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Note For Quadripartitioned Neutrosophic Offset, Pentapartitioned Neutrosophic Offset, And Heptapartitioned Neutrosophic Offset, Takaaki Fujita

Neutrosophic Systems with Applications

Neutrosophic sets assign each element three independent membership values—truth, indeterminacy, and falsity—and their partitioned extensions introduce additional components subject to sum-bounded constraints. Notable examples include quadripartitioned, pentapartitioned, and heptapartitioned neutrosophic sets. The O set concept further extends this framework by allowing membership values outside the standard [0, 1] interval, including negative values and values exceeding one. In this paper, we introduce and analyze the Quadripartitioned Neutrosophic O set, Pentapartitioned Neutrosophic O set, and Heptapartitioned Neutrosophic O set. These extensions are intended to enhance the expressiveness of neutrosophic theory and to stimulate further research in neutrosophic and fuzzy uncertainty.

Local-Neutrosophic Logic And Local-Neutrosophic Sets: Incorporating Locality With Applications, Florentin Smarandache, Takaaki Fujita

Branch Mathematics and Statistics Faculty and Staff Publications

The study of uncertainty has been a significant area of research, with concepts such as fuzzy sets [87], fuzzy graphs [51], and neutrosophic sets [58] receiving extensive attention. In Neutrosophic Logic, indeterminacy often arises from real-world complexities. This paper explores the concept of locality as a key factor in determining indeterminacy, building upon the framework introduced by F. Smarandache in [73]. Locality refers to processes constrained within a specific region, where an object or system is directly influenced by its immediate surroundings. In contrast, nonlocality involves effects that transcend spatial or temporal boundaries, where changes in one location have direct …

Neutrosophic Twofold Superhyperalgebra And Anti Superhyperalgebra, Takaaki Fujita, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

Neutrosophic Sets are conceptual frameworks designed to address uncertainty. A Neutrosophic TwoFold Algebra is a hybrid algebraic structure defined over a neutrosophic set, combining classical algebraic operations with neutrosophic components. Concepts such as Hyperalgebra and Superhyperalgebra extend classical Algebra using Power Sets and 𝑛-th powersets. Additionally, structures such as NeutroAlgebra and AntiAlgebra have been defined in recent y ears. This paper explores several related concepts, including TwoFold SuperhyperAlgebra and Anti SuperhyperAlgebra.

Some Types Of Hyperneutrosophic Set (4): Cubic, Trapozoidal, Q-Rung Orthopair, Overset, Underset, And Offset, Florentin Smarandache, Takaaki Fujita

Branch Mathematics and Statistics Faculty and Staff Publications

This paper builds upon the foundational work presented in [38–40]. The Neutrosophic Set provides a comprehensive mathematical framework for managing uncertainty, defined by three membership functions: truth, indeterminacy, and falsity. Recent advancements have introduced extensions such as the Hyperneutrosophic Set and the SuperHyperneutrosophic Set, which are specifically designed to address increasingly complex and multidimensional problems. The formal definitions of these sets are available in [30]. In this paper, we extend the Neutrosophic Cubic Set, Trapezoidal Neutrosophic Set, q-Rung Orthopair Neutrosophic Set, Neutrosophic Overset, Neutrosophic Underset, and Neutrosophic Offset using the frameworks of the Hyperneutrosophic Set and the SuperHyperneutrosophic Set. Furthermore, …

Some Types Of Hyperneutrosophic Set (3): Dynamic, Quadripartitioned, Pentapartitioned, Heptapartitioned, M-Polar, Takaaki Fujita, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

This paper builds upon the foundation established in [50, 51]. The Neutrosophic Set provides a robust mathematical framework for handling uncertainty, defined by three membership functions: truth, indeterminacy, and falsity. Recent developments have introduced extensions such as the Hyperneutrosophic Set and SuperHyperneutrosophic Set to tackle increasingly complex and multidimensional problems. In this study, we explore further extensions, including the Dynamic Neutrosophic Set, Quadripartitioned Neutrosophic Set, Pentapartitioned Neutrosophic Set, Heptapartitioned Neutrosophic Set, and m-Polar Neutrosophic Set, to address advanced challenges and applications.

Plithogenic Duplets And Plithogenic Triplets, Takaaki Fujita, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

A Neutrosophic Set is a mathematical framework that represents degrees of truth, indeterminacy, and falsehood to address uncertainty in membership values [41, 42]. In contrast, a Plithogenic Set extends this concept by incorporating attributes, their possible values, and the corresponding degrees of appurtenance and contradiction [50]. Among the related concepts of Neutrosophic Sets, Neutrosophic Duplets and Neutrosophic Triplets are well-known. This paper defines Plithogenic Duplets and Plithogenic Triplets as extensions of these concepts using the Plithogenic Set framework and briefly examines their relationship with existing concepts.

Some Types Of Hyperneutrosophic Set (2): Complex, Single-Valued Triangular, Fermatean, And Linguistic Sets, Takaaki Fujita, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

This paper is a continuation of the work presented in [35]. The Neutrosophic Set provides a mathematical framework for managing uncertainty, characterized by three membership functions: truth, indeterminacy, and falsity. Recent advancements have introduced extensions such as the Hyperneutrosophic Set and SuperHyperneutrosophic Set to address more complex and multidimensional challenges. In this study, we extend the Complex Neutrosophic Set, Single-Valued Triangular Neutrosophic Set, Fermatean Neutrosophic Set, and Linguistic Neutrosophic Set within the frameworks of Hyperneutrosophic Sets and SuperHyperneutrosophic Sets. Furthermore, we investigate their mathematical structures and analyze their connections with other set-theoretic concepts.

Some Types Of Hyperneutrosophic Set (1): Bipolar, Pythagorean, Double-Valued, Interval-Valued Set, Takaaki Fujita, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

The Neutrosophic Set is a mathematical framework designed to manage uncertainty, characterized by three membership functions: truth (T), indeterminacy (I), and falsity (F). In recent years, extensions such as the Hyperneutrosophic Set and SuperHyperneutrosophic Set have been introduced to address more complex scenarios. This paper proposes new concepts by extending Bipolar Neutrosophic Sets, Interval-Valued Neutrosophic Sets, Pythagorean Neutrosophic Sets, and Double-Valued Neutrosophic Sets using the frameworks of Hyperneutrosophic and SuperHyperneutrosophic Sets. Additionally, a brief analysis of these extended concepts is presented.

Some Types Of Hyperneutrosophic Set (7): Type-M, Nonstationary, Subset-Valued, And Complex Refined, Takaaki Fujita, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

This paper builds upon the foundational advancements introduced in [26,39–43]. TheNeutrosophic Set provides a versatile mathematical framework for addressing uncertainty through its three membership functions: truth, indeterminacy, and falsity [84]. Extensions such as the Hyperneutrosophic Set and the SuperHyperneutrosophic Set have been recently proposed to address increasingly complex and multidimensional problems. Detailed formal definitions of these concepts can be found in [33]. In this paper, we extend the Type-𝑚, Nonstationary, Subset-Valued, and Complex Refined Neutrosophic Sets using the Hyperneutrosophic Set and the SuperHyperneutrosophic Set frameworks.

Superhypergraph Neural Networks And Plithogenic Graph Neural Networks: Theoretical Foundations, Takaaki Fujita, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

Hypergraphs extend traditional graphs by allowing edges to connect multiple nodes, while superhypergraphs further generalize this concept to represent even more complex relationships. Neural networks, inspired by biological systems, are widely used for tasks such as pattern recognition, data classification, and prediction. Graph Neural Networks (GNNs), a well-established framework, have recently been extended to Hypergraph Neural Networks (HGNNs), with their properties and applications being actively studied. The Plithogenic Graph framework enhances graph representations by integrating multi-valued attributes, as well as membership and contradiction functions, enabling the detailed modeling of complex relationships. In the context of handling uncertainty, concepts such as …

Some Types Of Hyperneutrosophic Set (6): Multineutrosophic Set And Refined Neutrosophic Set, Takaaki Fujita, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

This paper builds on the foundational advancements introduced in [22, 29–32]. The Neutrosophic Set pro-vides a flexible mathematical framework for managing uncertainty by utilizing three membership functions: truth, indeterminacy, and falsity. Recent extensions, such as the HyperNeutrosophic Set and the SuperHy-perNeutrosophic Set, have been developed to address increasingly complex and multidimensional challenges. Comprehensive formal definitions of these concepts are provided in [26]. In this paper, we further extend various specialized classes of Neutrosophic Sets. Specifically, we explore extensions of the MultiNeutrosophic Set and the Refined Neutrosophic Set using HyperNeutrosophic Sets and 𝑛-SuperHyperNeutrosophic Sets, providing detailed analysis and examples.

Some Types Of Hyperneutrosophic Set (5): Support, Paraconsistent, Faillibilist, And Others, Takaaki Fujita, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

This paper builds upon the foundational advancements introduced in [14, 25–27]. The Neutrosophic Set offers a versatile mathematical framework for addressing uncertainty through its three membership functions: truth, indeterminacy, and falsity. Extensions such as the Hyperneutrosophic Set and the SuperHyperneutrosophic Set have been recently proposed to tackle increasingly sophisticated and multidimensional problems. Detailed formal definitions of these concepts can be found in [20]. In this paper, we extend various specialized classes of Neutrosophic Sets—namely, the Support Neutrosophic Set, Neutrosophic Intuitionistic Set (distinct from the Intuitionistic Fuzzy Set), Neutrosophic Paraconsistent Set, Neutrosophic Faillibilist Set, Neutrosophic Paradoxist Set, Neutrosophic Pseudo-Paradoxist Set, Neutrosophic …

Neutrosophic Treesoft Expert Set And Forestsoft Set, Takaaki Fujita, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

Concepts such as Fuzzy Sets [28,57],Neutrosophic Sets [42,44], and Plithogenic Sets [48] have been extensively studied to address uncertainty, finding diverse applications across various fields. The Soft Set provides a framework that associates each parameter with subsets of a universal set, enabling flexible approximations [31]. The TreeSoft Set extends the Soft Set by introducing hierarchical, tree-structured parameters, allowing for multi-level data representation [53]. In this paper, we revisit the concept of the Neutrosophic TreeSoft Set, which has been discussed in other studies [8, 34]. Additionally, we propose and examine the Neutrosophic TreeSoft Expert Set by incorporating the framework of the …

Reconsideration Of Neutrosophic Social Science And Neutrosophic Phenomenology With Non-Classical Logic, Takaaki Fujita, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

Body-Mind-Soul-Spirit Fluidity is a concept rooted in psychology and phenomenology, offering significant insights into human decision-making and well-being. Similarly, in social analysis and social sciences, frameworks such as PDCA, DMAIC, SWOT, and OODA have been established to enable structured evaluation and effective p roblem-solving. Furthermore, in phenomenology and social sciences, various logical systems have been developed to address specific objectives and practical applications. This paper extends these concepts using the Neutrosophic theory, revisiting their mathematical definitions and exploring their properties. The Neutrosophic Set, an extension of the Fuzzy Set, is a highly flexible framework that has been widely studied in …

On Neutrosophic Pδs-Irresolute Functions In Neutrosophic Topological Spaces, Bishnupada Debnath, Anjan Mukherjee

Neutrosophic Systems with Applications

In general topology the notion of pds-irresolute and aδs-irresolute functions were introduced by Beceren and Noiri. In the present paper, these concepts of pδs-irresolute (briefly, Npδs-irresolute) and aδs-irresolute (briefly, Naδs-irresolute) functions are explored for the first time in neutrosophic topological spaces (NTS) as generalized version. We proved that every Naδs-irresolute function is Npδs-irresolute function but not conversely. Some characterizations, counter examples, and fundamental features are also presented. By neutrosophic pre-open, neutrosophic δ-open, and neutrosophic δ-semi-open sets, some new fundamental properties of such functions are provided. Furthermore, under Npδs-irresolute functions, the behavior of neutrosophic semi-connected, neutrosophic pre-connected, neutrosophic pre-T2, neutrosophic δ-semi-T2, …

Neutrosophic Approach To Water Quality Assessment: A Case Study Of Gomati River, The Largest River In Tripura, India, Ajoy Kanti Das, Nandini Gupta, Carlos Granados, Rakhal Das, Suman Das

Neutrosophic Systems with Applications

This study addresses the complexity of assessing river water quality, a multifaceted process influenced by numerous water quality parameters (WQPs) characterized by inherent uncertainties and diverse judgment information from decision-makers. These uncertainties and diverse judgment information can be effectively represented and simulated using Neutrosophic sets. In this study, we propose an effective water pollution rating system, the Neutrosophic water quality index (NWQI), to derive a water pollution score (NWQI-score) for rating water pollution levels. We demonstrate the application of our methodology through an assessment of water quality indices for rating pollution in the Gomati River, the largest river in Tripura, …

Neutrosophic Approach To Water Quality Assessment: A Case Study Of Gomati River, The Largest River In Tripura, India, Ajoy Kanti Das, Nandini Gupta, Carlos Granados, Rakhal Das, Suman Das

Neutrosophic Systems with Applications

This study addresses the complexity of assessing river water quality, a multifaceted process influenced by numerous water quality parameters (WQPs) characterized by inherent uncertainties and diverse judgment information from decision-makers. These uncertainties and diverse judgment information can be effectively represented and simulated using Neutrosophic sets. In this study, we propose an effective water pollution rating system, the Neutrosophic water quality index (NWQI), to derive a water pollution score (NWQI-score) for rating water pollution levels. We demonstrate the application of our methodology through an assessment of water quality indices for rating pollution in the Gomati River, the largest river in Tripura, …

Single-Valued Neutrosophic Mcdm Approaches Integrated With Merec And Ram For The Selection Of Uavs In Forest Fire Detection And Management, Mai Mohamed, Amira Salam, Jun Ye, Rui Yong

Neutrosophic Systems with Applications

In recent times, the world has experienced a rise in the frequency of forest fires. These fires cause severe economic damage and pose a significant threat to human lives. Therefore, it is essential to search for solutions that can help combat fires and detect them early. Once a fire reaches a certain level, it becomes challenging to control it. Various systems have been proposed to collect data and detect forest fires, such as satellites and other traditional methods. However, these solutions have been ineffective in terms of cost, coverage of large areas, accuracy, and the safety of human lives. To …

Single-Valued Neutrosophic Mcdm Approaches Integrated With Merec And Ram For The Selection Of Uavs In Forest Fire Detection And Management, Mai Mohamed, Amira Salam, Jun Ye, Rui Yong

Neutrosophic Systems with Applications

In recent times, the world has experienced a rise in the frequency of forest fires. These fires cause severe economic damage and pose a significant threat to human lives. Therefore, it is essential to search for solutions that can help combat fires and detect them early. Once a fire reaches a certain level, it becomes challenging to control it. Various systems have been proposed to collect data and detect forest fires, such as satellites and other traditional methods. However, these solutions have been ineffective in terms of cost, coverage of large areas, accuracy, and the safety of human lives. To …

Nidus Idearum. Scilogs, Xiii: Structure / Neutrostructure / Antistructure, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

In this thirteenth book of scilogs – one may find topics on Neutrosophy, Plithogeny, Physics, Mathematics, Philosophy – email messages to research colleagues, or replies, notes, comments, remarks about authors, articles, or books, spontaneous ideas, and so on. It presents new types of soft sets and new types of topologies.

Exchanging ideas with Mohammad Abobala, Ishfaq Ahmad, Ibrahim M. Almanjahie, Fatimah Alshahrani, Nizar Altounji, Muhammad Aslam, Said Broumi, Victor Christianto, R. Diksh, Feng Liu, Frank Julian Gelli, Erick Gonzalez Caballero, Riad Hamido, Yaser Al-Hasan, Ahmed Hatip, Yasin Karmouta, Nivetha Martin, Preda Mihăilescu, V. Lakshmana Gomathi Nayagam, Ze Carlos Tiago de …

A Comparative Study On X-Ray Image Enhancement Based On Neutrosophic Set, Nihal N.Mostafa, Amit Krishan Kumar, Yasir Ali

Sustainable Machine Intelligence Journal

Medical image noise, ambiguity, and fuzziness pose challenges to the medical image analysis process. To lessen these problems, fuzzy sets are utilized; however, they frequently ignore the pixel's spatial context. The neutrosophic set (NS) is employed in picture denoising to get over these restrictions. Neutronosophic theory, in particular the NS Bilateral filter, NS Wiener filter, NS Median filter, NS gaussian filter, and NS rank-ordered filter, is covered in this paper. The Lung-cancer dataset of X-Ray images is used to assess the performance of different denoising techniques. The effectiveness of NS-based denoising techniques over conventional techniques is demonstrated through comparisons with …

Ct Image Segmentation Using Optimization Techniques Under Neutrosophic Domain, Doaa El-Shahat, Nourhan Talal, Jun Ye, Wen-Hua Cui

Neutrosophic Systems with Applications

In this paper, we introduce a hybrid technique between optimization algorithms and neutrosophic theory. This new hybridization can deal with uncertainties in brain computed tomography (CT) images in three different memberships very effectively. To prove the real-time application of this theory, a new segmentation method for brain CT medical images is presented. The grayscale medical image suffers from uncertainties and inconsistencies in the gray levels due to their bad luminance. The proposed technique addressed this problem by performing neutrosophic operations on gray levels based on the S membership function.

Q-Rung Neutrosophic Sets And Topological Spaces, Michael Gr. Voskoglou, Florentin Smarandache, Mona Mohamed

Neutrosophic Systems with Applications

The concept of the q-rung orthopair neutrosophic set is introduced in this paper, and fundamental properties of it are studied. Also, the ordinary notion of topological space is extended to the q-rung orthopair neutrosophic environment, as well as the fundamental concepts of convergence, continuity, compactness, and Hausdorff topological space. All these generalizations are illustrated by suitable examples.

Survey Of Planar And Outerplanar Graphs In Fuzzy And Neutrosophic Graphs, Takaaki Fujita, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

As many readers may know, graph theory is a fundamental branch of mathematics that explores networks made up of nodes and edges, focusing on their paths, structures, and properties [196]. A planar graph is one that can be drawn on a plane without any edges intersecting, ensuring planarity. Outerplanar graphs, a subset of planar graphs, have all their vertices located on the boundary of the outer face in their planar embedding. In recent years, outerplanar graphs have been formally defined within the context of fuzzy graphs. To capture uncertain parameters and concepts, various graphs such as fuzzy, neutrosophic, Turiyam, and …

Ermakov Equations Can Be Derived From Zel’Dovich Pancake, And They Are Cold And Nonlocal Through Using Neutrosophic Venn Diagram, Victor Christianto, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

As we argue in a previous article, the labyrinthine worlds of Jorge Luis Borges are more than captivating narratives; they are portals to a deeper understanding of existence. By weaving elements of science-fiction fantasy with philosophical and ethical inquiries, Borges's short stories bridge the seemingly disparate realms of physics and the humanities, offering fertile ground for contemporary physics research. The present-day universe consists of galaxies, galaxy clusters, one-dimensional filaments and two-dimensional sheets or pancakes, all of which combine to form the cosmic web. The so called ”Zeldovich pancakes”, are very difficult to observe, because their overdensity is only slightly greater …

Q-Rung Neutrosophic Sets And Topological Spaces, Michael Gr. Voskoglou, Florentin Smarandache, Mona Mohamed

Branch Mathematics and Statistics Faculty and Staff Publications

The concept of q-rung orthopair neutrosophic set is introduced in this paper and fundamental properties of it are studied. Also the ordinary notion of topological space is extended to q-rung orthopair neutrosophic environment, as well as the fundamental concepts of convergence, continuity, compactness and Hausdorff topological space. All these generalizations are illustrated by suitable examples.

A Neutrosophic Approach For B-Spline Curve By Using Interpolation Method, Siti Nur Idara Binti Rosli, Mohammad Izat Emir Bin Zulkifly

Neutrosophic Systems with Applications

This study introduces a B-spline curve interpolation model based on the neutrosophic set technique. To begin, the neutrosophic notion is used to define the neutrosophic control point relation. After that, the neutrosophic control point is combined with the B-spline basis function. Besides, the neutrosophic B-spline curve interpolation model is illustrated using the interpolation method. Furthermore, an example and the methods are shown for creating the right curve.

Ranking And Analysis The Strategies Of Crowd Management To Reduce The Risks Of Crushes And Stampedes In Crowded Environments And Ensure The Safety Of Passengers, Waleed Tawfiq Al-Nami

Neutrosophic Systems with Applications

Public gatherings, transit hubs, stadiums, and crowded retail malls are just a few examples of places where crowd management has become an urgent issue in recent years. Effective crowd management strategies have been required due to the increasing population, urbanization, and frequency of large-scale meetings. These strategies are used in dynamic, sometimes chaotic, circumstances to protect people and facilitate their free movement. The purpose of this study is to analyze and rank various strategies for crowd management to reduce the risks of crushes and stampedes, improve security, and facilitate smoother traffic flow. This study used the single-valued neutrosophic set to …

An Investigative Study On Quick Switching System Using Fuzzy And Neutrosophic Poisson Distribution, Uma G, Nandhitha S

Neutrosophic Systems with Applications

Stephens and Larson (1967) stated the sampling system as an allotted grouping of two or three sampling plans and the rules for switching between the plans for sentencing the lots of manufactured products. Quick Switching System (QSS) by Romboski (1969) is a sampling system with reference to the Single Sampling Plan (SSP) involves normal and tightened plans by adopting a switching rule. QSS provides quality protection and a reduction in the cost of inspection. The idea of fuzzy logic is adopted in the system to handle situations of fraction non-confirmation, uncertainty, or vagueness present in the parameters. The sampling plans …

An Investigative Study On Quick Switching System Using Fuzzy And Neutrosophic Poisson Distribution, Uma G, Nandhitha S

Neutrosophic Systems with Applications

Stephens and Larson (1967) stated the sampling system as an allotted grouping of two or three sampling plans and the rules for switching between the plans for sentencing the lots of manufactured products. Quick Switching System (QSS) by Romboski (1969) is a sampling system with reference to the Single Sampling Plan (SSP) involves normal and tightened plans by adopting a switching rule. QSS provides quality protection and a reduction in the cost of inspection. The idea of fuzzy logic is adopted in the system to handle situations of fraction non-confirmation, uncertainty, or vagueness present in the parameters. The sampling plans …