International Journal of Neutrosophic Science

This article presents a focused investigation of type-1 neutrosophic sets, derived from classical sets by introducing an indeterminacy component, I. type-1 neutrosophic sets generalize classical set theory by incorporating four-valued logic, which was generated by Boolean logic in our work. This work will appear in the future. As we know, a neutrosophic set is based on a many-valued logic defined by three independent membership functions: truth, indeterminacy, and falsehood. This work systematically re-examines and consolidates foundational research conducted between 2024 and 2025, isolating type-1 structures from the broader frameworks of type-2 and type-3 neutrosophic sets for clearer axiomatic and theoretical development. We establish core concepts, terminology, operations, and properties specific to type-1 neutrosophic sets, constructing and analyzing the type-1 neutrosophic Cartesian product. In addition, we introduce and investigate the properties of type-1 neutrosophic ordered pairs and their corresponding products. This foundation formally defines type-1 neutrosophic relations and neutrosophic partially ordered relations, establishing their core properties. Furthermore, the article explores type-1 neutrosophic functions, detailing their various types, including injective,surjective, and bijective functions and their respective properties. A significant focus is placed on invertible neutrosophic functions, where we examine the conditions for invertibility and prove key related theorems.By focusing exclusively on type-1, we aim to create a more dynamic and effective foundation for application across diverse neutrosophic fields, including neutrosophic algebra, number theory, and logic. This focused approach is intended to open new research pathways within the neutrosophic science.

Energy consumption in large-scale distributed computing has become a first-order design constraint, affecting operational costs, carbon emissions, and service reliability. This paper proposes a hybrid optimization framework that combines Linear Programming (LP) for feasible solution seeding with a Hybrid Genetic–Simulated Annealing (HGSA) metaheuristic for global search. The objective is to minimize total energy while preserving Quality of Service (QoS) and Service-Level Agreement (SLA) constraints. We adopt a widely used server power model that relates power to utilization and extend it with an optional carbon-aware objective that weights power by time- and location-varying grid carbon intensity. Decision variables include task–node–time assignments and, optionally, per-host frequency states for dynamic voltage and frequency scaling (DVFS). The proposed HGSA leverages LP-based seeding to accelerate convergence, applies crossover and mutation operators to explore the search space, and uses simulated annealing to refine solutions and escape local optima. We evaluate the approach using Google Cluster traces and CloudSim Plus, reporting standard metrics such as total energy (kWh), carbon emissions (kgCO₂e) when applicable, SLA violations (%), and makespan. A percentage-reduction indicator quantifies improvements over baselines (e.g., Round Robin and First-Fit). The framework is designed to be reproducible and extensible, with an experimental template specifying workload preprocessing, simulator configuration, and evaluation protocols. Results demonstrate consistent reductions in energy alongside improved utilization balancing, while respecting SLA constraints; when carbon-aware weighting is enabled, the scheduler further shifts flexible work to cleaner intervals without compromising throughput. The contributions include: (i) a unified energy/carbon objective with explicit constraints; (ii) an LP-seeded HGSA tailored to task scheduling; (iii) a dataset-driven evaluation recipe using realistic traces; and (iv) a practical measurement protocol that reports both absolute values and percentage reductions to facilitate cross-study comparison.

The increasing adoption of large-scale machine learning (ML) applications has exposed critical performance limitations in current data processing pipelines, particularly due to the separation between relational query execution and ML inference. This separation introduces redundant computations, excessive data materialization, and inefficient utilization of GPU Matrix Processing [10] resources. In this paper, we present a unified execution framework that integrates relational query processing and machine learning prediction by representing both as linear algebra operations. Leveraging algebraic properties such as associativity and distributivity, we introduce an operator fusion [8] strategy that enables query operators and ML models to be jointly executed on GPU Matrix Processing [10] architectures. This approach reduces intermediate data movement and enables end-to-end pipeline execution within a single linear algebra runtime. We analyze the computational complexity of the proposed fusion strategy and discuss its applicability to star-schema workloads commonly found in analytical systems. Experimental insights from prior studies indicate that linear algebra–based query execution combined with operator fusion [8] can yield substantial performance improvements over conventional GPU Matrix Processing [10]-accelerated pipelines, while maintaining scalability and portability. The proposed framework provides a viable foundation for future data-intensive systems that aim to unify analytics and machine learning on heterogeneous computing platforms. [1–3,14–16] This work unifies relational query processing and ML inference within a single algebraic runtime on GPUs, rather than coupling independent GPU-accelerated stages, thereby enabling cross-stage optimization and eliminating redundant materialization. Unlike existing GPU-accelerated databases and tensor-based query processors, the proposed framework provides a system-level unification of relational analytics and machine learning inference, rather than treating them as isolated or sequential stages. The framework is backend-agnostic and applicable to modern tensor runtimes and heterogeneous accelerator platforms, making it suitable for next-generation data-intensive systems.

In this paper, we introduce some concepts : soft group point, soft group set, soft group topology ,define soft group Fréchet space and soft group Hausdorff space in soft group topological spaces, study a relation between FG - topological space and soft group topological space with examples. Finally we introduce a new generalized definition called NSG- sets study the relations between it and the related sets.

We introduce a neutrosophic framework to assess corruption across micro, meso, and macro levels and illustrate it with a public, fully synthetic dataset covering five Central Asian societies (2020–2025). The framework models the proposition “High Corruption” with three independent degrees: Truth (T ), Indeterminacy (I), and Falsity (F), which need not sum to one. We propose a summary index—the Neutrosophic Evidence Risk Index (NERI)—that couples evidence for and against high corruption with indeterminacy. Empirically, we document three stylized patterns in the synthetic data: (i) a moderate decline in country-level NERI over time for most countries; (ii) a negative association between region-year e-service adoption and bribe solicitation; and (iii) a negative association between digital government capacity and T at the country-year level. For example, the average bribe-solicitation rate is 0.047 overall, 0.198 without e-services (95% CI 0.175–0.220) vs. 0.019 with e-services (95% CI 0.015–0.022), implying a risk difference of -0.179 and a relative risk of 0.094.

Mathematically, this study aims to analyze the dynamic linkage between economic growth and sustainable development by employment of integrated econometric–neutrosophic approach. Standard econometric models typically fail to address the risk, ambiguity and multi-dimensionality of sustainability indicators. In comparison, the neutrosophic approach – based on truth, indeterminacy and falsity – provides a solid tool for expressing uncertainty and vagueness with respect to socio-economic assessments. The article creates the ability to use quantitative data together with indeterminacy level (neutrosophic decision making) for evaluating a more complete effort of the sustainability–growth continuum, i.e., beyond only measurable results we evaluate confidence and indeterminacy embedded within them which can be seen by policy makers. Empirical evidence comes from a transition economy characterized by the significant structural reforms and modernization over recent years that clearly shows how strong economic growth can be accompanied by continuing environmental pressures. We compare the official statistics with regards to GDP growth and CO2 emissions per capita that are predicted from 2018 till 2023, in order to analyze whether environmental sustainability develops in line with economic development. Results show that the economy is resilient and growing consistently, while environmental performance is mixed, indicating partial decoupling of growth from sustainability.

The Gulf Cooperation Council (GCC) banking sector has experienced rapid digital transformation, regulatory shifts, and disruptions in recent years, especially during periods of crisis and recovery. Despite extensive studies on banking efficiency, there remains uncertainty and inconsistency regarding which bank-specific factors most influence performance. Traditional models often assume deterministic relationships, overlooking the indeterminate and ambiguous nature of real-world decision environments. Guided by Neutrosophic theory, this study reinterprets efficiency as a state influenced simultaneously by degrees of truth, falsity, and indeterminacy, acknowledging that the impact of Artificial Intelligence (AI) and Data-Driven Decision Making (DDDM) on efficiency may vary across contexts and times. The study analyzes 43 banks from six GCC countries between 2010 and 2024. In the first stage, efficiency is estimated using Data Envelopment Analysis (DEA). In the second stage, panel regression models are applied to examine the influence of bank-specific factors, including AI adoption, capital adequacy (CAR), asset quality (NPL), returns (ROA, ROI), branch footprint, and bank age. Within a Neutrosophic theoretical lens, these relationships are interpreted not as fixed or absolute but as having degrees of certainty and uncertainty that coexist within the decision environment. Findings reveal significant variation in efficiency across countries and banks. AI adoption, CAR, and ROA show strong positive associations with efficiency (high truth-values), while NPLs exhibit negative effects (high falsity values). ROI and branch footprint demonstrate mixed or indeterminate influences, suggesting that their roles depend on contextual and temporal factors. This perspective highlights how efficiency drivers in the GCC banking sector cannot be fully captured by binary or crisp evaluations. By applying Neutrosophic theory, this study provides a novel conceptual understanding of banking efficiency under uncertainty. It recognizes managerial and policy decisions are often made in environments where information is incomplete, contradictory, or evolving. The Neutrosophic interpretation enhances the explanatory depth of traditional efficiency analyses and offers a more flexible lens for understanding how digital transformation and AI adoption contribute to organizational performance amid indeterminacy.

Multisets have been the subject of extensive research, and their usefulness has been recognized in various areas such as computation, database management, and more. This study aims to explore certain properties of neutro-topological spaces by introducing a multi-neutro-topological space. Many fundamental features of interior, the exterior, the closure, and the boundary in a neutro-topological space are found to be preserved in a multi-neutro-topological space with the incorporation of multisets.

This study addresses the inherent challenges of uncertainty, vagueness, and imprecision in real-world decision-making, particularly focusing on the problem small-scale farmer’s face in optimally selecting short-term crops across diverse planting seasons. The central challenge is the absence of a systematic framework to evaluate multiple, often conflicting, criteria such as initial investment, expected yield, market demand, water and soil requirements, specific fertilizer needs, and pest susceptibility. To overcome this, a robust Multi-Criteria Decision-Making (MCDM) framework is introduced, integrating Cubic Spherical Neutrosophic Sets (CSNS) with Neutrosophic Hyper Soft Sets (NHSS). The research proposes the cubic spherical neutrosophic Bonferroni mean operator as a novel geometric representation for aggregating neutrosophic sets, which enables a more refined modeling of uncertainty and indeterminacy in complex environments. Cubic Spherical Neutrosophic Sets embed neutrosophic information within a spherical structure using interval-based (Truth, Indeterminacy, Falsity) triplets and a radius, offering robust aggregation and ranking capabilities. Neutrosophic hypersoft sets further enhance logical expressiveness by associating each multi-parameter tuple with a neutrosophic triplet, effectively managing complex multi-attribute decision-making tasks with deep interdependencies. The applicability and effectiveness of this approach are demonstrated through a practical case study involving the selection of the most suitable crop for different climatic zones (Pattams) in Tamil Nadu, considering agricultural, environmental, and economic factors. Expert linguistic assessments are converted into neutrosophic values and aligned with seasonal cropping patterns. A subsequent sensitivity analysis confirms the robustness of the model, revealing a perfect correlation between the outcomes of different decision-making methods and thereby validating the consistency and reliability of the proposed approach. This context-aware, data-driven tool aims to enhance decision-making, improve resource utilization, reduce risks, and promote agricultural sustainability and improved farmer livelihoods.

In this paper, we utilize the concept of point-wise independent set of closed operators that enabled us to find atomic solutions of the non-homogeneous α−fractional abstract Cauchy problem of order n. The proposed fractional abstract Cauchy problem is Anu(nα)(t) + An−1u((n−1)α)(t) + · · · + A1u(α)(t) + A◦u(t) = f (t) where the involved operators An, An−1, · · · , A◦ are closed and linear on a given Banach space and the unknown function u(t) is assumed to be n-times α−differentiable. Beyond the deterministic setting, we indicate how the atomic-solution framework extends naturally when coefficients, data, or initial states are modeled as neutrosophic (single-valued) quantities, thereby accommodating uncertainty and indeterminacy at the operator or forcing level.

A multi-neutrosophic set is a collection in which each element has a vector of truth indeterminacy, and falsity membership degree, rather than a Neutrosophic set. These vectors may correspond to multiple criteria, perspectives, or layers of information. Multi-neutrosophic sets are a more adaptive strategy for handling ambiguity in complex systems because they broaden neutrosophic sets and allow for better modeling of uncertain information. In this study, we have proposed the fundamental structure of multi-neutrosophic BCI/BCK Algebra and extended it to the category of multi-neutrosophic BCI(BCK) algebras. Theoretical results are presented along with examples. This study advances algebraic structure to multi-neutrosophic set and provides novel directions for future research in non-classical logic.

This paper presents an innovative generalization of intuitionistic fuzzy Q-subalgebras (IF-Q-S) by incorporating the structure of q-Rung Orthopair fuzzy sets (q-ROFS), which are distinguished by their independen membership and non-membership functions. It inserts and investigates q-Rung Orthopair fuzzy Q-subalgebras (q-ROFQ-S), demonstrating that this model is equivalent to a combination of a fuzzy Q-subalgebra (F-Q-S) and an anti-fuzzy Q-subalgebra (AF-Q-S). The study’s notable contributions include the definition of the nil radical and an exploration of its properties under homomorphisms. Additionally, it establishes that the union of q-ROFQ-subalgebras can itself form such a subalgebra under particular commutative conditions. Expanding the concept to the realm of ideals, the paper defines q-Rung Orthopair fuzzy Q-ideals (q-ROFQ-I) and proves that every q-regular q-ROFQ-S is inherently a q-ROFQ-I. This work offers a robust and versatile algebraic framework for addressing approximation in complex nonlinear systems.

Neutrosophic Logic is a neonate field of research in which all propositions are considered to have the percentage of truth in a subset I, F, and T. Neutrosophic set (NS) has been positively utilized for indeterminate data processing, and proven benefits for addressing the indeterminacy data information and is still a method nominated for classification application and data analysis. Soft set (SS) is a powerful device for handling the uncertainty of information in a parametric situation. On the other hand, the concept of interval-valued neutrosophic soft sets (IVNSSs) is a novel generality of the neutrosophic soft sets (NSSs) to the NSs once the authors incorporate the important features of IVNS and soft sets (SSs) in one method. Therefore, this method operated to offer decision-makers with flexibility in the procedure of understanding unclear information. From the scientific viewpoint, the procedure of estimating this higher performance IVNSS vanishes. Q-neutrosophic SSs are fundamentally NSSs considered by 3 independent 2D membership functions that represents indeterminacy, falsity and uncertainty. Therefore, it is used to 2D inconsistent, imprecise and indeterminate data, which seem in most real world challenges.  The usage of robo-readers for analyzing news texts is the advanced technology trend in financial technology. A considerable effort has been invested to develop refined financial orientation that is applied to inspect how financial sentiments related to future performance of the company. Recently, the financial sentiment analysis (SA) has become a more and more related subfield within text analytics that addresses the computational analysis of subjectivity and opinion in texts. Most of the methods have concentrated on particular fields, utilizing type-based corpora as training data for machine learning (ML) methods that classify the input text as both negative and positive. In this manuscript, we develop a Semantic Orientation Identification Framework in Economic Text Using Q-neutrosophic soft matrix under Interval-valued (SOIFET-IVQNSM) model for financial SA. The aim of the paper is to propose an innovative approach for identifying semantic orientation in economic texts to enhance financial sentiment and prediction accuracy. Primarily, the input text data is preprocessed utilizing diverse preprocessing levels like removal of stop words, tokenization, stemming, spelling correction, and lemmatization to make it suitable for further processing. Besides, the word embedding process is mainly executed by the term frequency-inverse document frequency (TF-IDF) model to transform economic text into meaningful vector representation. For classification purpose, the proposed SOIFET-IVQNSM model designs a Q-neutrosophic soft matrix under Interval-valued (IV-Q-NSM) model. The simulation validation of the SOIFET-IVQNSM algorithm is tested on a benchmark database, and the results are measured under several metrics. The simulation result highlighted the improvement of the SOIFET-IVQNSM system in semantic orientation identification.

In this paper, we conduct a comprehensive study of neutrosophic subalgebras of various types within the framework of Sheffer stroke UP-algebras (SUP-algebras). Specifically, we introduce and characterize (∈, ∈), (∈, ∈ ∨q), and (q, ∈ ∨q)-neutrosophic subalgebras based on neutrosophic ∈-subsets, q-subsets, and (∈ ∨q)-subsets. Necessary and sufficient conditions are established for these subsets to form subalgebras under the Sheffer stroke operation. Several theorems demonstrate how these types interrelate and differ in their structural properties, with illustrative examples provided. Furthermore, we identify the conditions under which certain canonical subsets, such as X1 0 = {x ∈ X | T (x) > 0, I(x) > 0, F (x) < 1}, form subalgebras across differ- ent neutrosophic configurations. These results offer a unified perspective and deeper insight into the algebraic behavior of neutrosophic systems in the context of SUP-algebras.

The main goal of this article is to study the division of symbolic n-plithogenic numbers using the identification method and n-plithogenic AH-isometry. In particular, we discuss the division of symbolic 2-plithogenic numbers and 3-plithogenic numbers, and we generalize these divisions. Additionally, we prove the validity of the formulas using AH-isometry and provide four worked examples to enhance understanding.