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)

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

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 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)

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 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 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 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)

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)

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.

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.

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.

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 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).

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.

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.

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 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.

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

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 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)

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

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)

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)

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)

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)

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 paper, we defined (I,T) − standard neutrosophic rough sets based on an implicator I and a t-norm T on I; lower and upper approximations of standard neutrosophic sets in a standard neutrosophic approximation are defined. Some properties of (I,T) − standard neutrosophic rough sets are investigated. We consider the case when the neutrosophic components (truth, indeterminacy, and falsehood) are totally dependent, single-valued, and hence their sum is ≤ 1.

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.

In this paper, we presented antoher concept of neutrosophic open sets called neutrosophic semi-α-open sets and studied their fundamental poperties in neutrosophic topological spaces. We also present neutrospohic semi-α-interior and neutrosophic semi-α-closure and study some of their fundamental properties.

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)

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 work introduces the framework for selecting architecture in 5G networks, considering various technological, performance, economic, and operational factors. With the emergence of 5G technology, the architecture selection process has become pivotal in meeting diverse requirements for ultra-high-speed connectivity, low latency, scalability, and diverse service demands. The evaluation comprehensively analyses different architecture options, including centralized, distributed, cloud-based, and virtualized architectures. Factors such as network performance, scalability, cost-effectiveness, security, and compatibility are considered within a multi-criteria decision-making framework. Findings reveal each architecture (...) option's diverse strengths and limitations, emphasizing the importance of aligning architectural choices with specific use cases, deployment scenarios, and business objectives. We used the concept of multi-criteria decision-making (MCDM) methodology to control the various criteria. We used the single-valued neutrosophic set (SVNS) to deal with vague and inconsistent data. The SVNS is integrated with the hyperSoft set. The proposed methodology uses a single-valued neutrosophic hyperSoft set (SVNHSS) to select the best 5G architecture. We used the VIKOR method as an MCDM method to rank the alternatives. The proposed framework provides stakeholders with a structured methodology to evaluate and prioritize architecture options, facilitating informed decision-making in the complex landscape of 5G network deployments. (shrink)

In this paper, multi attribute decision making problem based on grey relational analysis in neutrosophic cubic set environment is investigated. In the decision making situation, the attribute weights are considered as single valued neutrosophic sets. The neutrosophic weights are converted into crisp weights. Both positve and negative GRA coefficients, and weighted GRA coefficients are determined. Hamming distances for weighted GRA coefficients and standard (ideal) GRA coefficients are determined. The relative closeness coefficients are derived in order to rank (...) the alternatives. The relative closeness coefficients are designed in ascending order. Finally, a numerical example is solved to demonstrate the applicability of the proposed approach. (shrink)

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.

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)

In this paper authors for the first time define a new notion called neutrosophic lattices. We define few properties related with them. Three types of neutrosophic lattices are defined and the special properties about these new class of lattices are discussed and developed. This paper is organised into three sections. First section introduces the concept of partially ordered neutrosophic set and neutrosophic lattices. Section two introduces different types of neutrosophic lattices and the final section studies (...) neutrosophic Boolean algebras. Conclusions and results are provided in section three. (shrink)

In this paper we introduce the notion of filter on the neutrosophic crisp set, then we consider a generalization of the filter’s studies. Afterwards, we present the important neutrosophic crisp filters. We also study several relations between different neutrosophic crisp filters and neutrosophic topologies. Possible applications to database systems are touched upon.

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 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.

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.

In this paper, we first defined soft intervalvalued neutrosophic rough sets(SIVN- rough sets for short) which combines interval valued neutrosophic soft set and rough sets and studied some of its basic properties. This concept is an extension of soft interval valued intuitionistic fuzzy rough sets( SIVIF- rough sets). Finally an illustartive example is given to verfy the developped algorithm and to demonstrate its practicality and effectiveness.

In the real word, in most cases, everything (an attribute, event, proposition, theory, idea, person, object, action, etc.) is evaluated in general by many sources (called experts), not only one. The more sources evaluate a subject, the better accurate result (after fusioning all evaluations). That’s why, in this paper, we straightforwardly extend the Refined Neutrosophic Set to the MultiNeutrosophic Set, and we show that the last two are isomorphic. A MultiNeutrosophic Set is a Neutrosophic Set whose all elements’ (...) degrees of truth/indeterminacy/falsehood are evaluated by many (Multi) sources. Afterwards, we introduce a total order on the set of n-plets of the form (p, r, s), we build the operators on the (p, r, s)-plets, and show several applications of the MultiNeutrosophic Sets. Several particular cases of the MultiNeutrosophic Sets are presented: such as MultiFuzzy Set, MultiIntuitionistic Fuzzy Set, MultiPicture Fuzzy Set, and other Multi(Fuzzy Extension) Set. (shrink)

The paper proposes a new technique for dealing with multi-attribute decision making problems through an extended TOPSIS method under neutrosophic cubic environment. Neutrosophic cubic set is the generalized form of cubic set and is the hybridization of a neutrosophic set with an interval neutrosophic set. In this study, we have defined some operation rules for neutrosophic cubic sets and proposed the Euclidean distance between neutrosophic cubic sets. In the decision making situation, the rating of (...) alternatives with respect to some predefined attributes are presented in terms of neutrosophic cubic information where weights of the attributes are completely unknown. In the selection process, neutrosophic cubic positive and negative ideal solutions have been defined. An extended TOPSIS method is then proposed for ranking the alternatives and finally choosing the best one. Lastly, an illustrative example is solved to demonstrate the decision making procedure and effectiveness of the developed approach. (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)

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)