Both neutrosophic sets theory and rough sets theory are emerging as powerful tool for managing uncertainty, indeterminate, incomplete and imprecise information.In this paper we develop an hybrid structure called “ rough neutrosophic sets” and studied their properties.

Today, soft computing is a field that is used a lot in solving real-world problems, such as problems in economics, finance, banking... With the aim to serve for solving the real problem, many new theories and/or tools which were proposed, improved to help soft computing used more efficiently. We can mention some theories as fuzzy sets theory (L. Zadeh, 1965), intuitionistic fuzzy set (K Atanasov, 1986), neutrosophic set (F. Smarandache 1999). In this paper, we introduce a new notion of (...) sup port-neutrosophic set (SNS), which is the combination a neutrosophic set with a fuzzy set. So, SNS set is a direct extension of fuzzy set and neutrosophic sets (F. Smarandache). Then, we define some operators on the support-neutrosophic sets, and investigate some properties of these operators. (shrink)

In this article, we apply the notion of interval neutrosophic sets to ideal theory in BCK/BCI-algebras.

Smarandache (1995) defined the notion of neutrosophic sets, which is a generalization of Zadeh's fuzzy set and Atanassov's intuitionistic fuzzy set. In this paper, we first develop some similarity measures of neutrosophic sets. We will present a method to calculate the distance between neutrosophic sets (NS) on the basis of the Hausdorff distance. Then we will use this distance to generate a new similarity measure to calculate the degree of similarity between NS. Finally we will prove some (...) properties of the proposed similarity measures. (shrink)

This special issue of the journal “Neutrosophic Sets and Systems” compiles a selection of notable works from the IX Ibero-American Biometry Meeting, held in Quito, Ecuador. The collection focuses on the application of neutrosophic and plithogenic approaches to data science and multivariate analysis. The papers explore advanced studies in neutrosophy, which is a branch of philosophy examining the origin, nature, and scope of neutralities. Neutrosophic theory generalizes classic logic by introducing a degree of indeterminacy, alongside degrees of (...) truth and falsity. The articles within this volume demonstrate the application of these concepts to diverse fields, including financial dynamics, psychological treatments, cryptocurrency regulation, sentiment analysis, and the evaluation of public policies. (shrink)

In this paper, we first propose the concept of divergence measure on neutrosophic sets. We also provide some formulas for the divergence measure for neutrosophic sets. After that, we investigate the properties of proposed neutrosophic divergence measure. Finally, we also apply these formulas in medical problem and the classification problem.

In the literature, new types of neutrosophic sets have been introduced in the meantime by the growing neutrosophic community. We present a few: Pythagorean Neutrosophic Set, Fermatean Neutrosophic Set, Generalized Fermatean Neutrosophic Set, n-power Neutrosophic Set, Cubic Spherical Neutrosophic Set, Spherical Neutrosophic Set, n-HyperSpherical Neutrosophic Set, Refined n-HyperSpherical Neutrosophic Set.

In this paper we have defined neutrosophic ideals, neutrosophic interior ideals, netrosophic quasi-ideals and neutrosophic bi-ideals (neutrosophic generalized bi-ideals) and proved some results related to them. Furthermore, we have done some characterization of a neutrosophic LA-semigroup by the properties of its neutrosophic ideals. It has been proved that in a neutrosophic intra-regular LA-semigroup neutrosophic left, right, two-sided, interior, bi-ideal, generalized bi-ideal and quasi-ideals coincide and we have also proved that the set of (...) neutrosophic ideals of a neutrosophic intra-regular LA-semigroup forms a semilattice structure. (shrink)

A rough fuzzy set is the result of the approximation of a fuzzy set with respect to a crisp approximation space. It is a mathematical tool for the knowledge discovery in the fuzzy information systems. In this paper, we introduce the concepts of rough standard neutrosophic sets and standard neutrosophic information system, and give some results of the knowledge discovery on standard neutrosophic information system based on rough standard neutrosophic sets.

In this paper, we introduce the concepts of Pythagorean fuzzy valued neutrosophic set (PFVNS) and Pythagorean fuzzy valued neutrosophic (PFVNV) constructed by considering Pythagorean fuzzy values (PFVs) instead of numbers for the degrees of the truth, the indeterminacy and the falsity, which is a new extension of intuitionistic fuzzy valued neutrosophic set (IFVNS). By means of PFVNSs, the degrees of the truth, the indeterminacy and the falsity can be given in Pythagorean fuzzy environment and more sensitive evaluations (...) are made by a decision maker in decision making problems compared to IFVNSs. In other words, such sets enable a decision maker to evaluate the degrees of the truth, the indeterminacy and the falsity as PFVs to model the uncertainty in the evaluations. (shrink)

This book explores the development and importance of these mathematical approaches in managing uncertainty and imprecision in decision-making, and focuses on the practical applications of these theories in decision-making, demonstrating how they improve uncertainty modeling and facilitate more effective decision-making across various fields. By leveraging these frameworks, decision-makers can better navigate complexity, leading to more accurate and dependable outcomes. Ultimately, it highlights the progression from fuzzy to intuitionistic fuzzy to neutrosophic set theories as a significant advancement in capturing and (...) analyzing uncertainty within decision-making contexts. (shrink)

In this short note we show that the so-called Ambiguous Set (2019) is a subclass of the Double Refined Indeterminacy Neutrosophic Set (2017) and is a particular case of the Refined Neutrosophic Set (2013). Also, the Ambiguous Set is similar to the Quadripartitioned Neutrosophic Set (2016), and Belnap’s Four-Valued Logic (1975).

The purpose of this paper is to introduce and study the characteristic function of a neutrosophic set. After given the fundamental definitions of neutrosophic set operations generated by the characteristic function of a neutrosophic set ( Ng for short), we obtain several properties, and discussed the relationship between neutrosophic sets generated by Ng and others. Finally, we introduce the neutrosophic topological spaces generated by Ng . Possible application to GIS topology rules are touched upon.

In this paper, we propose some transfor mations based on the centroid points between single valued neutrosophic numbers. We introduce these trans formations according to truth, indeterminacy and falsity value of single valued neutrosophic numbers. We propose a new similarity measure based on falsity value between single valued neutrosophic sets. Then we prove some properties on new similarity measure based on falsity value between falsity value between single valued neutrosophic sets. Furthermore, we propose similarity measure based (...) on falsity value between single valued neutrosophic sets based on the centroid points of transformed single valued neutrosophic numbers. We also apply the proposed similarity measure between single valued neutrosophic sets to deal with pattern recognition problems. (shrink)

In this paper, we introduce the plithogenic set (as generalization of crisp, fuzzy, intuitionistic fuzzy, and neutrosophic sets), which is a set whose elements are characterized by many attributes (parameters)’ values. An attribute value v has a corresponding (fuzzy, intuitionistic fuzzy, or neutrosophic) degree of appurtenance d(x,v) of the element x, to the set P, with respect to some given criteria. In order to obtain a better accuracy for the plithogenic aggregation operators in the plithogenic set, and for (...) a more exact inclusion (partial order), a (fuzzy, intuitionistic fuzzy, or neutrosophic) contradiction (dissimilarity) degree is defined be-tween each attribute value and the dominant (most important) attribute value. The plithogenic intersection and union are linear combinations of the fuzzy operators tnorm and tconorm, while the plithogenic complement, inclusion (inequality), equality are influenced by the attribute values contradiction (dissimilarity) degrees. This article offers some examples and applications of these new concepts in our everyday life. (shrink)

This book introduces readers to the concept of the neutrosophic set which can deal with dynamic and complex decision-making problems. With the complexity of the socio-economic environment, today's decision-making is one of the most notable ventures, whose mission is to decide the best alternative under numerous known or unknown criteria. This book provides a large amount of theoretical and practical information about the latest research in the field, allowing readers to gain an extensive understanding of both the fundamentals and (...) applications of neutrosophic sets to solve different kinds of decision-making problems and mathematical programming such as medical diagnosis, pattern recognition, construction problems, technology selection etc. (shrink)

Gathering the attitudes of the examined respondents would be very significant in some evaluation models. Therefore, a multiple criteria approach based on the use of the neutrosophic set is considered in this paper. An example of the evaluation of restaurants is considered at the end of this paper with the aim to present in detail the proposed approach.

Cross entropy measure is one of the best way to calculate the divergence of any variable from the priori one variable. We define a new cross entropy measure under interval neutrosophic set environment.

This study addresses the challenges of uncertainty, vagueness, and imprecision in real-world decision-making, particularly for small-scale farmers selecting short-term crops. The central problem is the lack of a systematic framework to evaluate conflicting criteria such as investment, yield, market demand, and soil requirements. To solve this, a Multi-Criteria Decision-Making (MCDM) framework is introduced, which integrates Cubic Spherical Neutrosophic Sets (CSNS) and Neutrosophic Hyper Soft Sets (NHSS). The research also proposes a novel cubic spherical neutrosophic Bonferroni mean operator (...) for aggregating neutrosophic sets. The effectiveness of this approach is demonstrated through a case study in Tamil Nadu, focusing on selecting the most suitable crop for different climatic zones. A sensitivity analysis confirms the model's robustness and reliability. The tool aims to improve resource utilization, reduce risks, and promote agricultural sustainability. (shrink)

Interval bipolar neutrosophic set is a significant extension of interval neutrosophic set where every element of the set comprises of three independent positive membership functions and three independent negative membership functions. In this study, we first define correlation coefficient, and weighted correlation coefficient measures of interval bipolar neutrosophic sets and prove their basic properties. Then, we develop a new multi-attribute decision making strategy based on the proposed weighted correlation coefficient measure. Finally, we solve an investment problem with (...) interval bipolar neutrosophic information and comparison is given to demonstrate the applicability and effectiveness of the proposed strategy. (shrink)

In this paper, we define a new cosine similarity between two interval valued neutrosophic sets based on Bhattacharya’s distance [19]. The notions of interval valued neutrosophic sets (IVNS, for short) will be used as vector representations in 3D-vector space. Based on the comparative analysis of the existing similarity measures for IVNS, we find that our proposed similarity measure is better and more robust. An illustrative example of the pattern recognition shows that the proposed method is simple and effective.

Neutrosophy considers a proposition, theory, event, concept, or entity, "A" in relation to its opposite, "Anti A" and that which is not A, "Non-A", and that which is neither "A" nor "Anti-A", denoted by "Neut-A". Neutrosophy is the basis of neutrosophic logic, neutrosophic probability, neutrosophic set, and neutrosophic statistics.

In this paper, we define projection and bidirectional projection measures between interval rough neutrosophic sets and prove their basic properties. Then two new multi attribute decision making strategies are proposed based on interval rough neutrosophic projection and bidirectional projection measures respectively. Then the proposed methods are applied for solving multi attribute decision making problems. Finally, a numerical example is solved to show the feasibility, applicability and effectiveness of the proposed strategies.

This paper introduces a refined single-valued neutrosophic set (RSVNS) and presents a similarity measure of RSVNSs. Then a multicriteria decision-making method with RSVNS information is developed based on the similarity measure of RSVNSs. By the similarity measure between each alternative and the ideal solution (ideal alternative), all the alternatives can be ranked and the best one can be selected as well. Finally, an actual example on the selecting problems of construction projects demonstrates the application and effectiveness of the proposed (...) method. (shrink)

We have introduced for the first time the degree of dependence (and consequently the degree of independence) between the components of the fuzzy set, and also between the components of the neutrosophic set in our 2006 book’s fifth edition [1]. Now we extend it for the first time to the refined neutrosophic set considering the degree of dependence or independence of subcomponets.

his book introduces Type-k Neutrosophic Sets as a recursive extension of classical neutrosophic logic. While standard neutrosophic sets represent each proposition through three independent components—truth, indeterminacy, and falsity—Type-k Neutrosophic Sets allow each of these components to be recursively characterized by further neutrosophic triplets. This creates a hierarchical epistemic structure capable of representing nested uncertainty, meta-uncertainty, contradiction, ignorance, and paraconsistent states that cannot be adequately captured by classical probability, fuzzy logic, or intuitionistic fuzzy models. -/- The (...) book develops the formal foundations of this recursive framework, proves its expressive hierarchy, and extends it through single-valued neutrosophic tensors and plithogenic tensors for multi-criteria decision-making. It also explores connections with neutrosophic paraconsistent logic and applies the framework to the epistemic auditing of Large Language Models. Particular attention is given to hallucination detection, the limitations of softmax-based confidence, and the need for triadic and recursive models of truth, falsity, and indeterminacy in AI evaluation. -/- By combining non-classical logic, tensorial structures, decision theory, and artificial intelligence auditing, the work proposes a new formal and computational framework for understanding uncertainty in complex epistemic systems. It is intended for researchers in neutrosophic logic, philosophy of logic, formal epistemology, artificial intelligence, paraconsistent reasoning, and decision sciences. (shrink)

This book introduces Type-k Neutrosophic Sets, a recursive generalization of classical neutrosophic sets designed to represent nested and higher-order uncertainty. Unlike traditional neutrosophic frameworks, where truth, indeterminacy, and falsity are represented as independent scalar values, the Type-k model recursively assigns a complete neutrosophic triplet to each component, producing increasingly expressive structures capable of modeling meta-uncertainty and epistemic self-assessment. The work establishes the formal mathematical foundations of Type-k neutrosophic representations, proves their strict expressive hierarchy, and develops (...) recursive aggregation operators that preserve compatibility with classical single-valued neutrosophic systems. The book further extends neutrosophic theory through tensorial frameworks, including single-valued neutrosophic tensors and plithogenic tensors, enabling advanced multi-criteria decision-making and multilinear analysis. A neutrosophic paraconsistent logic is introduced to address contradiction and indeterminacy within formal reasoning systems. The theoretical developments are applied to the epistemic auditing of large language models, where extensive empirical studies demonstrate the limitations of conventional probabilistic approaches and reveal the necessity of extended neutrosophic representations for capturing hallucinations, paradoxes, contradictions, and structural uncertainty. The volume concludes with open problems and future research directions connecting neutrosophic mathematics, artificial intelligence auditing, decision sciences, and paraconsistent reasoning. (shrink)

In this paper, we generalize the definition of Neutrosophic sets and present a method for extending crisp functions on Neutrosophic sets and study some properties of such extended functions.

In this paper, one extends the single-valued complex neutrosophic set to the subsetvalued complex neutrosophic set, and afterwards to the subset-valued complex refined neutrosophic set.

The Neutrosophic Set offers a robust mathematical framework for handling uncertainty by incorporating three key membership functions: truth, indeterminacy, and falsity [36]. To address the growing complexity of real-world problems, advanced extensions such as the HyperNeutrosophic Set and the SuperHyperNeutrosophic Set have been developed. These higher-dimensional models provide an enhanced representation of uncertainty and are formally defined in [20, 21, 22, 24, 23, 37, 19]. In addition, the TreeNeutrosophic Set and ForestNeutrosophic Set are well-established extensions of the Neutrosophic (...) Set, broadening its applicability to structured and hierarchical data[16]. This paper explores the practical applications of the Neutrosophic Set in livestock management [28, 11, 2], highlighting its potential to enhance decision-making processes in this field. Additionally, we introduce several new variations of the Neutrosophic Set and the ForestNeutrosophic Set [16], further extending their theoretical framework and practical applicability. (shrink)

In this paper, we have developed an Excel package to be utilized for calculating neutrosophic data and analyze them. The use of object oriented programming techniques and concepts as they may apply to the design and development a new framework to implement neutrosophic data operations, the c# programming language, NET Framework and Microsoft Visual Studio are used to implement the neutrosophic classes. We have used Excel as it is a powerful tool that is widely accepted and used (...) for statistical analysis. Figure 1 shows Class Diagram of the implemented package. Figure 2 presents a working example of the package interface calculating the complement. Our implemented Neutrosophic package can calculate Intersection, Union, and Complement of the neutrosophic set. Figure 3 presents our neutrosophic package capability to draw figures of presented neutrosophic set. Figure 4 presents charting of Union operation calculation, and figure 5 Intersection Operation. neutrosophic set are characterized by its efficiency as it takes into consideration the three data items: True, Intermediate, and False. -/- . (shrink)

Clustering plays an important role in data mining, pattern recognition, and machine learning. Single-valued neutrosophic sets are useful means to describe and handle indeterminate and inconsistent information that fuzzy sets and intuitionistic fuzzy sets cannot describe and deal with. To cluster the data represented by single-valued neutrosophic information, this article proposes single-valued neutrosophic clustering methods based on similarity measures between SVNSs. First, we define a generalized distance measure between SVNSs and propose two distance-based similarity measures of SVNSs. (...) Then, we present a clustering algorithm based on the similarity measures of SVNSs to cluster single-valued neutrosophic data. Finally, an illustrative example is given to demonstrate the application and effectiveness of the developed clustering methods. (shrink)

This book introduces and develops neutrosophy as a unifying philosophical and mathematical framework for the study of neutralities and their interactions with truth and falsity. Extending beyond classical, fuzzy, and intuitionistic logics, the work formalizes neutrosophic logic, in which every proposition is characterized by three independent components: truth (T), indeterminacy (I), and falsity (F). These components are defined over a non-standard unit interval, allowing values beyond conventional probabilistic constraints and enabling the modeling of incomplete, inconsistent, and paradoxical information. The (...) volume systematically generalizes several foundational domains, including logic, set theory, probability, and statistics, by introducing corresponding neutrosophic structures such as neutrosophic sets, neutrosophic probability, and neutrosophic statistics. The framework accommodates contradictions, uncertainty, and vagueness in a unified mathematical structure, overcoming limitations of existing theories that cannot adequately represent indeterminacy or simultaneous truth and falsity. The book also establishes formal definitions based on non-standard analysis and hyper-real numbers, providing a rigorous mathematical foundation for neutrosophic components and operations. By integrating philosophical insight with mathematical formalism, the work opens new directions for research across disciplines, including artificial intelligence, decision theory, physics, and social sciences, where complex and uncertain phenomena require more expressive representational models. (shrink)

The objective of the paper is to implement and validate diagnosis in the medical field via refined neutrosophic fuzzy logic (RNFL). As such, we have proposed a Max-Min composition (MMC) method in RNFL. This method deals with the diagnosis under certain constraints like uncertainty and indeterminacy. Further, we have considered the diagnosis problems to validate the sensitivity analysis of the novel multi attribute decision-making technique. Finally, we gave the graphical representations and compared the obtained results with other existing measures (...) in refined neutrosophic fuzzy sets. (shrink)

The concept of neutrosophic locally compact and neutrosophic compact open topology are introduced and some interesting propositions are discussed.

This Paper combines interval- valued neutrouphic sets and rough sets. It studies roughness in interval- valued neutrosophic sets and some of its properties. Finally we propose a Hamming distance between lower and upper approximations of interval valued neutrosophic sets.

Bipolar neutrosophic sets are the extension of neutrosophic sets and are based on the idea of positive and negative preferences of information. Projection measure is a useful apparatus for modelling real life decision making problems. In the paper, we define projection, bidirectional projection and hybrid projection measures between bipolar neutrosophic sets. Three new methods based on the proposed projection measures are developed for solving multi-attribute decision making problems. In the solution process, the ratings of performance values of (...) the alternatives with respect to the attributes are expressed in terms of bipolar neutrosophic values. We calculate projection, bidirectional projection, and hybrid projection measures between each alternative and ideal alternative with bipolar neutrosophic information. All the alternatives are ranked to identify the best alternative. Finally, a numerical example is provided to demonstrate the applicability and effectiveness of the developed methods. Comparison analysis with the existing methods in the literature in bipolar neutrosophic environment is also performed. (shrink)

Since the world is full of indeterminacy, the Neutrosophics found their place into contemporary research. We now introduce for the first time the notions of Neutrosophic Crisp Sets and Neutrosophic Topology on Crisp Sets. We develop the 2012 notion of Neutrosophic Topological Spaces and give many practical examples. Neutrosophic Science means development and applications of Neutrosophic Logic, Set, Measure, Integral, Probability etc., and their applications in any field. It is possible to define the neutrosophic (...) measure and consequently the neutrosophic integral and neutrosophic probability in many ways, because there are various types of indeterminacies, depending on the problem we need to solve. Indeterminacy is different from randomness. Indeterminacy can be caused by physical space, materials and type of construction, by items involved in the space, or by other factors. In 1965 [51], Zadeh generalized the concept of crisp set by introducing the concept of fuzzy set, corresponding to the situation in which there is no precisely defined set;there are increasing applications in various fields, including probability, artificial intelligence, control systems, biology and economics. Thus, developments in abstract mathematics using the idea of fuzzy sets possess sound footing. In accordance, fuzzy topological spaces were introduced by Chang [12] and Lowen [33]. After the development of fuzzy sets, much attention has been paid to the generalization of basic concepts of classical topology to fuzzy sets and accordingly developing a theory of fuzzy topology [1-58]. In 1983, the intuitionistic fuzzy set was introduced by K. Atanassov [55, 56, 57] as a generalization of the fuzzy set, beyond the degree of membership and the degree of non-membership of each element. In 1999 and 2002, Smarandache [71, 72, 73, 74] defined the notion of Neutrosophic Sets, which is a generalization of Zadeh’s fuzzy set and Atanassov's intuitionistic fuzzy set. Some neutrosophic concepts have been investigated by Salama et al. [61-70]. Forwarding the study of neutrosophic sets, this book consists of seven chapters, targeting to:  generalize the previous studies in [1-59], and[91-94] so to define the neutrosopic crisp set and neutrosophic set concepts;  discuss their main properties; A. A. Salama & Florentin Smarandache  introduce and study some concepts of neutrosophic crisp and neutrosophic topological spaces and deduce their properties;  deduce many types of functions and give the relationships between different neutrosophic topological spaces, which helps to build new properties of neutrosophic topological spaces;  stress once more the importance of Neutrosophic Ideal as a nontrivial extension of neutrosophic set and neutrosophic logic [71, 72, 73, 74];  propose applications on computer sciences by using neutrosophic sets. (shrink)

The purpose of this paper is to introduce new types of neutrosophic crisp sets with three types 1, 2, 3. After given the fundamental definitions and operations, we obtain several properties, and discussed the relationship between neutrosophic crisp sets and others. Also, we introduce and study the neutrosophic crisp point and neutrosophic crisp relations. Possible applications to database are touched upon.

Neutrosophic set is a new mathematical tool for handling problems involving imprecise, indetermi nacy and inconsistent data. Pseudo-BCI algebra is a kind of non-classical logic algebra in close connection with various non-commutative fuzzy logics. Recently, we applied neutrosophic set theory to pseudo-BCI al gebras. In this paper, we study neutrosophic filters in pseudo-BCI algebras. The concepts of neutrosophic regular filter, neutrosophic closed filter and fuzzy regular filter in pseudo-BCI algebras are introduced, and some basic properties (...) are discussed. Moreover, the relationships among neutrosophic regular filter, fuzzy filters and anti-grouped neutrosophic filters are prese nted, and the results are proved: a neutrosophic filter (fuzzy filter) is a neutrosophic regular filter (fuzzy regular filter), if and only if it is both a neutrosophic closed filter (fuzzy closed filter) and an anti-grouped neutrosophic filter (fuzzy anti-grouped filter). (shrink)

The neutrosophic sets were known since 1999, and because of their wide applications and their great flexibility to solve the problems, we used these the concepts to define a new types of neutrosophic crisp closed sets and limit points in neutrosophic crisp topological space, namly [neutrosophic crisp Gem sets and neutrosophic crisp Turig points] respactvely, we stady their properties in details and join it with topological concepts. Finally we used [neutrosophic crisp Gem sets and (...) neutrosophic crisp Turig points] to introduce of topological concepts as: neutrosophic crisp closed (open) sets, neutrosophic crisp closure, neutrosophic crisp interior, neutrosophic crisp extrior and neutrosophic crisp boundary which are fundamental for further reserch on neutrosophic crisp topology and will setrengthen the foundations of theory of neutrosophic topological spaces. (shrink)

In this paper, we introduce and study the concept of" neutrosophic closed set" and" neutrosophic continuous function". Possible application to GIS topology rules are touched upon.

This book provides a comprehensive and systematic survey of modern set-theoretic frameworks developed to model uncertainty, vagueness, indeterminacy, and partial truth in real-world phenomena. It synthesizes a wide spectrum of classical and advanced theories, including fuzzy sets, intuitionistic fuzzy sets, neutrosophic sets, plithogenic sets, and numerous related extensions. The work organizes these concepts across multiple structural and conceptual dimensions—such as interval, granular, probabilistic, and complex extensions—highlighting their mathematical foundations, interrelations, and distinguishing characteristics. By unifying diverse approaches under a coherent (...) analytical perspective, the book clarifies how different uncertainty models complement or generalize one another. It also explores emerging directions in set theory, including hybrid and dynamic frameworks, offering insights into their applicability in fields such as decision-making, control systems, artificial intelligence, and data analysis. Serving both as a reference and a conceptual guide, the book contributes to advancing the theoretical understanding and practical deployment of uncertainty-oriented mathematical models. (shrink)

This research monograph introduces the theory of Neutrosophic Super Hyper Sets (NSHS), a novel mathematical framework that integrates neutrosophic set theory with the hierarchical structure of iterated power sets to model uncertainty in multi-level and nested systems. Building upon the evolution from classical, fuzzy, intuitionistic fuzzy, and neutrosophic sets, the authors address situations in which information is associated not only with individual elements but also with collections, collections of collections, and higher-order hierarchical structures. The work develops the (...) foundational theory of NSHS, including definitions, set-theoretic operations, algebraic properties, lattice structures, topology, relations, mappings, similarity measures, entropy, and distance metrics. Advanced extensions such as interval-valued, bipolar, plithogenic, and multi-polar neutrosophic super hyper sets are also formulated. The monograph establishes rigorous mathematical results concerning partial orders, distributive lattices, aggregation operators, score and accuracy functions, and topological structures, while demonstrating practical applications in decision-making, medical diagnosis, environmental risk assessment, data mining, machine learning, knowledge graphs, network science, and complex systems analysis. By combining hierarchical modeling with the expressive representation of truth, indeterminacy, and falsity, the NSHS framework provides a powerful tool for analyzing uncertainty in increasingly interconnected and multi-layered scientific, technological, and social systems. (shrink)