Neutrosophic Probability

Many decision problems depend on linguistic opinions that exhibit uncertain and directional (phase/ angle) information, which standard models capture only partially. Yet efficient decision support requires a combined treatment of these features. Existing complex picture fuzzy sets and 2-tuple linguistic term sets do not jointly encode linguistic evaluations on the complex plane with complete aggregation and ranking procedures. This manuscript introduces an inventive notion, termed 2-tuple linguistic complex picture fuzzy sets, showcasing progress and expansion beyond traditional complex picture fuzzy sets and 2-tuple linguistic term sets. The distinctive feature of 2TLCPFS lies in its remarkable capacity to address both uncertainty and periodicity simultaneously. This is achieved by integrating a phase term that extends the 2TLCPFS range from the real plane to the complex plane, encompassing the unit disk. We explain and authenticate the operational characteristics of 2TLCPFS. Moreover, a novel aggregation operator, such as the 2TLCPF weighted averaging operator, is introduced within the 2TLCPFS framework. The application of this operator, combined with the TOPSIS method, is demonstrated in the context of a multicriteria decision-making problem. A numerical example reinforces this demonstration, underscoring the validity and efficacy of the proposed method. Additionally, this article explores situations where both decision-makers and criterion weights are undisclosed, underscoring the benefits of the proposed generalized aggregation technique. The work probes the impact of weight perturbations and conducts a comparison analysis, which validates the model's robustness and superiority.

The intuitionistic hesitant fuzzy set (IHFS) is an enriched version of hesitant fuzzy sets (HFSs) that can cover both fuzzy sets (FSs) and intuitionistic fuzzy sets (IFSs). By assigning membership and non-membership grades as subsets of [0, 1], the IHFS can model and handle situations more proficiently. Another related theory is the theory of set pair analysis (SPA), which considers both certainties and uncertainties as a cohesive system and represents them from three aspects: identity, discrepancy, and contrary. In this article, we explore the suitability of combining the IHFS and SPA theories in multi-attribute decision making (MADM) and present the hybrid model named intuitionistic hesitant fuzzy connection number set (IHCS). To facilitate the design of a novel MADM algorithm, we first develop several averaging and geometric aggregation operators on IHCS. Finally, we highlight the benefits of our proposed work, including a comparative examination of the recommended models with a few current models to demonstrate the practicality of an ideal decision in practice. Additionally, we provide a graphical interpretation of the devised attempt to exhibit the consistency and efficiency of our approach.

In this paper, we discussed some saturated classes of ℋ -commutative semigroups, left (right) regular semigroups, medial semigroups, and paramedial semigroups. The results of this paper significantly extend the long standing result about normal bands that normal bands were saturated and, thus, significantly broaden the class of saturated semigroups.

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.

This volume of “Neutrosophic Sets and Systems” is an international journal that features publications on advanced studies in neutrosophy, neutrosophic sets, neutrosophic logic, neutrosophic probability, and neutrosophic statistics. The document highlights neutrosophy as a branch of philosophy that examines the origin, nature, and scope of neutralities, considering every notion or idea <A> along with its opposite <antiA> and a spectrum of neutralities <neutA>. This approach generalizes classical and fuzzy logic by introducing a degree of indeterminacy. The included papers demonstrate applications of these concepts in diverse fields, such as vocational training, sustainable tourism, medical diagnosis, employee mental health, social media information dissemination, and humanitarian logistics.

This volume of “Neutrosophic Sets and Systems” is an international journal in information science and engineering that features publications on advanced studies in neutrosophy, neutrosophic sets, neutrosophic logic, and neutrosophic statistics. The document states that neutrosophy is a new branch of philosophy that studies the origin, nature, and scope of neutralities, as well as their interactions with different ideational spectra. The theory considers every notion <A> along with its opposite <antiA> and a spectrum of neutralities <neutA> in between them. The included papers demonstrate applications of these concepts in various fields, such as optimizing teaching quality in English translation programs, evaluating intangible cultural heritage for tourism, analyzing green policy dynamics in coal mining, modeling the impact of exercise on mental health, and evaluating competitiveness in the power and electronics technology industry. The journal is indexed in databases such as SCOPUS and Google Scholar.

Volume 80 of “Neutrosophic Sets and Systems” showcases cutting-edge research and applications of neutrosophic theory in various scientific and engineering disciplines. The papers in this volume present novel frameworks and methodologies for addressing problems characterized by uncertainty, vagueness, and imprecision. A significant focus is on advanced multi-criteria decision-making (MCDM) approaches, with applications ranging from evaluating e-commerce sites and teaching quality in diverse university subjects to credit evaluation in construction projects. The volume also includes studies on theoretical extensions of neutrosophic concepts, such as neutrosophic hypersoft sets, plithogenic sets, and hypersoft semi-connected spaces. Furthermore, applied research is featured on topics including stock price prediction using hybrid models, assessing the performance of social institutions, and mapping barriers to sustainable fashion consumption. This collection demonstrates the versatility of neutrosophic theory as a powerful tool for modeling and solving complex, real-world problems.

This special issue of “Neutrosophic Sets and Systems”, Volume 62, presents a collection of articles that highlight the transformative impact of neutrosophic advancements on research in Latin America. The contributions explore the application of neutrosophic principles to address complex issues across diverse domains, including education, law, and healthcare6. The papers within this volume demonstrate how neutrosophic methods, which consider degrees of truth, indeterminacy, and falsity, provide a nuanced approach to understanding ambiguous, inconsistent, and contradictory situations. The research showcases how neutrosophic modeling and analysis can lead to more adaptive and informed decision-making in various contexts, from healthcare resource allocation to corporate social responsibility and international diplomacy. This volume stands as a testament to the growing influence of neutrosophy in the region and its potential to revolutionize traditional research paradigms.

Volume 53 of “Neutrosophic Sets and Systems” (2023) gathers a diverse range of studies advancing both the theoretical foundations and applied dimensions of neutrosophy. The issue presents new developments in algebraic structures, topology, probability, statistics, and graph theory under neutrosophic frameworks, as well as innovative models such as hypersoft sets, cubic sets, and type-2 neutrosophic systems. Applications span multiple domains, including decision-making, medical diagnosis, urban traffic management, vehicular malfunction detection, queuing systems, supply chain inventory modeling, and image processing. Several contributions introduce computational algorithms, aggregation operators, and optimization methods, highlighting the growing integration of neutrosophy with engineering, computer science, and applied mathematics. Collectively, this volume underscores the flexibility of neutrosophic approaches in addressing uncertainty, indeterminacy, and inconsistency across interdisciplinary contexts.

Volume 52 of “Neutrosophic Sets and Systems” (2022) is a special issue dedicated to exploring the progress and perspectives of neutrosophics in Latin America. It brings together a wide range of contributions applying neutrosophic theory, statistics, and decision-making models to pressing legal, social, educational, medical, and political issues in the region. The papers demonstrate how neutrosophic methods—by explicitly integrating truth, indeterminacy, and falsity—can provide more nuanced analyses of complex and uncertain phenomena, from human rights and justice systems to healthcare, education, governance, and sustainable development. This collection highlights both methodological advances and applied studies, showcasing the versatility of neutrosophy as a tool for addressing regional challenges and promoting interdisciplinary dialogue. The issue serves as a landmark reference for scholars and practitioners interested in the expansion of neutrosophic approaches across Latin American contexts and beyond.

In this paper we basically make two propositions - firstly a non-linear process that is primarily fuelled by mass cognitive dissonance could generate systematic deviations between the theoretical and market prices of long-term options, and secondly such deviations are best reconciled in terms of neutrosophic rather than ruled-based reasoning, especially in the context of the users of automated trading systems designed to generate trading signals based on analysis of information  from conflicting sources.

Prioritizing software requirements is one of the most significant and complex tasks for software developers. It involves a multi-criteria decision process to balance the benefit of each requirement and its cost, considering different factors and dimensions, many of which are qualitative. Although numerous prioritization methods have been developed to date, the appropriate treatment in cases of the uncertainty and indeterminacy inherent in human decisions is limited. In the present work, the neutrosophic TOPSIS is proposed as a method of prioritizing requirements. This multi-criteria and multi-expert method uses linguistic terms associated with Single Value Neutrosophic Sets and allows the inclusion of aspects such as the importance of the criteria and the weight of expert evaluations. A case study is used to show the applicability of the proposal.

The Cooperative of Taxis and Vans of Puyo in Ecuador is dedicated to the transportation of people and minor loads. Due to the considerable number of members of this cooperative, it is difficult to determine the presence of each of the participants in the meetings. That is why it was decided to implement a facial recognition system that allows identifying the presence of members in each moment. However, in order to apply this system, certain requirements are needed for guaranteeing its success. This paper aims to apply the neutrosophic Analytic Hierarchy Process (NAHP) technique for analyzing the prioritization of requirements necessary to implement a facial recognition system in the cooperative. Neutrosophic AHP permits including the indetermination incorporated in neutrosophic models and additionally experts can provide their evaluations based on linguistic terms, which results in greater ease and effectiveness to evaluate.

Neutrosophy is a branch of philosophy proposed in 1998 by Romanian philosopher and mathematician F. Smarandache, which studies the origin, nature, and scope of neutrality and its interaction with different spectra of ideas. Because it is a relatively new field, specific scientific production on this subject may be limited, but as it develops, new research and contributions are likely to emerge. This is why it is becoming more necessary to undertake studies to understand the evolution, impact, and scope of research, especially in a context where it had never been examined in depth before: in the Latin American region. This research aims to describe the scientific production of Latin American authors in the Scopus database referring to Neutrosophic Science in the period 2019-2023 by carrying out a bibliometric study of a descriptive nature. The scientific production in Neutrosophy assumes patterns of behavior that vary from international patterns with a marked social focus. Although productivity levels have not increased in recent years, the impact and recognition from the international community have grown significantly, meaning the evolution of a process of transformation.

ملخَّص: يصف هذا المقال كيفية انتقال النظام العشري لكتابة الأعداد من الهند إلى أوروبا مرورا بالبلدان العربية حتى نهاية القرن الخامس عشر. فأقدم ما لدينا من تفاصيل عن الحساب الهندي هو ما ألفه الخوارزمي بالعربية حوالي سنة 825م. لا نتوفر على أي مخطوطة عربية لهذا العمل. كل ما نعرفه عنه هو إعادة صياغة مبكرة لترجمة لاتينية. قبل بضع سنوات [من 2001]، لم تكن هاته المخطوطة معروفة سوى جزئيا ويعود عهدها للقرن الثاني عشر (Cambridge, UL, Ii.6.5) . ولم يتوفّر نصّها الكامل إلا عبر مخطوطة أخرى أتاحت دراسةً أَدَقَّ لعمل الخوارزمي (نيويورك، الجمعية الإسبانية الأمريكية، HC 397/726). تقدم مقالتنا هاته خطوطا عريضة لمحتويات المخطوطة المكتملة وتناقش بعض طرقها الخاصة في العرض.

Bien que la place du roman français dans la vie littéraire des Lumières en Pologne ait déjà donné matière à de nombreux débats et études, la richesse d un thème aussi profond que la «réécriture» nous invite aujourd hui à réviser le dialogue qui s établit à l époque, entre ces deux cultures litté- raires distinctes et géographiquement distantes. «Imitation»: telle était l étiquette collée aux écrits romanesques de Krajewski, Kropiñski ou Bernartowicz auteurs s inspirant de textes étrangers, notamment français, dans la langue polonaise. Néanmoins l étude des rapports intertextuels ne se borne pas à un simple constat d «imitation»: elle demande à être poussée plus loin[...

With a growing emphasis on developing transverse competencies in students, many educational institutions are trying to revolutionize the traditional system of education. One among others is TEC de Monterrey in Mexico which refines its educational models so that the students can be nourished with the skills that prepare them not only for the present but for years to come. Among skill development transverse competencies are considered a key to the successful career development of a student that emphasizes on critical thinking aspect. So mainly the the educational models try to inculcate or enhance the critical thinking of students. Towards this goal, TEC introduced the TEC21 model that is oriented towards critical thinking skill development of the students. The current work in this regard utilizes real-time student data collected using the eOpen instrument and takes into consideration experts' opinions to analyze the TEC21 model. The determinate factors are taken from the TEC21 model and indeterminate factors are identified from the literature. As real-time data especially educational data contain much uncertain, indeterminate, and unknown information, therefore this research introduces the use of Neutrosophic Logic to address the uncertainty of the data. Through the use of neutrosophy, we have shown how indeterminate factors when considered for analysis give a better representation of information. We have also compared neutrosophic cognitive maps with used earlier fuzzy cognitive maps to show their effectiveness in this regard. Overall system developed using NCM gives a better analysis of educational models like TEC21. This is the only work now that has utilized neutrosophy to understand and analyze the TEC21 model.

A special issue of the International Journal in Information Science and Engineering “Neutrosophic Sets and Systems” (vol. 69/2024) is dedicated to the Neutrosophic approaches in research, on the occasion of the international and multidisciplinary conference held at the Universidad César Vallejo in Lima, Peru, on July 8 and 9. This event marks a significant milestone, as it is the first time that the Andean region and Latin America host scholars and researchers dedicated to studying various theoretical and applicative issues in the expansive and diverse field of Neutrosophic approaches.
Since its conception, Neutrosophic theory has proven to be an interdisciplinary and innovative field, notably growing with the introduction of several generalizations of Neutrosophic Sets, such as Plithogenic Sets, Hypersoft Sets, IndetermSoft Sets, SuperHyperSoft Sets, and MultiAlism. These advanced conceptualizations have further expanded the versatility and application range of Neutrosophic theory, allowing its adoption in an ever-increasing spectrum of disciplines.
The conference, with its international and multidisciplinary character, has brought together experts and scholars from various fields, providing a unique platform for the discussion and exchange of ideas on the multiple applications of Neutrosophic approaches.
This special issue also addresses how scientific production in Neutrosophy focuses on social issues specific to Latin American philosophy. In the regional context of Latin America, it is possible to state that Neutrosophic tools and knowledge are used for the identification, analysis, and resolution of social problems, offering unique approaches or distinctive contributions to the field of Neutrosophy, influenced by its cultural and philosophical context.
Neutrosophic science in Latin America shows a clear pattern of how scientific production addresses social problems, standing out for its innovative approaches that reflect the cultural and philosophical particularities of the region. This approach has allowed Neutrosophy not only to advance in theoretical terms but also to provide practical and contextually relevant solutions to social challenges.
This special issue compiles works presented at the conference, reflecting the richness and diversity of current research in this field. We hope that these articles not only contribute to the advancement of knowledge in Neutrosophic theory but also inspire new research and applications in multiple disciplines.

بالرغم من حضوره في الحياة اليومية منذ عدة قرون بأشكال مختلفة، إلا أن اختراعه لم يكن بالسهولة التي نتعامل فيها معه اليوم. فهناك ثقافات لم تعرفه أصلاً، وهناك ثقافات قاومت دخوله إليها في واعتبرته أمراً بالغ التعقيد كما كان الأمر في أوروبا القرن الثالث عشر. ولكنه يشغل مكانة الصدارة في الحضارة السائدة اليوم.

The selection of an optimal investment sector is of critical importance not only for individual financial success but also to drive economic development. The allocation of capital into sectors with high
potential for growth, innovation, and job creation is key. In addressing the complexity of decision-making scenarios associated with investment sector exploration, we introduce a novel data structure known as Picture fuzzy hypersoft set (PFHSSs). This specialized approach within computational intelligence and decision-
making aims to categorize data into various attributes and sub-attributes, considering the significant role of neutrality. The study stems from the need for a comprehensive framework (PFHSSs) that can effectively handle intricate decision-making scenarios involving attributes, subattributes, and nuanced factors such as neutrality. Traditional tools such as TOPSIS and its extensions of fuzzy sets, while robust in Multiple Criteria Decision Making (MCDM), may face challenges in modeling and analyzing decision-making information
within a PFHSSs environment. The rationale behind this study lies in enhancing the accuracy and efficiency of decision-making processes when dealing with complex, fuzzy, and multi-criteria data. By introducing newly proposed distances and similarity measures tailored to PFHSSs, and constructing a PFHSSs-TOPSIS method, we aim to address the limitations faced by existing models in the PFHSSs environment. The application of Hamming distance-based similarity measures further distinguishes our method by determining
the weights assigned to each decision maker. The proposed PFHSSs-TOPSIS method is practically applied in designing an optimal investment sector exploration tool for investors. This method has the potential to establish a crucial connection between alternatives and attributes, providing value across various fields and industries. The research emphasizes bridging the gap in decision-making scenarios where alternatives and attributes need to be effectively connected and analyzed, thereby contributing to the advancement of decision-
making processes in complex domains

The survey is a research procedure used in sociology to determinethe thoughts and feelings of a social group at a given time and context. Within the survey, the questionnaire is consideredas avery useful instrument used to measure the state ofopinionsof social groups. Although it has been demonstratedthat fuzzy responses to questionnaires are more appropriate than crispones, there may be indeterminacy and thus fuzzy processingdoesnot accurately capture the thought that the respondent wants to express, due to doubts, unclear and vague thoughts, among others. Modelingsuch scenarioby means ofneutrosophic sets providesrespondentsa greater range of possible responses and hence it is more appropriate. In this paper,we propose a method to design single-valued neutrosophic sets from questionnaires to social groups. This method, inspired by another fuzzyone, allowsus to createmembership functions of truthfulness, indeterminacy and falseness through experimental data, which will let usfindthe...

In this paper we carry out an analysis of the humanistic and higher education components in students of legal careers. Neutrosophic statistics are used to demonstrate the importance of these components. Thus, person&#39;s life is better, insofar as it is filled with meaning and continuous reflection during his (her) existence. For that reason this study aims to analyze humanistic and higher education components of Law School students for their inclusion into the practical activities from an integral perspective.

The aging of population includes a social group that has increased in recent years. Properly attending to the conditions of the elderly represents a problem to be addressed by Public Health institutions. A pathology that occurs very frequently is the cognitive impairment. Selecting the correct treatment for this type of condition is subject to the set of criteria that characterize the patient. Problems of this nature have been addressed by science from the implementation of artificial intelligence. The present research aims to develop a method for the selection of non-pharmacological treatments for mild cognitive impairment in older adults. The method operates using the neutrosophic AHP to model indeterminacy. The result is a case study with the implementation of the proposed method as an example of applicability.

Abstract. The purpose of this research was to determine from a trialist perspective what factors currently affect the labor inclusion of people from indigenous cultures in Ecuadorian society and to propose strategies for their mitigation. A broad and generic analysis is carried out to assess whether the norms and the social reality evident in the country effectively adhere to each other. We used a fusion of PESTEL method and Neutrosophic Cognitive Maps, since during the mathematical exercise that endorses the analysis, indeterminacy is incorporated into the modeling of the causal relationships between the factors analyzed. Finally, the figure of indigenous labor inclusion is declared as an unfinished topic. Therefore, the priority factor to consider is the establishment of policies favoring the employment of indigenous people. As a solution, this paper ends with the proposal of compliant strategies to promote improvements in this area.

En la actualidad existen dificultades en el país para lograr la sostenibilidad, lo que provoca una situación que repercute negativamente en la gestión ambiental empresarial. Por ello, para completar la implementación del Sistema de Gestión Ambiental de la Empresa Eléctrica Holguín, es necesario establecer indicadores de desempeño operacional como paso para sentar las bases dentro de la evaluación del desempeño ambiental en general. Lo cual es el objetivo de la presente investigación. Con ello s e podrá medir la evaluación del desempeño ambiental en los procesos y ayudará a minimizar su impacto ambiental negativo en la zona donde se encuentra, así como potenciar los impactos positivos. Para realizar una selección efectiva de los indicadores operativos para determinar un camino a seguir, se utilizaron las técnicas TOPSIS y AHP Saaty en sus versiones neutrosóficas. Esto garantiza que se tenga en cuenta la incertidumbre de la subjetividad dentro del proceso de toma de decisiones, incluyendo las indeterminaciones. Palabras clave: AHP Saaty-TOPSIS neutrosófico, evaluación del desempeño ambiental, indicadores operativos.

Currently, there are difficulties in the country for achieving sustainability, which causes a situation that has a negative impact on business environmental management. For this reason, in order to complete the implementation of the Environmental Management System of the Holguín Electric Company, it is necessary to establish operational performance indicators as a step to lay the foundations within the evaluation of environmental performance in general. Which is the objective of the present investigation. With this, it will be possible to measure the evaluation of environmental performance in the processes and it will help to minimize its negative environmental impact in the area where it is located as well as to enhance the positive impacts. In order to make an effective selection of the operational indicators to determine a path to follow, TOPSIS and AHP Saaty techniques were used in their neutrosophic versions. This guarantees that the uncertainty of subjectivity is taken into account within the decision-making process, including indeterminacies.

Prioritizing software requirements is one of the most significant and complex tasks for software developers. It involves a multi-criteria decision process to balance the benefit of each requirement and its cost, considering different factors and dimensions, many of which are qualitative. Although numerous prioritization methods have been developed to date, the appropriate treatment in cases of the uncertainty and indeterminacy inherent in human decisions is limited. In the present work, the neutrosophic TOPSIS is proposed as a method of prioritizing requirements. This multi-criteria and multi-expert method uses linguistic terms associated with Single Value Neutrosophic Sets and allows the inclusion of aspects such as the importance of the criteria and the weight of expert evaluations. A case study is used to show the applicability of the proposal.

Abstract. Democracy in Ecuador is based on the election of the highest authorities through popular election and the territorial division over which the elected has functions and powers to fulfill in case of being elected. The candidates make commitments on which they would work, as well as the concrete actions that they would take in office if elected. However, there are inconsistencies between the government plans proposed by the candidates and the obligations that are required by law to comply with the low level of competencies. This research proposes a solution to the posed problem with the development of a method to evaluate the competencies of mayoral candidates in the execution of government plans. The proposed method bases its operation on a multi-expert multi-criteria approach with the use of neutrosophic numbers.

In today's world, people's need for sustainable energy is crucial. In India, people utilizing 73% of electricity from fossil fuels is the biggest hazard to the environment. In order to partly gratify the nation, policymakers have decided to invest in sustainability renewable energy sources rather than non-renewable resources for the production of energy. There five distinct kinds of green energy sources. The research key contribution is selecting the best renewable energy sources among the five different categories. Therefore, here we form a mathematical fuzzy model to identify the decision received from hesitating knowledge in the form of an Picture Q-rung orthopair fuzzy set (PQROFS) is engaged to handle the uncertain data in decision makers' thoughts. A gathering of relevant judgment Multi-Attribute Decision Analysis (MADA) is employed. Attributes play a prominent role in DM for ranking the alternatives. Here we have twelve attributes and five alternatives. An objective CRiteria Importance Through Intercriteria Correlation (CRITIC) method is employed for attributes priority determination and EXtension of the PROMethee-II (EXPROM-II) method an method is determining the alternative from the preference between each alternative and novel methodology. Ultimately, to validate the proposed results, compared the proposed results with other existing MCDA method results.

Enterprise Resource Planning (ERP) and success factors are systems that enable effective and efficient management of businesses by facilitating coordination and interaction among different units. Therefore, the decision to select the right and applicable ERP implementation and success factors for businesses is a crucial matter. However, due to the cost of ERP implementations and the lengthy adaptation periods for businesses, selecting the relevant systems and success factors requires careful consideration. Identifying ERP implementations and success factors that are suitable for the business processes and structure of the company is a significant decision-making problem, which can be considered as a gap in the literature. In this regard, this study used the Neutrosophic Decision Making Trial and Evaluation Laboratory (DEMATEL) method to analyze the critical success factors for ERP in production companies that had more than 10 employees. The results show that project management and t...

In this paper, the theory of soft topological space generated by L-soft sets is introduced. As a continuation of the study of operations on L-soft sets, the aim of this paper is to introduce new soft topologies using restricted and extended intersections on L-soft sets and to study the differences of these soft topologies.

In this paper, the theory of soft topological space generated by L-soft sets is in troduced. As a continuation of the study of operations on L-soft sets, the aim of this paper is  to introduce new soft topologies using restricted and extended intersections on L-soft sets and to  study the diferences of these soft topologies.

The Picture fuzzy linguistic set (PFLS) is an extension of the intuitionistic fuzzy set (IFS) and linguistic variables (LVs), which has been applied successfully in the process of decision making. Considering the lack of closeness of extant PFLS operations and the interrelationship among input attributes do not consider. In this paper, for the sake of addressing those limitations, we firstly redefine some novel operational laws for PFLS by introducing linguistic scale functions and the related properties are studied. Then, a novel score function and accuracy function are also defined to compare PFLSs. Subsequently, in consideration of the superiority of the Muirhead Mean (MM) operator in capturing the interaction relationship between the input parameters, we extend the MM operator to the Picture fuzzy linguistic context and then propose Picture fuzzy linguistic weighted MM operator and its dual form in a new light. After that, these operators have adopted to build two novel models to solve multiple attribute decision-making (MADM) problems. Finally, a practical example for the selection of the innovative "Mobike" sharing bike design is provided to illustrate the practicality and effectiveness of proposed approaches. Keywords Picture fuzzy linguistic set • Muirhead Mean (MM) operator • Linguistic scale functions • Multiple attribute decision making Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Recently, there has been increasing interest in competency-based education. Additionally, neutrosophic cognitive maps and its application in decision making have become a topic of significant importance for researchers and practitioners. In this paper, a framework 33based on static analysis of neutrosophic cognitive maps applied to competencies modelling and prioritization in presented. A case study based on modelling and prioritization of transversal competencies in system engineering is developed. The paper ends with conclusion and future research directions. .

The Sinos River watershed is one of the most polluted water basins in Brazil with great efforts for its recovery through integral management. PESTEL is an analysis for the study of the external variables with influence in the efficiency of the organization or project. This paper presents a model to address problems encountered in the measurement and evaluation process of PESTEL analysis taking into account interdependencies among sub-factors and modeling uncertainty and indeterminacy in Sinos river basin. A Neutrosophic Cognitive Maps was used for modeling the integrated structure of PESTEL sub-factors. A quantitative analysis was developed based on static analysis and neutrosophic numbers. To demonstrate the applicability of the proposal in the Sinos river external factor analysis a case study is developed. Interdependencies among sub-factors were includes and uncertainty and indeterminacy were modeled in a practical way. Sub-factor was ranked and reduced, with Ecological, Technolo...

In the Latin American region, public management systems are driven almost exclusively by a formal mechanism. This is a difficulty when implementing study methodologies that provide objective data for the analysis and evaluation of the effectiveness, relevance and sustainability of the policies established by the states governments. Given that the political-social contexts of each nation determine a series of subjective and imprecise factors, and that the main officials and social actors that rule the development of strategic plans regarding public services are inscribed within a certain reality for internal and external incidents; a neutrosophic analysis of the fuzzy, gray and / or undefined zones of the relations between the political system and the administrative system becomes evident. Neutrosophy, as a discipline that studies undefined areas between opposing ideas or sentences, offers a theoretical framework suitable for the analysis of the previously described problematic. This...

In this paper, we propose the introduction of a neutrosophic chi-square-test for consistency, incorporating neutrosophic statistics. Our aim is to modify the existing chi-square-test for consistency in order to analyze imprecise data. We present a novel test statistic for the neutrosophic chi-square-test for consistency, which accounts for the uncertainties inherent in the data. To evaluate the performance of the proposed test, we compare it with the traditional chi-square-test for consistency based on classical statistics. By conducting a comparative analysis, we assess the e ciency and e ectiveness of our proposed neutrosophic chi-square-test for consistency. Furthermore, we illustrate the application of the proposed test through a numerical example, demonstrating how it can be utilized in practical scenarios. Through this implementation, we aim to provide empirical evidence of the improved performance of our proposed test when compared to the traditional chi-square-test for consistency based on classical statistics. We anticipate that the proposed neutrosophic chi-square-test for consistency will outperform its classical counterpart, o ering enhanced accuracy and reliability when dealing with imprecise data. This advancement has the potential to contribute significantly to the field of statistical analysis, particularly in situations where data uncertainty and imprecision are prevalent.

Scaling system can be considered as range-base measurement system, it's a fatal tool used in all human activities in daily-bases, also, all business domains and sectors heavily use scaling systems in all business process specially in decisions-making as one of the main critical business activities, despite the fact that, there is no scientific base for calculate an unified scale system ranges, all provided scales or ranges are determined based on expert opinions', an enhanced scale system using single-valued neutrosophic set SVNS is offered that suggest a scientific methods for defining ranges in scaling systems, in addition, a new crisp value functions "De-neutrosophication" for converting both Simplified Neutrosophic Number SNN, and SVNS to them equivalent crisp values using distance measure based on Euclidean space are proposed, Finally, the offered framework and methods are implemented with numerical examples for best prove and validate of the framework and proposed methods.

Beyond the predominant paradigm of an essentially rational human cognition, based on the classical binary logic, we want to propose some reflections that are organized around the intuition that the representations we have of the world are weighted with appreciations, for example affective ones. resulting from our integration into a social environment. We see these connotations as essentially ternary in nature, depending on the concepts underlying neutrosophy: either positive, negative or neutral. This form of representation would then influence the very nature of the cognitive process, which in complex real-world situations, has to deal with problems of a combinatorial nature leading to a number of cases too large for our abilities. Forced to proceed by shortcuts on the basis of heuristics, cognition would use these assessments of the representations it manipulates to decide whether partial solutions are attractive for solving the problem or on the contrary are judged negative and are then quickly rejected. There is still the case of a neutral weighting that allows processing to continue. Thus a neutrosophical conception of our representations of the world explains how our cognition functions in its treatment of combinatorial problems in the form of producing processing accelerating heuristics, both in terms of partial solutions selection and processing optimization.

Hypersoft set is the generalization of the soft set as it converts single attribute function to multi-attribute function. The core purpose of this study is to make the existing literature regarding fuzzy parameterized soft set in line with the need for multi-attribute function. We first conceptualize the fuzzy parameterized hypersoft set along with some of its fundamentals. Then we propose a decision-making-based algorithm with the help of this theory. Moreover, an illustrative example is presented which depicts its validity for successful application to the problems involving vagueness and uncertainties.

Multi-attributes decision-making problem in dynamic neutrosophic environment is an open and highly-interesting research area with many potential applications in real life. The concept of the dynamic interval-valued neutrosophic set and its application for the dynamic decision-making are proposed recently, however the interdependence among criteria or preference is not dealt with in the proposed operations to well treat interdependence problems. Therefore, the definitions, mathematical operations and its properties are mentioned and discussed in detail. Then, Choquet integral-based distance between dynamic inteval-valued neutrosophic sets is defined and used to develop a new decision making model based on the proposed theory. A practical application of proposed approach is constructed and tested on the data of lecturers' performance collected from Vietnam National University (VNU) to illustrate the efficiency of new proposal.

In this paper, we study Smarandache special definite groups. We give necessary and sufficient conditions for a group to be Smarandache special definite group(S-special definite group). Moreover we study Smarandache special definite subgroups, Smarandache special definite maximal ideals, Smarandache special definite minmal ideals.

The paper develops a novel analysis of mutual interactions between topology and soft topology. It is known that each soft topology produces a system of crisp (parameterized) topologies. The other way round is also possible. Namely, one can generate a soft topology from a system of crisp topologies. Different methods of producing soft topologies are discussed by implementing two formulas. Then, the relationships between the resulting soft topologies are obtained. With the help of an example, it is demonstrated that one formula is more constructible than the other. Now, it is reasonable to ask which (topological) properties of a soft topology can be transferred to the set of crisp topologies, or the opposite. To address this question, we consider the standard separation axioms and show how well these axioms can be preserved when moving from a system of crisp topologies to the soft topology generated by it and contrariwise. Additionally, our findings extend and disprove some results from the literature.

In this paper, we introduce a novel fuzzy structure named "fuzzy primal". We study the essential properties and discuss basic operations on it. A fuzzy operator (•) ⋄ on the family of all fuzzy sets is introduced here by applying the q-neighborhood structure to a primal fuzzy topological space along with the Lukasiewicz disjunction. We explore the main characterizations of (•) ⋄. Then, we define another fuzzy operator, symbolized by Cl ⋄ , with the utilization of (•) ⋄. These fuzzy operators are studied in order to deduce a new fuzzy topology from the original one. Such a new fuzzy topology is called primal fuzzy topology. The fundamental structure, particularly a fuzzy base that generates primal fuzzy topologies, as well as many relationships between different fuzzy primals and fuzzy topologies, are also analyzed. Lastly, the concept of compatibility between fuzzy primals and fuzzy topologies is introduced, and some equivalent conditions related to this are examined. It is shown that if a fuzzy primal is compatible with a fuzzy topology, then the fuzzy base that generates the primal fuzzy topology is itself a fuzzy topology.