This paper develops a topological framework based on type-1 neutrosophic sets, extending the study of uncertainty beyond classical and intuitionistic fuzzy structures. We investigate fundamental operators—interior, closure, and exterior—on type-1 neutrosophic sets X[I], and establish separation axioms for type-1 topological spaces Ti[I], i = 0, 1, . . . , 4). Furthermore, we analyze continuous and contra type-1 neutrosophic functions, exploring their structural properties within this framework. By isolating type-1 neutrosophic sets and formalizing their topological behavior, our results provide new insights into neutrosophic topology and lay the groundwork for broader applications in mathematical analysis, algebra, and related fields. \
The present manuscript delves into the ideology of best proximity point under the purview of neutrosophic fuzzy metric space. For this purpose, the concept of a new type of mapping named α-proximal admissible mapping has been defined under the realm of the said space. Also for the contractive condition, the pre-existent ideologies of Ciric and rational contraction has been combined. Notably, for the sake of best proximity, concept of α-ψ proximal contraction has also been considered. Here α and ψ can also be regarded as control functions. Additionally, some examples and applications are also deduced in support of the established results, for some particular conditions. To the best of known literatures, the results are established for the first time in the mentioned area.
This paper addresses the problem of utilizing auxiliary information to estimate the population mean under neutrosophic stratified random sampling. Within the framework of neutrosophic statistics, we propose a neutrosophic combined logarithmic type estimator that effectively accounts for the uncertainty and indeterminacy inherent in survey sampling. The suggested estimator’s bias and mean squared error (MSE) are expressed up to the first order of approximation, and ideal circumstances for reducing the MSE are determined. Neutroposophic combined mean, neutrosophic combined ratio, and neutrosophic combined difference estimators are theoretically compared with the suggested estimator.The analytical findings show that in some real-world situations, the suggested estimator performs better than its conventional competitors. Additionally, line and radar chart representations and empirical data analysis are used to assess the performance of the suggested estimator. The results provide useful information for practitioners and survey statisticians who use Neutrosophic stratified random sampling when uncertainty is present.
Decision-making in complex environments often involves ambiguity, inconsistency, and incomplete information—conditions inadequately addressed by classical fuzzy or even interval-valued neutrosophic systems. Building upon the earlier foundations established by Kamal Nasir and Priyanka (2025) in their works on the Multi-Expert, Multi-Criteria Neutrosophic Fuzzy Assignment Problem and its Interval-Valued extension, this study introduces a Generalized Neutrosophic Fuzzy Assignment Model (GNFAM) designed to operate effectively under deep uncertainty.The proposed framework extends the neutrosophic structure to generalized neutrosophic numbers, enabling the representation of varying degrees of truth, indeterminacy, and falsity in flexible forms (single-valued, interval-valued, or polygonal). A multi-expert, multi-criteria evaluation mechanism is integrated through a Priority-Optimized Aggregation Mean, allowing the model to synthesize heterogeneous expert judgments across diverse decision dimensions. The decision process is optimized using a generalized score function, facilitating precise ranking and assignment under uncertain and conflicting data.A real-world case study illustrates the model’s applicability and robustness, demonstrating superior performance compared to conventional fuzzy and interval-valued neutrosophic assignment frameworks. The proposed GNAM framework thus advances neutrosophic decision science by providing a unified, extensible, and computationally efficient tool for multi-criteria decision-making under deep uncertainty.
In this paper, the fourth dimension of neutrosophy, that of "Turiyam," is examined as it symbolizes the "absolute" or "transcendental" essence that preexists and permeates the truth triad, indeterminacy, and the falsity trio. Although the current state of Neutrosophic Sets works perfectly to represent the notion of "ambiguity," they are inefficient in incorporating the essence wherein such truth measures oscillate. Thus, the proposed model, Fuzzy Neutrosophic Turiyam Set (FNTS), will have its corresponding algebraic operator functions and "Law of Turiyam Non-Exclusion." Moreover, models involving distance function in four-dimensional space as well as the hyper-polyhedron will be considered. Quantum computing applications and modeling of socioeconomic "Black Swan" events validate that the proposed FNTS model is significantly more stable in chaotic situations than the current models proposed for neutrosophic settings.
Assessing the digital maturity of businesses plays a crucial role in their development in the digital environment. Neutrosophic sets have been widely applied in many problems to address uncertainty in complex realities. The Combined Compromise Solution (CoCoSo) techniques have been used to address multi-criteria decision making problem. In this paper, an hybrid model combining the CoCoSo technique, entropy, and score function is used to address multi-criteria decision making problem under interval Neutrosophic environment. Information entropy is used to manage the weighted values of criteria in the decision-making model, and the score function is used to transform the evaluation into interval value Neutrosophic in the CoCoSo technique. Finally, the digital maturity level of a business is evaluated using proposed integrated model.
In this study, we define the basic operations of IndetermSoft Operators, viz., ∂−subset, ∂−superset, ∂−equal, ∂−union, and ∂−intersection, which helps define the fundamental terms of an IndetermSoft Set, including the equality of two IndetermSoft Sets, subset, superset, and complement of an IndetermSoft Set. In addition, we define some operations, such as union, intersection, ‘AND’ and, ‘OR’ operations of two IndetermSoft Sets and also verify the validity of De Morgan’s law in IndetermSoft Set theory.
In this paper, we introduced some types of fuzzy neutrosophic strongly category sets and their characterizations and examples are obtained. Also we discuss fuzzy neutrosophic μ – strongly Baire Spaces and its properties are to be described.
In this paper, we investigate the algebraic structure of Q-neutrosophic soft quasigroups as an extension of Q-neutrosophic soft sets to non-associative systems. We establish several fundamental properties of these structures. In particular, we prove that the intersection of two Q-neutrosophic soft quasigroups is itself a Q-neutrosophic soft quasigroup, whereas their union is not necessarily closed under quasigroup operations. The conditions under which the product, left division, and right division of two Q-neutrosophic soft quasigroups remain Q-neutrosophic soft quasigroups, particularly within entropic quasigroups, are established. We further derive necessary and sufficient conditions for a Q-neutrosophic soft groupoid to become a Q-neutrosophic soft quasigroup and examine structural properties such as idempotency, unipotency, 3-power associativity, and n-power associativity in relation to membership degrees. Using these results, we construct a decision-making algorithm that models uncertainty through truth-, indeterminacy-, and falsity-membership functions. An application to medical decision processes demonstrates how quasigroup operations can systematically combine indeterminate and interacting clinical information. The findings show that Q-neutrosophic soft quasigroups provide a rigorous mathematical framework for analyzing indeterminate, inconsistent, and non-associative data across two universal sets, offering enhanced modeling capabilities for complex real-world systems..
Neutrosophy represents a comprehensive philosophical and mathematical framework for the study of neutrality, indeterminacy, and the interaction between opposites. The subsequent development of Refined Neutrosophy and Refined Neutrosophication significantly expanded the applicability of neutrosophic concepts to decision sciences, artificial intelligence, engineering, economics, and uncertainty modeling. This paper demonstrates that several influential theories developed independently in decision sciences and systems analysis—namely Regret Theory, Grey System Theory, and Three-Way Decision Theory—can be interpreted as particular cases of Neutrosophication. A formal Neutrosophication Operator is introduced to show how binary frameworks are transformed into triadic neutrosophic structures. We further demonstrate that Three-Way Decision models arise naturally from Neutrosophic Probability and that n-Way Decision models emerge as particular cases of Refined Neutrosophy. The proposed framework establishes Neutrosophy as a unifying meta-theory capable of integrating multiple approaches to uncertainty, partial knowledge, and decision-making under indeterminate conditions.
This paper proposes an extension of the Neutrosophic Theory of Evolution into the field of evolutionary biogeography. Using the freshwater crab Potamon fluviatile as a case study, we analyze how evolutionary, involutionary, and indeterminate processes interact across geological, ecological, and spatial scales. The species originated from marine ancestors in the ancient Tethys Ocean and later colonized freshwater ecosystems across the Mediterranean basin. A long-isolated population inhabiting subterranean canals beneath the ruins of Rome exhibits unusual body size and longevity, suggesting a form of island-like gigantism. We argue that such evolutionary phenomena can be interpreted within a neutrosophic framework where adaptation involves simultaneous processes of evolution, involution, and indeterminacy. This study represents the first formal applications of neutrosophic theory to evolutionary biogeography.
We introduce a new class of open sets in neutrosophic topological spaces, called neutrosophic I-open (NI-open) sets, together with their dual NI-closed sets, defined via neutrosophic δ-interior and δ-closure operators. We establish basic stability (arbitrary unions of NI-open sets and arbitrary intersections of NI-closed sets), give characterizations in terms of the associated interior/closure operators, and prove a decomposition formula expressing every NI-open set as the union of its N δS-interior and N δβ-interior. We further locate NI-openness within the existing landscape by showing that N δS-open, N δP -open, N-eopen, and Ne∗-open sets are, in general, contained in NI-open sets, while the converses may fail; illustrative examples are provided. Finally, we present a neutrosophic multi-criteria assessment for supplier selection that combines the neutrosophic score with its negative counterpart to rank alternatives under ambiguity and indeterminacy, including a worked example with per-strategy rankings and an assignment rule.
We introduce and study two classes of neutrosophic continuous mappings: the neutrosophic irresolute δ-continuous mappings (NIr δ–CM) and the neutrosophic contra δ-continuous mappings (NC δ–CM). We establish their fundamental properties and provide characterizations in terms of preimages of δ-open and δ-closed sets. The interplay between the two notions is analyzed through implication chains, (non-)equivalences under mild hypotheses, and stability results under composition, subspaces, and products. We then extend the framework to the setting by defining neutrosophic irresolute δ-continuous mappings (NIr δ–CM) and neutrosophic contra δ-continuous mappings (NC δ–CM), showing how core properties lift to the case and where genuinely new phenomena arise. Throughout, examples and counterexamples are provided to separate the classes and to illustrate the sharpness of the obtained results.
In modern engineering systems, uncertainties and incomplete information often pose significant challenges in accurately evaluating system reliability. Traditional probabilistic and fuzzy approaches are insufficient to effectively capture indeterminacy and inconsistency inherent in real-world environments. To address these limitations, this study proposes a neutrosophic framework for fuzzy reliability analysis of a robotic system using single-valued neutrosophic sets (SVNS). In the proposed approach, the reliability of each system component is represented by a neutrosophic triplet characterized by truth, indeterminacy, and falsity membership degrees. A fuzzy success fault tree model is developed to analyze the reliability structure of the robotic system, incorporating both series and parallel configurations of components such as motors, sensors, rollers, and bearings. The overall system reliability is computed using neutrosophic aggregation operators. Furthermore, to ensure a comprehensive and consistent evaluation, three decision-making measures namely score, accuracy, and certainty functions are employed. These functions establish a total order on neutrosophic numbers, enabling complete and unambiguous assessment of system reliability. A numerical example of a robotic system is presented to demonstrate the applicability and effectiveness of the proposed methodology. The results indicate that the neutrosophic-based approach provides a more flexible, robust, and realistic representation of uncertainty compared to conventional fuzzy reliability models. The inclusion of the certainty function enhances the decision-making process by ensuring completeness in ranking and improving the reliability evaluation framework.
The classical Lyapunov matrix equation (LME) is important for checking system stability, but it assumes we know exact numbers. It cannot handle unclear, conflicting, or missing information. Most of real-world application of control systems have uncertainty and unpredictability that makes it hard to analyze the stability precisely using conventional mathematical models. The classical Lyapunov matrix equation is important in determining the system stability but it fails to accommodate vague, inconsistent, or incomplete information. Therefore, this study introduces a solution for the Fully Fuzzy Neutrosophic Lyapunov Matrix Equation (FFNLME) to address the limitation. The proposed model extends the classical model of LME into the neutrosophic fuzzy domain by incorporating three independent membership components which are Truth (T ), Indeterminacy (I), and Falsity (F ). Left right-triangular neutrosophic fuzzy numbers (LR-TriNFN) is utilized to represent uncertain system parameters. Associated linear systems (ALS) and score function method are employed for deneutrosophication, allowing for computational analysis while preserving the embedded uncertainty. The study demonstrates that the FFNLME framework provides a more comprehensive and flexible representation of uncertain systems, enhancing the robustness of stability analysis compared to classical or fuzzy Lyapunov equations. This formulation can be applied to both linear and nonlinear systems in control engineering, offering an effective means to evaluate stability under ambiguous or incomplete information. Future research may focus on developing optimized numerical algorithms and extending this approach to large-scale and time-varying systems.
This paper investigates the analytical properties of generalized discrete probability series, Neutrosophic Poisson distributions, and sigmoid functions. We introduce a novel Advanced Differential-Sigmoid Operator based on Mersenne polynomials to enhance analytical precision and structural adaptability. Utilizing this operator, we derive several functional inequalities, growth estimates, and coefficient bounds for associated classes of analytic univalent functions. Furthermore, we explore convex combinations and the geometric behavior of the sigmoid function within this framework. By integrating neutrosophic logic, special functions, and operator theory, this research provides a unified framework that advances the treatment of uncertainty and supports practical applications in statistical modeling, signal processing, and neural networks.
In this paper, we introduce a method for determining the generalized inverse (g-inverse) and the Moore-Penrose inverse of Neutrosophic Hyper Soft Rough Fuzzy Matrices (NHSRFMs), along with the necessary conditions. Furthermore, no algorithm currently exists to find the g-inverse of an NHSRFM. In this study, we present an algorithm to evaluate the g-inverse of an NHSRFM. Several properties and results related to the g-inverse of NHSRFMs are explored. This paper concludes with an application of the g-inverse, supported by numerical examples that illustrate the theorems, algorithm, and application.
Accurate segmentation of brain tumors in MRI scans is important for diagnosis and treatment, but noise, overlapping tissues, low contrast, and unclear boundaries make it difficult for traditional image processing methods. Our idea is to present a robust method for brain tumor segmentation using Refined Neutrosophic Set (NRS). This approach builds on the traditional neutrosophic framework by breaking down truth, indeterminacy, and falsity into multiple subcomponents for better handling of uncertainty. The input MRI images is transformed into the neutrosophic refined set domain, and then iterative refinement of truth and indeterminacy is applied. This method effectively handles uncertainty, reduces ambiguity, and produces clearer, more accurate tumor boundaries. Experimental results on benchmark MRI datasets demonstrate that the NRS-based method achieves higher Dice similarity scores and lower Hausdroff distance.
Artificial Intelligence systems are increasingly required to operate in environments characterized by incomplete, ambiguous, and contradictory information. Traditional probabilistic and fuzzy-logic approaches often compress uncertainty into a single scalar value, limiting their ability to distinguish between ignorance, conflict, and evidential support. This paper explores the conceptual and practical relationship between Neutrosophic Logic and modern Artificial Intelligence architectures. The neutrosophic framework represents knowledge through three independent dimensions—Truth (T), Indeterminacy (I), and Falsity (F)—thereby providing a richer representation of uncertainty than conventional binary or probabilistic models. The study examines how neutrosophic triplets can be implemented as software primitives and applied in Retrieval-Augmented Generation (RAG) systems, multi-agent architectures, cybersecurity workflows, and AI-assisted decision-making. Particular attention is given to the role of indeterminacy as a measurable and actionable variable that enables systems to recognize incomplete knowledge, avoid hallucinations, and improve reliability. The paper argues that the integration of Neutrosophic principles into AI engineering offers a promising pathway toward more robust, transparent, and trustworthy intelligent systems capable of operating under real-world uncertainty.
This research presents a systematic review of studies in mathematics education and problem-solving that address pedagogical work with children with a diagnosis of attention deficit and hyperactivity disorder. Through a historical-logical methodological analysis and a rigorous bibliographic search, significant contributions to the development of mathematical learning in this population were identified. The study highlights the main predictors of difficulties in mathematical performance, associated behavioral aspects, effective pedagogical models, and successful cases and didactic strategies that have shown significant progress, particularly in teaching through mathematical problem-solving. Additionally, a neutrosophic logic framework was applied to assess the degree of certainty, indeterminacy, and contradiction within each evidence category, providing a tripartite epistemic characterization of the reviewed literature.
