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. (shrink)

In this paper we recall our concepts of n th-Power Set of a Set, SuperHyperOperation, SuperHyperAxiom, SuperHyperAlgebra, and their corresponding Neutrosophic SuperHyperOperation, Neutrosophic SuperHyperAxiom and Neutrosophic SuperHyperAlgebra. In general, in any field of knowledge, one actually encounters SuperHyperStructures (or more accurately (m, n)- SuperHyperStructures).

The n-th PowerSet is used in defining the SuperHyperOperation, SuperHyperAxiom, and their corresponding Neutrosophic SuperHyperOperation, Neutrosophic SuperHyperAxiom in order to build the SuperHyperAlgebra and Neutrosophic SuperHyperAlgebra. In general, in any field of knowledge, one in fact encounters SuperHyperStructures. Also, six new types of topologies have been introduced in the last years (2019-2022), such as: Refined Neutrosophic Topology, Refined Neutrosophic Crisp Topology, NeutroTopology, AntiTopology, SuperHyperTopology, and Neutrosophic SuperHyperTopology. Neutrosophic set has been derived from a new branch of philosophy, namely Neutrosophy. (...) Neutrosophic set is capable of dealing with uncertainty, indeterminacy and inconsistent information. Neutrosophic set approaches are suitable to modeling problems with uncertainty, indeterminacy and inconsistent information in which human knowledge is necessary, and human evaluation is needed. Neutrosophic set theory firstly proposed in 1998 by Florentin Smarandache, who also developed the concept of single valued neutrosophic set, oriented towards real world scientific and engineering applications. Since then, the single valued neutrosophic set theory has been extensively studied in books and monographs introducing neutrosophic sets and its applications, by many authors around the world. Also, an international journal - Neutrosophic Sets and Systems started its journey in 2013. This first volume collects original research and applications from different perspectives covering different areas of neutrosophic studies, such as decision-making, neutroalgebra, neutro metric, and some theoretical papers. (shrink)

The domain of neutrosophic set theory has been undergoing significant and rapid expansion. New approaches, theories, strategies, and optimization techniques are developed at a swift rate. An important development in neutrosophic theory is its integration with existing set theories, such as soft set theory, rough set theory and more. Neutrosophic set has been extended to neutrosophic hyperset which is vast and extends across multiple disciplines and problem domains, from decision support and Artificial Intelligence (AI) to data analysis and complex systems (...) modelling. A neutrosophic hypersoft set extends the flexibility of neutrosophic logic and soft set theory by considering multidimensional or hypersoft sets to address real-world complexities where information is incomplete, uncertain, or inconsistent. Neutrosophic theories have emerged as essential tools in a diverse range of disciplines, including, MCDM, data mining, biomedical research, social studies, and beyond. The fourth volume of “New Trends in Neutrosophic Theories and Applications” focuses on theories, strategies, optimizing techniques for MCDM within neutrosophic frameworks. Some topics deal with introducing of Pythagorean hypersoft sets with possibility degree, quadripartitioned neutrosophic Lie-ideal of Lie-algebra, quadripartitioned neutrosophic quasi coincident topological space, neutrosophic supra-open set in neutrosophic supra topological space and neutrosophic soft matrices. Some topics deal with medical diagnosis, organ transplantation success using neutrosophic superhyperstructure and artificial intelligence. Some topics deal with revenue management, social situation. Some topics deal with MCDM in single valued neutrosophic set environment, rough set environment, and interval trapezoidal neutrosophic environment. (shrink)

SuperHyperScapes: An Ontological Atlas is a visual–conceptual exploration situated within the theoretical framework of oUTER-aRT, extending its principles into a structured yet open-ended system of image production. Composed of 77 generative visual fields, the album organizes reality into phased domains—ranging from perception and cognition to matter, society, and abstraction—each articulated through a distinct visual grammar. Grounded in neutrosophic logic, the project rejects binary representation and instead constructs images as spaces of simultaneous affirmation, negation, and indeterminacy. These “scapes” are not representations (...) of reality, but ontological propositions: dynamic environments where contradictions coexist, structures destabilize, and meaning emerges through tension rather than resolution. By integrating superhyperstructural thinking with recursive and self-referential visual systems, SuperHyperScapes proposes an expanded ontology of art—one that transcends medium, dissolves disciplinary boundaries, and situates artistic practice within a continuum between system and anomaly, order and breakdown. As both atlas and anti-atlas, the work maps not what reality is, but how it can be simultaneously constructed, deconstructed, and reimagined. (shrink)

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. (shrink)

This book develops a comprehensive and unified theory of SuperHyperStructures and their uncertain extensions, introducing a new paradigm for modeling multi-level, hierarchical, and non-deterministic systems. Extending classical structures and hyperstructures, the framework generalizes operations from single-valued mappings to higher-order mappings of the form f:X^n→P^k (X), enabling the representation of sets of sets and recursively organized relational systems. SuperHyperStructures were introduced by Florentin Smarandache. At its core, the theory addresses fundamental limitations of traditional mathematical models, which are often inadequate (...) for describing systems characterized by hierarchy, interdependence, and uncertainty. By integrating concepts from set theory, hyperstructures, n-ary operations, and neutrosophic logic, the book establishes a rigorous foundation for Uncertain SuperHyperStructures, incorporating truth, indeterminacy, and falsity within a unified formalism. The work systematically develops the theory across multiple domains, including SuperHyperAlgebra, SuperHyperGeometry, SuperHyperLogic, SuperHyperGraphs, and SuperHyperCalculus, and introduces corresponding uncertain and neutrosophic extensions. A comparative framework is provided to position SuperHyperStructures relative to classical structures and hyperstructures, highlighting their increased expressive power and structural depth. Beyond theoretical development, the book demonstrates the applicability of the framework to a wide range of disciplines, including artificial intelligence, complex systems, data science, and physics. In these contexts, SuperHyperStructures enable the modeling of systems-of-systems, multi-agent interactions, hierarchical networks, and multi-scale phenomena under uncertainty. The final chapters explore advanced directions such as SuperHyperCategory Theory and outline open problems, emphasizing the foundational nature of the theory and its potential to support a broad research program. Overall, this work proposes a conceptual and mathematical shift—from flat, deterministic representations to hierarchical, multi-level, and uncertainty-aware structures—offering a unified approach to complexity across scientific domains. (shrink)

The n-th PowerSet of a Set {or Pn(S)} better describes our real world, because a system S (which may be a company, institution, association, country, society, set of objects/plants/animals/beings, set of concepts/ideas/propositions, etc.) is formed by sub-systems, which in their turn by sub-sub-systems, and so on. We prove that the SuperHyperFunction is a generalization of classical Function, SuperFunction, and HyperFunction. And the SuperHyperAlgebra, SuperHyperGraph are part of the SuperHyperStructure. Almost all structures in our real world are Neutrosophic SuperHyperStructures since (...) they have indeterminate/incomplete/uncertain/conflicting data. (shrink)

This book is the fifth volume in the series of Collected Papers on Advancing Uncertain Combinatorics through Graphization, Hyperization, and Uncertainization: Fuzzy, Neutrosophic, Soft, Rough, and Beyond. This volume specifically delves into the concept of Various SuperHyperConcepts, building on the foundational advancements introduced in previous volumes. The series aims to explore the ongoing evolution of uncertain combinatorics through innovative methodologies such as graphization, hyperization, and uncertainization. These approaches integrate and extend core concepts from fuzzy, neutrosophic, soft, and rough set theories, (...) providing robust frameworks to model and analyze the inherent complexity of real-world uncertainties. At the heart of this series lies combinatorics and set theory—cornerstones of mathematics that address the study of counting, arrangements, and the relationships between collections under defined rules. Traditionally, combinatorics has excelled in solving problems involving uncertainty, while advancements in set theory have expanded its scope to include powerful constructs like fuzzy and neutrosophic sets. These advanced sets bring new dimensions to uncertainty modeling by capturing not just binary truth but also indeterminacy and falsity. In this fifth volume, the exploration of Various SuperHyperConcepts provides an innovative lens to address uncertainty, complexity, and hierarchical relationships. It synthesizes key methodologies introduced in earlier volumes, such as hyperization and neutrosophic extensions, while advancing new theories and applications. From pioneering hyperstructures to applications in advanced decision-making, language modeling, and neural networks, this book represents a significant leap forward in uncertain combinatorics and its practical implications across disciplines. -/- The book is structured into 17 chapters, each contributing unique perspectives and advancements in the realm of Various SuperHyperConcepts and their related frameworks: Chapter 1 introduces the concept of Body-Mind-Soul-Spirit Fluidity within psychology and phenomenology, while examining established social science frameworks like PDCA and DMAIC. It extends these frameworks using Neutrosophic Sets, a flexible extension of Fuzzy Sets, to improve their adaptability for mathematical and programming applications. The chapter emphasizes the potential of Neutrosophic theory to address multi-dimensional challenges in social sciences. Chapter 2 delves into the theoretical foundation of Hyperfunctions and their generalizations, such as Hyperrandomness and Hyperdecision-Making. It explores higher-order frameworks like Weak Hyperstructures, Hypergraphs, and Cognitive Hypermaps, aiming to establish their versatility in addressing multi-layered problems and setting a foundation for further studies. Chapter 3 extends traditional decision-making methodologies into HyperDecision-Making and n-SuperHyperDecision-Making. By building on approaches like MCDM and TOPSIS, this chapter develops frameworks capable of addressing complex decision-making scenarios, emphasizing their applicability in dynamic, multi-objective contexts. Chapter 4 explores integrating uncertainty frameworks, including Fuzzy, Neutrosophic, and Plithogenic Sets, into Large Language Models (LLMs). It proposes innovative models like Large Uncertain Language Models and Natural Uncertain Language Processing, integrating hierarchical and generalized structures to advance the handling of uncertainty in linguistic representation and processing. Chapter 5 introduces the Natural n-Superhyper Plithogenic Language by synthesizing natural language, plithogenic frameworks, and superhyperstructures. This innovative construct seeks to address challenges in advanced linguistic and structural modeling, blending attributes of uncertainty, complexity, and hierarchical abstraction. Chapter 6 defines mathematical extensions such as NeutroHyperstructures and AntiHyperstructures using the Neutrosophic Triplet framework. It formalizes structures like neutro-superhyperstructures, advancing classical frameworks into higher-dimensional realms. Chapter 7 explores the extension of Binary Code, Gray Code, and Floorplans through hyperstructures and superhyperstructures. It highlights their iterative and hierarchical applications, demonstrating their adaptability for complex data encoding and geometric arrangement challenges. Chapter 8 investigates the Neutrosophic TwoFold SuperhyperAlgebra, combining classical algebraic operations with neutrosophic components. This chapter expands upon existing algebraic structures like Hyperalgebra and AntiAlgebra, exploring hybrid frameworks for advanced mathematical modeling. Chapter 9 introduces Hyper Z-Numbers and SuperHyper Z-Numbers by extending the traditional Z-Number framework with hyperstructures. These extensions aim to represent uncertain information in more complex and multidimensional contexts. Chapter 10 revisits category theory through the lens of hypercategories and superhypercategories. By incorporating hierarchical and iterative abstractions, this chapter extends the foundational principles of category theory to more complex and layered structures. Chapter 11 formalizes the concept of n-SuperHyperBranch-width and its theoretical properties. By extending hypergraphs into superhypergraphs, the chapter explores recursive structures and their potential for representing intricate hierarchical relationships. Chapter 12 examines superhyperstructures of partitions, integrals, and spaces, proposing a framework for advancing mathematical abstraction. It highlights the potential applications of these generalizations in addressing hierarchical and multi-layered problems. Chapter 13 revisits Rough, HyperRough, and SuperHyperRough Sets, introducing new concepts like Tree-HyperRough Sets. The chapter connects these frameworks to advanced approaches for modeling uncertainty and complex relationships. Chapter 14 explores Plithogenic SuperHyperStructures and their applications in decision-making, control, and neuro systems. By integrating these advanced frameworks, the chapter proposes innovative directions for extending existing systems to handle multi-attribute and contradictory properties. Chapter 15 focuses on superhypergraphs, expanding hypergraph concepts to model complex structural types like arboreal and molecular superhypergraphs. It introduces Generalized n-th Powersets as a unifying framework for broader mathematical applications, while also touching on hyperlanguage processing. Chapter 16 defines NeutroHypergeometry and AntiHypergeometry as extensions of classical geometric structures. Using the Geometric Neutrosophic Triplet, the chapter demonstrates the flexibility of these frameworks in representing multi-dimensional and uncertain relationships. Chapter 17 establishes the theoretical groundwork for SuperHyperGraph Neural Networks and Plithogenic Graph Neural Networks. By integrating advanced graph structures, this chapter opens pathways for applying neural networks to more intricate and uncertain data representations. (shrink)