structural patterns recur across these four families of uncertainty-oriented models. In this book, we present a comprehensive, large-scale survey of Fuzzy, Intuitionistic Fuzzy, Neutro-sophic, and Plithogenic Sets. Our aim is to offer r eaders a s ystematic overview of existing develop-ments and, through this unified exposition, to foster new insights, further conceptual extensions, and additional applications across a wide range of disciplines.

A Neutrosophic Set assigns to each element independent degrees of truth, indeterminacy, and falsity, thereby extending classical uncertainty models. It is widely recognized as a generalization of both Fuzzy Sets and Intuitionistic Fuzzy Sets. Neutrosophic and fuzzy logics have been employed across sociology, psychology, the social sciences, biology, chemistry, physics, and philosophy. This paper revisits and systematizes these applications of Neutrosophic Logic, presenting rigorous mathematical definitions alongside concrete, domain–specific ex am ples. Our aim is tw ofold: to promote broader understanding and dissemination of neutrosophic methods, and to provide a clear foundation that enables future theoretical development and practical deployment. By organizing existing concepts into a coherent framework and illustrating their use through reproducible constructions, we seek to broaden the scope of applications and to stimulate further research across disciplines.

Este artículo propone un marco de pluralismo lógico aplicado al análisis causal en inteligencia artificial (IA) y ciencias sociales. Basado en la tesis pluralista de que no existe un único concepto monolítico de causalidad, sino una familia de conceptos relacionados, confrontamos la premisa humeana de que “no observamos la causalidad” con el contraargumento de Anscombe de que, de hecho, percibimos acciones causales como empujar, golpear y cortar. Sobre esta base, presentamos un estudio ilustrativo que compara el Análisis Comparativo Cualitativo de conjuntos nítidos (csQCA), el de conjuntos difusos (fsQCA) y el neutrosófico (nQCA). De esta comparación, derivamos implicaciones metodológicas y sustantivas. Metodológicamente, encontramos que: (i) la dicotomización estricta (csQCA) puede subestimar las relaciones causales; (ii) la pertenencia gradual (fsQCA) captura más eficazmente la fuerza y relevancia empírica; y (iii) la descomposición neutrosófica (nQCA) considera explícitamente la verdad, la indeterminación y la falsedad, ofreciendo así un diagnóstico pluralista de la relación causal. Enriquecemos aún más este análisis incorporando una visión pluralista de la causalidad, que abarca la producción mecánica, la diferencia contrafactual, las capacidades/disposiciones y los mecanismos.

The unprecedented migration of populations to urban areas has created major challenges for municipalities and service providers. To address these issues, decision-makers must embrace smart city and Society 5.0 paradigms, both of which focus on adaptability and sustainable development. Artificial intelligence (AI) plays a pivotal role by expanding service capacity, enabling automation, and processing vast data to align urban development with the UN’s Sustainable Development Goals (SDGs). This paper develops a multi-criteria analysis system designed to support decision-making in complex socio-technological contexts. Using cognitive mapping and the Decision-Making Trial and Evaluation Laboratory (DEMATEL) technique within a neutrosophic environment, the study analyzes cause-and-effect relationships among critical factors that influence businesses’ adaptation to AI, smart cities, and Society 5.0. The results offer a holistic decision-support framework to guide policymakers, businesses, and urban planners toward inclusive, sustainable, and technologically integrated societies.

This study addresses the challenges of uncertainty, vagueness, and imprecision in real-world decision-making, particularly for small-scale farmers selecting short-term crops. The central problem is the lack of a systematic framework to evaluate conflicting criteria such as investment, yield, market demand, and soil requirements. To solve this, a Multi-Criteria Decision-Making (MCDM) framework is introduced, which integrates Cubic Spherical Neutrosophic Sets (CSNS) and Neutrosophic Hyper Soft Sets (NHSS). The research also proposes a novel cubic spherical neutrosophic Bonferroni mean operator for aggregating neutrosophic sets. The effectiveness of this approach is demonstrated through a case study in Tamil Nadu, focusing on selecting the most suitable crop for different climatic zones. A sensitivity analysis confirms the model's robustness and reliability. The tool aims to improve resource utilization, reduce risks, and promote agricultural sustainability.

This study addresses the ambiguities in empirical findings on artificial intelligence (AI) in education by proposing a new methodological framework that combines neutrosophic stance detection and Fuzzy Set Qualitative Comparative Analysis (fsQCA). This approach explicitly models truth, indeterminacy, and falsity, allowing for the synthesis of contradictory research. The authors evaluated four causal hypotheses related to AI-based learning, using a survey of 24 university participants to explore the necessary conditions for perceived learning improvements. The results indicate that the digital divide is a perfectly necessary condition for effective AI-enhanced learning. The findings also reveal that AI feedback and AI platform use are necessary, though not sufficient, for learning improvements, but generate significant uncertainty. The study concludes that the neutrosophic-fsQCA framework is a viable technique for synthesizing ambiguous findings and provides empirical evidence that digital equity and high-quality design are crucial for the successful integration of AI in education.

A multitude of uncertainty models, such as fuzzy, neutrosophic, and plithogenic sets, have emerged in the domain of fuzzy sets and their extensions. This diversity, while reflecting an active research community, can lead to conceptual overlap. To address this, the authors propose a unified framework for the systematic management of these "Uncertain Sets" by defining the Uncertain Model, the U-Structure, and the U-Off Structure. The paper revisits and extends the definitions of HyperUncertain, SuperUncertain, SuperHyperUncertain, Tree Uncertain, and Forest Uncertain structures, along with their "Off" variants. To encompass systems beyond Uncertain Sets, the concepts of Functorial Set and Functorial Structure are also introduced. This comprehensive approach aims to establish a robust foundation for organizing and applying uncertainty models across various mathematical and applied domains.

"Neutrosophic Computing and Machine Learning" (NCML) es una revista académica dedicada a la publicación de estudios avanzados en neutrosofía, conjuntos neutrosóficos, lógica neutrosófica, probabilidad neutrosófica, estadística neutrosófica y enfoques neutrosóficos para el aprendizaje automático y sus aplicaciones en diversos campos. La revista da la bienvenida a trabajos profesionales en inglés o español que incluyan un breve resumen de un problema y los resultados obtenidos. Este número (36) de "Neutrosophic Computing and Machine Learning" (NCML) presenta una colección focalizada de artículos que exploran la aplicación práctica de la teoría neutrosófica en varios dominios. Los trabajos abordan desafíos clave en áreas como el diagnóstico médico, el análisis de riesgos y el análisis de redes sociales, demostrando cómo los enfoques neutrosóficos pueden manejar eficazmente la incertidumbre y la indeterminación. Un tema central es el desarrollo de modelos neutrosóficos novedosos para resolver problemas del mundo real. El número presenta investigaciones sobre mapas cognitivos neutrosóficos para la evaluación multicriterio, un enfoque integrado neutrosófico-TOPSIS para la selección de personal y un modelo neutrosófico único para la detección del cáncer de mama. Además, los artículos profundizan en el uso de conjuntos neutrosóficos refinados para analizar el sentimiento en las redes sociales y su aplicación en la clasificación de rasgos de personalidad. Esta colección de investigaciones demuestra la utilidad en expansión de la computación neutrosófica en la creación de soluciones más robustas y matizadas para sistemas complejos.

"Neutrosophic Computing and Machine Learning" (NCML) es una revista académica dedicada a la publicación de estudios avanzados en neutrosofía, conjuntos neutrosóficos, lógica neutrosófica, probabilidad neutrosófica, estadística neutrosófica y enfoques neutrosóficos para el aprendizaje automático y sus aplicaciones en diversos campos. La revista da la bienvenida a trabajos profesionales en inglés o español que incluyan un breve resumen de un problema y los resultados obtenidos. Este número (37) de "Neutrosophic Computing and Machine Learning" (NCML) presenta una colección diversa de investigaciones que demuestran la extensa aplicación de la teoría neutrosófica para abordar complejos desafíos sociales, legales y de salud, con un enfoque particular en temas de Ecuador. Los artículos utilizan una variedad de métodos neutrosóficos, incluyendo evaluación multicriterio, mapas cognitivos y procesos analíticos, para analizar problemas del mundo real. Las contribuciones clave incluyen un análisis legal y social, con varios trabajos que aplican métodos neutrosóficos para evaluar los desafíos legales relacionados con la telemedicina, evaluar el impacto de las herramientas digitales en el derecho notarial y analizar los factores que influyen en los homicidios utilizando datos abiertos. El número también examina problemas como la violencia de género y la migración, los derechos de las víctimas menores de violencia sexual y los desafíos legales del "porno de venganza". En cuanto a las aplicaciones médicas y de salud, la investigación profundiza en los campos de la medicina y la odontología, con artículos sobre la evaluación de la eficacia de las estrategias para prevenir la hepatitis B en mujeres embarazadas, el análisis de los factores de riesgo para la diabetes tipo II y la optimización del tratamiento para el Alzheimer. Otros estudios se centran en la salud animal, como la evaluación del impacto de la brucelosis y la mastitis en el ganado y la evaluación del raquitismo canino. El número también incluye estudios ambientales y económicos, con aplicaciones de la lógica neutrosófica para evaluar el impacto de los plásticos oceánicos y analizar la contaminación del agua. Adicionalmente, los trabajos evalúan temas económicos como la rentabilidad hotelera y el impacto del impuesto a la moneda en el desarrollo económico de Ecuador.

"Neutrosophic Computing and Machine Learning" (NCML) es una revista académica dedicada a la publicación de estudios avanzados en neutrosofía, conjuntos neutrosóficos, lógica neutrosófica, probabilidad neutrosófica, estadística neutrosófica y enfoques neutrosóficos para el aprendizaje automático y sus aplicaciones en diversos campos. La revista da la bienvenida a trabajos profesionales en inglés o español que incluyan un breve resumen de un problema y los resultados obtenidos. Este número (38) de "Neutrosophic Computing and Machine Learning" (NCML) es una edición especial dedicada a explorar la "Neutrosofía aplicada a los dilemas bioéticos: una mirada transdisciplinaria desde la medicina, el derecho y la educación". Los artículos del número utilizan una variedad de métodos neutrosóficos para abordar una amplia gama de problemas complejos, principalmente en el contexto de la sociedad ecuatoriana. El número se centra en gran medida en los desafíos legales y sociales, presentando análisis neutrosóficos de temas críticos como los derechos de la familia y personales, la eficacia legal de la adopción homoparental, la revictimización de las víctimas de violencia sexual y la protección de los derechos de los niños durante los procedimientos de divorcio y cuando nacen en penitenciarías. También se abordan temas relacionados con el sistema de justicia, como la eficacia de la justicia restaurativa, la despenalización del aborto, el impacto de los medios en los "juicios paralelos" y la exclusión de las mujeres indígenas del sistema de justicia. En cuanto a los asuntos civiles y públicos, se analizan los derechos de los migrantes, la discriminación contra los Testigos de Jehová, la seguridad jurídica del comercio digital y las cuestiones procesales relacionadas con los casos de pensión alimenticia y la adjudicación de bienes abandonados. Además de los asuntos legales, la investigación se extiende a otros campos, demostrando la versatilidad de los enfoques neutrosóficos. Los artículos evalúan la reforma del sistema penitenciario, analizan las lagunas legales en la gestión de áreas de conservación y evalúan el derecho a la educación en Ecuador. La investigación en salud y bienestar aplica métodos neutrosóficos para evaluar la eficacia de los adhesivos dentales y el impacto de la leucemia viral felina. En los campos de negocios y economía, los trabajos exploran estrategias comerciales para cafeterías especializadas, analizan la viabilidad financiera para pequeñas empresas y proponen un modelo de innovación gastronómica para la cocina vegetariana. El número también incluye un análisis de la aplicación de la inteligencia artificial en la lucha contra el terrorismo y un estudio sobre el uso de la lógica neutrosófica para evaluar los marcos legales para el ciberacoso.

"Neutrosophic Computing and Machine Learning" (NCML) es una revista académica dedicada a la publicación de estudios avanzados en neutrosofía, conjuntos neutrosóficos, lógica neutrosófica, probabilidad neutrosófica, estadística neutrosófica y enfoques neutrosóficos para el aprendizaje automático y sus aplicaciones en diversos campos. La revista da la bienvenida a trabajos profesionales en inglés o español que incluyan un breve resumen de un problema y los resultados obtenidos. Este número (39) de "Neutrosophic Computing and Machine Learning" (NCML) presenta una amplia gama de investigaciones que demuestran la vasta aplicación de la teoría neutrosófica en múltiples campos, con un enfoque significativo en la resolución de problemas complejos del mundo real que implican incertidumbre e indeterminación. La colección comienza con una discusión fundamental sobre el concepto de indeterminación en las teorías neutrosóficas. A partir de esta base teórica, los trabajos aplican métodos neutrosóficos a una amplia variedad de dominios. En medicina y salud, la investigación incluye la evaluación de la función respiratoria en atletas profesionales, la comparación de procedimientos quirúrgicos para la obesidad mórbida y una revisión sistemática del impacto del sistema neuroendocrino en la función inmunológica. El número también profundiza en el análisis legal y social, con artículos sobre el poder discrecional en el derecho tributario, los aspectos legales de la familia y las pensiones alimenticias, y la responsabilidad del estado con respecto a las víctimas de feminicidio. Un tema importante en este volumen es el uso de modelos neutrosóficos para el análisis estratégico y la toma de decisiones, especialmente en negocios y políticas públicas. Los trabajos cubren la aplicación de modelos neutrosóficos PEST-FODA para pequeñas y medianas empresas (PyMEs) y para analizar la industria audiovisual cubana. Otras contribuciones exploran tecnología y educación, como el uso de métodos neutrosóficos para la gestión de procesos municipales, el desarrollo de sistemas de recomendación para estudiantes y el análisis de expresiones emocionales en la educación de estudiantes con Trastorno del Espectro Autista (TEA). En general, el número muestra la versatilidad de la computación neutrosófica al proporcionar soluciones sólidas para sistemas complejos tanto en un contexto teórico como aplicado.

The Neutrosophic Set offers a robust mathematical framework for handling uncertainty by incorporating three key membership functions: truth, indeterminacy, and falsity [36]. To address the growing complexity of real-world problems, advanced extensions such as the HyperNeutrosophic Set and the SuperHyperNeutrosophic Set have been developed. These higher-dimensional models provide an enhanced representation of uncertainty and are formally defined in [20, 21, 22, 24, 23, 37, 19]. In addition, the TreeNeutrosophic Set and ForestNeutrosophic Set are well-established extensions of the Neutrosophic Set, broadening its applicability to structured and hierarchical data[16]. This paper explores the practical applications of the Neutrosophic Set in livestock management [28, 11, 2], highlighting its potential to enhance decision-making processes in this field. Additionally, we introduce several new variations of the Neutrosophic Set and the ForestNeutrosophic Set [16], further extending their theoretical framework and practical applicability.

The Neutrosophic Set provides a flexible mathematical framework for handling uncertainty by incorporating three distinct membership functions: truth, indeterminacy, and falsity [50]. In response to the growing complexity of real-world problems, advanced extensions such as the Hyperneutrosophic Set and the SuperHyperneutrosophic Set have been introduced. These extensions offer a higher-dimensional approach to modeling uncertainty and can be formally defined in [33, 34, 35, 37, 36, 51, 32]. Since the Hyperneutrosophic Set and the SuperHyperneutrosophic Set have only been recently defined, there remains a lack of concrete examples demonstrating their applicability. This paper aims to bridge this gap by exploring practical examples of these concepts in the field of agriculture. Additionally, we introduce two novel extensions, the Forest Hyperneutrosophic Set and the Forest SuperHyperneutrosophic Set, which further generalize the existing models to accommodate hierarchical and interconnected uncertainty structures in complex systems.