Neutrosophic Over-/Under-/Off-Set and -Logic were defined for the first time by Smarandache in 1995 and published in 2007. They are totally different from other sets/logics/probabilities. He extended the neutrosophic set respectively to Neutrosophic Overset {when some neutrosophic component is > 1}, Neutrosophic Underset {when some neutrosophic component is < 0}, and to Neutrosophic Offset {when some neutrosophic components are off the interval [0, 1], i.e. some neutrosophic component > 1 and (...) other neutrosophic component < 0}. This is no surprise with respect to the classical fuzzy set/logic, intuitionistic fuzzy set/logic, or classical/imprecise probability, where the values are not allowed outside the interval [0, 1], since our real-world has numerous examples and applications of over-/under-/off-neutrosophic components. (shrink)

We have defined the Neutrosophic Over- /Under-/Off-Set and -Logic for the first time in 1995 and published in 2007. During 1995-2016 we presented them to various national and international conferences and seminars ([16]-[37]) and did more publishing during 2007-2016 ([1]-[15]). These new notions are totally different from other sets/logics/probabilities. We extended the neutrosophic set respectively to Neutrosophic Overset {when some neutrosophic component is > 1}, to Neutrosophic Underset {when some neutrosophic component is < (...) 0}, and to Neutrosophic Offset {when some neutrosophic components are off the interval [0, 1], i.e. some neutrosophic component > 1 and other neutrosophic component < 0}. This is no surprise since our real-world has numerous examples and applications of over-/under-/off-neutrosophic components. (shrink)

The second volume of “Advancing Uncertain Combinatorics through Graphization, Hyperization, and Uncertainization: Fuzzy, Neutrosophic, Soft, Rough, and Beyond” presents a deep exploration of the progress in uncertain combinatorics through innovative methodologies like graphization, hyperization, and uncertainization. This volume integrates foundational concepts from fuzzy, neutrosophic, soft, and rough set theory, among others, to further advance the field. Combinatorics and set theory, two central pillars of mathematics, focus on counting, arrangement, and the study of collections under defined rules. Combinatorics excels (...) in handling uncertainty, while set theory has evolved with concepts such as fuzzy and neutrosophic sets, which enable the modeling of complex real-world uncertainties by addressing truth, indeterminacy, and falsehood. These advancements, when combined with graph theory, give rise to novel forms of uncertain sets in "graphized" structures, including hypergraphs and superhypergraphs. Innovations such as Neutrosophic Oversets, Undersets, and Offsets, as well as the Nonstandard Real Set, build upon traditional graph concepts, pushing both theoretical and practical boundaries. The synthesis of combinatorics, set theory, and graph theory in this volume provides a robust framework for addressing the complexities and uncertainties inherent in both mathematical and real-world systems, paving the way for future research and application. In the first chapter, “A Review of the Hierarchy of Plithogenic, Neutrosophic, and Fuzzy Graphs: Survey and Applications”, the authors investigate the interrelationships among various graph classes, including Plithogenic graphs, and explore other related structures. Graph theory, a fundamental branch of mathematics, focuses on networks of nodes and edges, studying their paths, structures, and properties. A Fuzzy Graph extends this concept by assigning a membership degree between 0 and 1 to each edge and vertex, representing the level of uncertainty. The Turiyam Neutrosophic Graph is introduced as an extension of both Neutrosophic and Fuzzy Graphs, while Plithogenic graphs offer a potent method for managing uncertainty. The second chapter, “Review of Some Superhypergraph Classes: Directed, Bidirected, Soft, and Rough”, examines advanced graph structures such as directed superhypergraphs, bidirected hypergraphs, soft superhypergraphs, and rough superhypergraphs. Classical graph classes include undirected graphs, where edges lack orientation, and directed graphs, where edges have specific directions. Recent innovations, including bidirected graphs, have sparked ongoing research and significant advancements in the field. Soft Sets and their extension to Soft Graphs provide a flexible framework for managing uncertainty, while Rough Sets and Rough Graphs address uncertainty by using lower and upper approximations to handle imprecise data. Hypergraphs generalize traditional graphs by allowing edges, or hyperedges, to connect more than two vertices. Superhypergraphs further extend this by allowing both vertices and edges to represent subsets, facilitating the modeling of hierarchical and group-based relationships. The third chapter, “Survey of Intersection Graphs, Fuzzy Graphs, and Neutrosophic Graphs”, explores the intersection graph models within the realms of Fuzzy Graphs, Intuitionistic Fuzzy Graphs, Neutrosophic Graphs, Turiyam Neutrosophic Graphs, and Plithogenic Graphs. The chapter highlights their mathematical properties and interrelationships, reflecting the growing number of graph classes being developed in these areas. Intersection graphs, such as Unit Square Graphs, Circle Graphs, and Ray Intersection Graphs, are crucial for understanding complex graph structures in uncertain environments. The fourth chapter, “Fundamental Computational Problems and Algorithms for SuperHyperGraphs”, addresses optimization problems within the SuperHypergraph framework, such as the SuperHypergraph Partition Problem, Reachability, and Minimum Spanning SuperHypertree. The chapter also adapts classical problems like the Traveling Salesman Problem and the Chinese Postman Problem to the SuperHypergraph context, exploring how hypergraphs, which allow hyperedges to connect more than two vertices, can be used to solve complex hierarchical and relational problems. The fifth chapter, “A Short Note on the Basic Graph Construction Algorithm for Plithogenic Graphs”, delves into algorithms designed for Plithogenic Graphs and Intuitionistic Plithogenic Graphs, analyzing their complexity and validity. Plithogenic Graphs model multi-valued attributes by incorporating membership and contradiction functions, offering a nuanced representation of complex relationships. The sixth chapter, “Short Note of Bunch Graph in Fuzzy, Neutrosophic, and Plithogenic Graphs”, generalizes traditional graph theory by representing nodes as groups (bunches) rather than individual entities. This approach enables the modeling of both competition and collaboration within a network. The chapter explores various uncertain models of bunch graphs, including Fuzzy Graphs, Neutrosophic Graphs, Turiyam Neutrosophic Graphs, and Plithogenic Graphs. In the seventh chapter, “A Reconsideration of Advanced Concepts in Neutrosophic Graphs: Smart, Zero Divisor, Layered, Weak, Semi, and Chemical Graphs”, the authors extend several fuzzy graph classes to Neutrosophic graphs and analyze their properties. Neutrosophic Graphs, a generalization of fuzzy graphs, incorporate degrees of truth, indeterminacy, and falsity to model uncertainty more effectively. The eighth chapter, “Short Note of Even-Hole-Graph for Uncertain Graph”, focuses on Even-Hole-Free and Meyniel Graphs analyzed within the frameworks of Fuzzy, Neutrosophic, Turiyam Neutrosophic, and Plithogenic Graphs. The study investigates the structure of these graphs, with an emphasis on their implications for uncertainty modeling. The ninth chapter, “Survey of Planar and Outerplanar Graphs in Fuzzy and Neutrosophic Graphs”, explores planar and outerplanar graphs, as well as apex graphs, within the contexts of fuzzy, neutrosophic, Turiyam Neutrosophic, and plithogenic graphs. The chapter examines how these types of graphs are used to model uncertain parameters and relationships in mathematical and real-world systems. The tenth chapter, “General Plithogenic Soft Rough Graphs and Some Related Graph Classes”, introduces and explores new concepts such as Turiyam Neutrosophic Soft Graphs and General Plithogenic Soft Graphs. The chapter also examines models of uncertain graphs, including Fuzzy, Intuitionistic Fuzzy, Neutrosophic, and Plithogenic Graphs, all designed to handle uncertainty in diverse contexts. The eleventh chapter, “Survey of Trees, Forests, and Paths in Fuzzy and Neutrosophic Graphs”, provides a comprehensive study of Trees, Forests, and Paths within the framework of Fuzzy and Neutrosophic Graphs. This chapter focuses on classifying and analyzing graph structures like trees and paths in uncertain environments, contributing to the ongoing development of graph theory in the context of uncertainty. (shrink)

The third volume of “Advancing Uncertain Combinatorics through Graphization, Hyperization, and Uncertainization: Fuzzy, Neutrosophic, Soft, Rough, and Beyond” presents an in-depth exploration of the cutting-edge developments in uncertain combinatorics and set theory. This comprehensive collection highlights innovative methodologies such as graphization, hyperization, and uncertainization, which enhance combinatorics by incorporating foundational concepts from fuzzy, neutrosophic, soft, and rough set theories. These advancements open new mathematical horizons, offering novel approaches to managing uncertainty within complex systems. Combinatorics, a discipline focused on (...) counting, arrangement, and structure, often faces challenges when uncertainty is present. Set theory, which underpins combinatorial problems, has evolved to tackle these challenges. The introduction of fuzzy and neutrosophic sets has expanded the toolkit for modeling uncertainty by incorporating elements of truth, indeterminacy, and falsehood into decision-making processes. These innovations seamlessly intersect with graph theory, providing new ways to represent uncertain structures through "graphized" forms such as hypergraphs and superhypergraphs. This volume also introduces advanced concepts like Neutrosophic Oversets, Undersets, and Offsets, which push the boundaries of classical graph theory and offer deeper insights into the mathematical and practical challenges posed by real-world systems. By blending combinatorics, set theory, and graph theory, the authors have created a robust framework for addressing uncertainty in both mathematical systems and their real-world applications. This foundation sets the stage for future breakthroughs in combinatorics, set theory, and related fields. Each chapter in this volume contributes both theoretical foundations and practical applications, demonstrating the power of integrating graph theory, set theory, and uncertainty models. The new ideas, algorithms, and mathematical tools presented here will drive the future of combinatorial research and its applications in uncertain environments. In the first chapter, “Introduction to Upside-Down Logic: Its Deep Relation to Neutrosophic Logic and Applications”, the authors present Upside-Down Logic, a novel logical framework that systematically transforms truths into falsehoods and vice versa, based on contextual shifts. Introduced by F. Smarandache, this paper provides a mathematical definition of Upside-Down Logic, including applications related to the Japanese language. The chapter also introduces Contextual Upside-Down Logic, an extension that adjusts logical connectives alongside flipped truth values, as well as Indeterm-Upside-Down Logic and Certain Upside-Down Logic to address indeterminacy. A simple algorithm is also proposed to demonstrate the computational aspects of this logic. In the second chapter, “Local-Neutrosophic Logic and Local-Neutrosophic Sets: Incorporating Locality with Applications”, the authors introduce Local-Neutrosophic Logic and Local-Neutrosophic Sets, which integrate the concept of locality into Neutrosophic Logic. By defining locality as the influence of immediate surroundings on an object or system, this chapter explores how it affects indeterminacy in real-world problems. The paper also examines potential applications and provides mathematical definitions for these new concepts. The third chapter, “A Review of Fuzzy and Neutrosophic Offsets: Connections to Some Set Concepts and Normalization Function”, extends the concept of offsets in uncertain set-theoretic frameworks, such as Fuzzy Sets, Neutrosophic Sets, and Plithogenic Sets. This chapter introduces several advanced types of offsets, including Nonstationary Fuzzy Offset, Multi-valued Plithogenic Offset, and Subset-valued Neutrosophic Offset, offering deeper insights into handling uncertainty in mathematical models. In the fourth chapter, “Review of Plithogenic Directed, Mixed, Bidirected, and Pangene OffGraph”, the authors build upon Plithogenic Graphs to propose extensions such as Plithogenic Directed OffGraph, Plithogenic BiDirected OffGraph, and Plithogenic Mixed OffGraph. These new concepts, including the Plithogenic Pangene OffGraph, are explored in detail, with a focus on their mathematical properties and potential applications in uncertain graph theory. The fifth chapter, “Short Note on Neutrosophic Closure Matroids”, explores the extension of matroid concepts into Neutrosophic and Turiyam Neutrosophic set theories, introducing Neutrosophic closure matroids. This concept integrates uncertainty, indeterminacy, and liberal states into matroid theory, enhancing its applicability in optimization and combinatorial problems. In the sixth chapter, “Some Graph Parameters for Superhypertree-width and Neutrosophic Tree-width”, the authors discuss graph parameters such as Superhypertree-width and Neutrosophic tree-width. These parameters play a crucial role in the study of graph characteristics, particularly in algorithms and real-world applications. The chapter explores the generalization of hypergraphs to SuperHyperGraphs and examines how these concepts extend tree-width parameters within the context of Neutrosophic logic. (shrink)

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.

In this paper, we introduce the integer programming in neutrosophic environment, by considering coffecients of problem as a triangulare neutrosophic numbers. The degrees of acceptance, indeterminacy and rejection of objectives are simultaneously considered. The Neutrosophic Integer Programming Problem (NIP) is transformed into a crisp programming model, using truth membership (T), indeterminacy membership (I), and falsity membership (F) functions as well as single valued triangular neutrosophic numbers. To measure the efficiency of the model, we solved several numerical (...) examples. (shrink)

In this paper, the cosine similarity measure of neutrosophic refined (multi-) sets is proposed and its properties are studied. The concept of this cosine similarity measure of neutrosophic refined sets is the extension of improved cosine similarity measure of single valued neutrosophic. Finally, using this cosine similarity measure of neutrosophic refined set, the application of medical diagnosis is presented.

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.

The Neutrosophic Precalculus and the Neutrosophic Calculus can be developed in many ways, depending on the types of indeterminacy one has and on the method used to deal with such indeterminacy. This article is innovative since the form of neutrosophic binomial factorial theorem was constructed in addition to its refrains. Two other important theorems were proven with their corollaries, and numerical examples as well. As a conjecture, we use ten (indeterminate) forms in neutrosophic calculus taking an (...) important role in limits. To serve article's aim, some important questions had been answered. (shrink)

Misinformation and biased information significantly impact public perception and political decisions, especially on critical issues such as climate change and environmental conservation. This study aims to understand how indeterminacy and contradiction influence public perception and policy formulation by applying neutrosophic theory to model the complexity and multi-dimensionality of ignorance. Using neutrosophic Likert scales, we capture a nuanced spectrum of opinions on the scientific certainty of human impact on climate change. The results are analyzed through a k-means clustering algorithm (...) to identify patterns and segment participants into groups based on their levels of truth, indeterminacy, and falsehood. This approach reveals deeper insights into public perceptions and aids in evaluating their implications for effective communication and policy-making. (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)

We use a neutrosophic set, instead of an intuitionistic fuzzy because the neutrosophic set is more general, and it allows for independent and partial independent components, while in an intuitionistic fuzzy set, all components are totally dependent. In this article, we present and demonstrate the concept of neutrosophic invariant subgroups. We delve into the exploration of this notion to establish and study the neutrosophic quotient group. Further, we give the concept of a neutrosophic normal subgroup (...) as a novel concept. (shrink)

The objective of this paper is to develop neutrosophic quadruple algebraic hyperstructures. Specifically, we develop neutrosophic quadruple semihypergroups, neutrosophic quadruple canonical hypergroups and neutrosophic quadruple hyperrings and we present elementary properties which characterize them.

The Vehicle Routing Problem (VRP) has been extensively studied by different researchers from all over the world in recent years. Multiple solutions have been proposed for different variations of the problem, such as Capacitive Vehicle Routing Problem (CVRP), Vehicle Routing Problem with Time Windows (VRP-TW), Vehicle Routing Problem with Pickup and Delivery (VRPPD), among others, all of them with deterministic times. In the last years, researchers have been interested in including in their different models the variations that travel times may (...) experience when exposed to all kind of phenomena, mainly vehicle traffic. This article addresses the VRP from this perspective, proposing the design and implementation of a genetic algorithm based on neutrosophic theory for calculating the fitness function of each route, considering the variability and uncertainty present in travel times. A deterministic genetic algorithm is also implemented with the average travel times to compare it with the neutrosophic algorithm using simulation. As conclusion, a deterministic algorithm does not necessarily generate the best solution in the real world, full of uncertainty. Also, the quantification of uncertainty using neutrosophic theory can be used in route planning, opening a broad and interesting field of research for future investigations. (shrink)

“Neutrosophics Knowledge” has been created for publications on advanced studies in neutrosophy, neutrosophic set, neutrosophic logic, neutrosophic probability, neutrosophic statistics that started in 1995 and their applications in any field, such as the neutrosophic structures developed in algebra, geometry, topology, etc. The submitted papers should be professional, in good English and Arabic, containing a brief review of a problem and obtained results. 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. (shrink)

In this paper, we introduce the concept of ""neutrosophic crisp neighborhoods system for the neutrosophic crisp point ". Added to, we introduce and study the concept of neutrosophic crisp local function, and construct a new type of neutrosophic crisp topological space via neutrosophic crisp ideals. Possible application to GIS topology rules are touched upon.

In this paper, we firstly review the neutrosophic set, and then construct two new concepts called neutrosophic implication of type 1 and of type 2 for neutrosophic sets. Furthermore, some of their basic properties and some results associated with the two neutrosophic implications are proven.

Big Data is a large-sized and complex dataset, which cannot be managed using traditional data processing tools. Mining process of big data is the ability to extract valuable information from these large datasets. Association rule mining is a type of data mining process, which is indented to determine interesting associations between items and to establish a set of association rules whose support is greater than a specific threshold. The classical association rules can only be extracted from binary data where an (...) item exists in a transaction, but it fails to deal effectively with quantitative attributes, through decreasing the quality of generated association rules due to sharp boundary problems. In order to overcome the drawbacks of classical association rule mining, we propose in this research a new neutrosophic association rule algorithm. The algorithm uses a new approach for generating association rules by dealing with membership, indeterminacy, and non-membership functions of items, conducting to an efficient decision-making system by considering all vague association rules. To prove the validity of the method, we compare the fuzzy mining and the neutrosophic mining. The results show that the proposed approach increases the number of generated association rules. (shrink)

In this book authors study neutrosophic super matrices. The concept of neutrosophy or indeterminacy happens to be one the powerful tools used in applications like FCMs and NCMs where the expert seeks for a neutral solution. Thus this concept has lots of applications in fuzzy neutrosophic models like NRE, NAM etc. These concepts will also find applications in image processing where the expert seeks for a neutral solution.

Neutrosophic LA-semigroup is a midway structure between a neutrosophic groupoid and a commutative neutrosophic semigroup. Rings are the old concept in algebraic structures. We combine the neutrosophic LA-semigroup and ring together to form the notion of neutrosophic LA-semigroup ring. Neutrosophic LAsemigroup ring is defined analogously to neutrosophic group ring and neutrosophic semigroup ring.

The field of neutrosophic set theory and its applications has been rapidly expanding, particularly since the introduction of the journal "Neutrosophic Sets and Systems." New theories, techniques, and algorithms are being developed at a very high rate. One of the most notable trends in neutrosophic theory is its hybridization with other set theories such as rough set theory, bipolar set theory, soft set theory, hesitant fuzzy set theory, and more. Various hybrid structures like rough neutrosophic sets, (...) neutrosophic soft set, single valued neutrosophic hesitant fuzzy sets, among others, have been proposed in a short period. Neutrosophic sets have proven to be crucial tools across a wide array of fields including data mining, decision making, e-learning, engineering, medical diagnosis, social sciences, and beyond. -/- The third volume in the series “New Trends in Neutrosophic Theories and Applications” focuses on theories, methods, and algorithms for decision making, as well as applications involving neutrosophic information. Some topics introduce new sets such as the Pythagorean neutrosophic vague soft set, the triangular fuzzy penta-partitioned neutrosophic set, interval-valued neutrosophic b-open sets, and interval-valued neutrosophic b-closed sets. Other topics present applications in medical diagnosis, non-preemptive neutrosophic priority queues with uneven services (labeled as NM/NM/1), AHP in an interval neutrosophic set environment, MAGDM in a triangular fuzzy neutrosophic number environment, MAGDM in a pentapartitioned neutrosophic environment, the entropy-ARAS strategy in a single-valued neutrosophic number environment, and the MABAC strategy in a rough neutrosophic set environment. (shrink)

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)

Neutrosophy has been introduced some years ago by Florentin Smarandache as a new branch of philosophy dealing with “the origin, nature and scope of neutralities, as well as their interactions with different ideational spectra”. A variety of new theories have been developed on the basic principles of neutrosophy: among them is neutrosophic logics, a family of many-valued systems that can be regarded as a generalization of fuzzy logics. In this paper we present a critical introduction to neutrosophic logics, (...) focusing on the problem of defining suitable neutrosophic propositional connectives and discussing the relationship between neutrosophic logics and other well-known frameworks for reasoning with uncertainty and vagueness, such as (intuitionistic and interval-valued) fuzzy systems and Belnap’s logic. (shrink)

In this paper, we introduce and investigate a new class of sets and functions between topological space called neutrosophic semi-supra open set and neutrosophic semi-supra open continuous functions respectively.

The notion of neutrosophic cubic set is originated from the hybridization of the concept of neutrosophic set and interval valued neutrosophic set. We define similarity measure for neutrosophic cubic sets and prove some of its basic properties. We present a new multi criteria group decision making method with linguistic variables in neutrosophic cubic set environment. Finally, we present a numerical example to demonstrate the usefulness and applicability of the proposed method.

Soft neutrosophic group and soft neutrosophic subgroup are generalized to soft neutrosophic bigroup and soft neutrosophic N-group respectively in this paper. Different kinds of soft neutrosophic bigroup and soft neutrosophic N-group are given. The structural properties and theorems have been discussed with a lot of examples to disclose many aspects of this beautiful man made structure.

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.

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

In this paper, we prove that Neutrosophic Statistics is more general than Interval Statistics, since it may deal with all types of indeterminacies (with respect to the data, inferential procedures, probability distributions, graphical representations, etc.), it allows the reduction of indeterminacy, and it uses the neutrosophic probability that is more general than imprecise and classical probabilities and has more detailed corresponding probability density functions. While Interval Statistics only deals with indeterminacy that can be represented by intervals. And we (...) respond to the arguments by Woodall et al. [1]. We show that not all indeterminacies (uncertainties) may be represented by intervals. Also, in some cases, we should better use hesitant sets (that have less indeterminacy) instead of intervals. We redirect the authors to the Plithogenic Probability and Plithogenic Statistics which are the most general forms of MultiVariate Probability and Multivariate Statistics respectively (including, of course, the Imprecise Probability and Interval Statistics as subclasses). (shrink)

The neutrosophic automorphisms of a neutrosophic groups G (I) , denoted by Aut(G (I)) is a neu-trosophic group under the usual mapping composition. It is a permutation of G (I) which is also a neutrosophic homomorphism. Moreover, suppose that X1 = X(G (I)) is the neutrosophic group of inner neutrosophic auto-morphisms of a neutrosophic group G (I) and Xn the neutrosophic group of inner neutrosophic automorphisms of Xn-1. In this paper, we show (...) that if any neutrosophic group of the sequence G (I), X1, X2, … is the identity, then G (I) is nilpotent. (shrink)

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)

In this paper, we define for the first time three neutrosophic actions and their properties. We then introduce the prevalence order on {T, I, F} with respect to a given neutrosophic operator “o”, which may be subjective - as defined by the neutrosophic experts; and the refinement of neutrosophic entities <A>, <neutA>, and <antiA> . Then we extend the classical logical operators to neutrosophic literal logical operators and to refined literal logical operators, and we define (...) the refinement neutrosophic literal space. (shrink)

In this article, we first provide a modified definition of SuperHyperGraphs (SHG) and we call it Restricted SuperHyperGraphs (R-SHG). We then generalize the R-SHG to the neutrosophic graphs and then define the corresponding trees. In the following, we examine the Helly property for subtrees of SuperHyperGraphs.

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 paper is devoted to present Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) method for multi-attribute group decision making under rough neutrosophic environment. The concept of rough neutrosophic set is a powerful mathematical tool to deal with uncertainty, indeterminacy and inconsistency. In this paper, a new approach for multi-attribute group decision making problems is proposed by extending the TOPSIS method under rough neutrosophic environment. Rough neutrosophic set is characterized by the upper and lower (...) approximation operators and the pair of neutrosophic sets that are characterized by truth-membership degree, indeterminacy membership degree, and falsity membership degree. In the decision situation, ratings of alternatives with respect to each attribute are characterized by rough neutrosophic sets that reflect the decision makers’ opinion. Rough neutrosophic weighted averaging operator has been used to aggregate the individual decision maker’s opinion into group opinion for rating the importance of attributes and alternatives. Finally, a numerical example has been provided to demonstrate the applicability and effectiveness of the proposed approach. (shrink)

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)

In this paper we define neutrosophic bi-LAsemigroup and neutrosophic N-LA-semigroup. Infact this paper is an extension of our previous paper neutrosophic left almost semigroup shortly neutrosophic LAsemigroup. We also extend the neutrosophic ideal to neutrosophic biideal and neutrosophic N-ideal. We also find some new type of neutrosophic ideal which is related to the strong or pure part of neutrosophy. We have given sufficient amount of examples to illustrate the theory of neutrosophic (...) bi-LA-semigroup, neutrosophic N-LAsemigroup and display many properties of them this paper. (shrink)

In this paper we extend the theory of neutrosophic rings and neutrosophic fields to soft sets and construct soft neutrosophic rings and soft neutrosophic fields. We also extend neutrosophic ideal theory to form soft neutrosophic ideal over a neutrosophic ring and soft neutrosophic ideal of a soft neutrosophic ring . We have given many examples to illustrate the theory of soft neutrosophic rings and soft neutrosophic fields and display many (...) properties of of these. At the end of this paper we gave soft neutrosophic ring homomorphism. (shrink)

Soft set theory is a general mathematical tool for dealing with uncertain, fuzzy, not clearly defined objects. In this paper we introduced soft neutrosophic loop,soft neutosophic biloop, soft neutrosophic N -loop with the discuission of some of their characteristics. We also introduced a new type of soft neutrophic loop, the so called soft strong neutrosophic loop which is of pure neutrosophic character. This notion also found in all the other corresponding notions of soft neutrosophic thoery. (...) We also given some of their properties of this newly born soft structure related to the strong part of neutrosophic theory. (shrink)

Soft set theory is a general mathematical tool for dealing with uncertain, fuzzy, not clearly defined objects. In this paper we introduced soft neutrosophic biLA-semigroup,soft neutosophic sub bi-LA-semigroup, soft neutrosophic N -LA-semigroup with the discuission of some of their characteristics. We also introduced a new type of soft neutrophic bi-LAsemigroup, the so called soft strong neutrosophic bi-LAsemigoup which is of pure neutrosophic character. This is also extend to soft neutrosophic strong N-LA-semigroup. We also given some (...) of their properties of this newly born soft structure related to the strong part of neutrosophic theory. (shrink)

In this paper we introduce the concept of interval valued neutrosophic soft topological space together with interval valued neutrosophic soft finer and interval valued neutrosophic soft coarser topology. We also define interval valued neutrosophic interior and closer of an interval valued neutrosophic soft set. Some theorems and examples are cites. Interval valued neutrosophic soft subspace topology are studied. Some examples and theorems regarding this concept are presented.

Science is the basis for managing the affairs of life and human activities, and living without knowledge is a form of wandering and a kind of loss. Using scientific methods helps us understand the foundations of choice, decision-making, and adopting the right solutions when solutions abound and options are numerous. Operational research is considered the best that scientific development has provided because its methods depend on the application of scientific methods in solving complex issues and the optimal use of available (...) resources in various fields, private and governmental work in peace and war, in politics and economics, in planning and implementation, and in various aspects of life. Its basic essence is to use the data provided for the issue under study to build a mathematical model that is the optimal solution. It is the basis on which decision makers rely in managing institutions and companies, and when operations research methods meet with the neutrosophic teacher, we get ideal solutions that take into account all the circumstances and fluctuations that may occur in the work environment over time. One of the most important operations research methods is the linear programming method. Which prompted us to reformulate the linear models, the graphical method, and the simplex method, which are used to obtain the optimal solution for linear models using the concepts of neutrosophic science. In this research, and as a continuation of what we presented previously, we will reformulate the modified simplex algorithm that was presented to address the difficulty that we were facing when applying the direct simplex algorithm. It is the large number of calculations required to be performed in each step of the solution, which requires a lot of time and effort. (shrink)

Neutrosophic set has been derived from a new branch of philosophy, namely Neutrosophy. Neutrosophic set is capable of dealing with uncertainty, indeterminacy and inconsistent information. Neutrosophic set approaches are suitable to modeling problems with uncertainty, indeterminacy and inconsistent information in which human knowledge is necessary, and human evaluation is needed. Neutrosophic set theory firstly proposed in 1998 by Florentin Smarandache, who also developed the concept of single valued neutrosophic set, oriented towards real world scientific and (...) engineering applications. Since then, the single valued neutrosophic set theory has been extensively studied in books and monographs introducing neutrosophic sets and its applications, by many authors around the world. (shrink)

One of the most important theories in linear programming is the dualistic theory and its basic idea is that for every linear model has dual linear model, so that solving the original linear model gives a solution to the dual model. Therefore, when we solving the linear programming model, we actually obtain solutions for two linear models. In this research, we present a study of the models. The neutrosophic dual and the binary simplex algorithm, which works to find the (...) optimal solution for the original and dual models at the same time. The importance of this algorithm is evident in that it is relied upon in several operations research topics, such as integer programming algorithms, some nonlinear programming algorithms, and sensitivity analysis in linear programming. (shrink)

Neutrosophic theory and its applications have been expanding in all directions at an astonishing rate especially after of the introduction the journal entitled “Neutrosophic Sets and Systems”. New theories, techniques, algorithms have been rapidly developed. One of the most striking trends in the neutrosophic theory is the hybridization of neutrosophic set with other potential sets such as rough set, bipolar set, soft set, hesitant fuzzy set, etc. The different hybrid structures such as rough neutrosophic set, (...) single valued neutrosophic rough set, bipolar neutrosophic set, single valued neutrosophic hesitant fuzzy set, etc. are proposed in the literature in a short period of time. Neutrosophic set has been an important tool in the application of various areas such as data mining, decision making, e-learning, engineering, medicine, social science, and some more. (shrink)

Neutrosophic set has been derived from a new branch of philosophy, namely Neutrosophy. Neutrosophic set is capable of dealing with uncertainty, indeterminacy and inconsistent information. Neutrosophic set approaches are suitable to modeling problems with uncertainty, indeterminacy and inconsistent information in which human knowledge is necessary, and human evaluation is needed. Neutrosophic set theory was firstly proposed in 1998 by Florentin Smarandache, who also developed the concept of single valued neutrosophic set, oriented towards real world scientific (...) and engineering applications. Since then, the single valued neutrosophic set theory has been extensively studied in books and monographs, the properties of neutrosophic sets and their applications, by many authors around the world. Also, an international journal - Neutrosophic Sets and Systems started its journey in 2013. Neutrosophic triplet was first defined in 2016 by Florentin Smarandache and Mumtaz Ali and they also introduced the neutrosophic triplet groups in the same year. For every element “x” in a neutrosophic triplet set A, there exist a neutral of “x” and an opposite of “x”. Also, neutral of “x” must be different from the classical neutral element. Therefore, the NT set is different from the classical set. Furthermore, a NT of “x” is showed by . This first volume collects original research and applications from different perspectives covering different areas of neutrosophic studies, such as decision making, Triplet, topology, and some theoretical papers. This volume contains three sections: NEUTROSOPHIC TRIPLET, DECISION MAKING, AND OTHER PAPERS. (shrink)

Many results have been obtained on isolated graphs and complete graphs. In this paper, a necessary and sufficient condition will be proved for a single valued neutrosophic graph to be an isolated single valued neutrosophic graph.

This study provides an innovative approach to neutrosophic algebraic structures by introducing a new structure called Neutrosophic Soft Cubic T-ideal (NSCTID), which combines T-ideal (TID) and neutrosophic Soft Cubic Sets (NSCSs) within the framework of PS-Algebra. Within the already-existing neutrosophic cubic structures, the addition of soft sets with the characteristics of TID makes this structure more desirable. The theoretical development of the proposed structure includes the application of fundamental ideas as union, intersection, the Cartesian product, and (...) homomorphism. We also introduce the notions of NSCTID-translation and NSCTID-multiplication to further enhance the structure of NSCTID. By applying the idea of translation and multiplication, we offer improved algorithm for neutrosophic cubic sets to deal with different parameters that are satisying the TID’s distinctive characteristics. Through this thorough research, we offer an elementary understanding of NSCTID and its capabilities, providing the way to new algebraic structures. (shrink)

Smarandache in 2019 has generalized the algebraic structures to NeutroAlgebraic structures and AntiAlgebraic structures. In this paper, authors, for the first time, define the NeutroAlgebra of neutrosophic triplets group under usual+ and x, built using {Zn, x}, n a composite number, 5 < n < oo, which are not partial algebras. As idempotents in Zn alone are neutrals that contribute to neutrosophic triplets groups, we analyze them and build NeutroAlgebra of idempotents under usual + and x, which are (...) not partial algebras. We prove in this paper the existence theorem for NeutroAlgebra of neutrosophic triplet groups. This proves the neutrals assocaited with neutrosophic triplet groups in { Zn, X} under product is a NeutroAlgebra of triplets. We also prove the non-existence theorem of NeutroAlgebra for neutrosophic triplets in case of Zn when n = 2p, 3p and 4p (for some primes p). Several open problems are proposed. Further, the NeutroAlgebras of extended neutrosophic triplet groups have been obtained. (shrink)

We introduce now for the first time the neutrosophic modal logic. The Neutrosophic Modal Logic includes the neutrosophic operators that express the modalities. It is an extension of neutrosophic predicate logic and of neutrosophic propositional logic.