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Advances in the medical industry has become a major trend because of the new developments in information technologies. This research offers a novel approach for estimating the smart medical devices (SMDs) selection process in a group decision making (GDM) in a vague decision environment. The complexity of the selected decision criteria for the smart medical devices is a significant feature of this analysis. To simulate these processes, a methodology that combines neutrosophics using bipolar numbers with Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) under GDM is suggested. Neutrosophics with TOPSIS approach is applied in the decision making process to deal with the vagueness, incomplete data and the uncertainty, considering the decisions criteria in the data collected by the decision makers (DMs). In this research, the stress is placed upon the choosing of sugar analyzing smart medical devices for diabetics' patients. The main objective is to present the complications of the problem, raising interest among specialists in the healthcare industry and assessing smart medical devices under different evaluation criteria. The problem is formulated as a multi criteria decision type with seven alternatives and seven criteria, and then edited as a multi criteria decision model with seven alternatives and seven criteria. The results of the neutrosophics with TOPSIS model are analyzed, showing that the competence of the acquired results and the rankings are sufficiently stable. The results of the suggested method are also compared with the neutrosophic extensions AHP and MOORA models in order to validate and prove the acquired results. In addition, we used the SPSS program to check the stability of the variations in the rankings by the Spearman coefficient of correlation. The selection methodology is applied on a numerical case, to prove the validity of the suggested approach.

Clustering is a crucial method for deciphering data structure and producing new information. Due to its significance in revealing fundamental connections between the human brain and events, it is essential to utilize clustering for cognitive research. Dealing with noisy data caused by inaccurate synthesis from several sources or misleading data production processes is one of the most intriguing clustering difficulties. Noisy data can lead to incorrect object recognition and inference. This research aims to innovate a novel clustering approach, named Picture-Neutrosophic Trusted Safe Semi-Supervised Fuzzy Clustering (PNTS3FCM), to solve the clustering problem with noisy data using neutral and refusal degrees in the definition of Picture Fuzzy Set (PFS) and Neutrosophic Set (NS). Our contribution is to propose a new optimization model with four essential components: clustering, outlier removal, safe semi-supervised fuzzy clustering and partitioning with labeled and unlabeled data. The effectiveness and flexibility of the proposed technique are estimated and compared with the state-of-art methods, standard Picture fuzzy clustering (FC-PFS) and Confidence-weighted safe semi-supervised clustering (CS3FCM) on benchmark UCI datasets. The experimental results show that our method is better at least 10/15 datasets than the compared methods in terms of clustering quality and computational time. [ABSTRACT FROM AUTHOR]

The objective of this study is to incorporate topological space into the realm of n-Cylindrical Fuzzy Neutrosophic Sets (n-CyFNS), which are the most novel type of fuzzy neutrosophic sets. In this paper, we introduce n-Cylindrical Fuzzy Neutrosophic Topological Spaces (n-CyFNTS), n-Cylindrical Fuzzy Neutrosophic (n-CyFN) open sets, and n-CyFN closed sets. We also defined the n-CyFN base, n-CyFN subbase, and some related theorems here. [ABSTRACT FROM AUTHOR]

This article discusses Neutrosophic Logic interpretation of the Schrodinger's cat paradox. We argue that this paradox involves some degree of indeterminacy (unknown) which Neutrosophic Logic could take into consideration, whereas [...]

The aim of this paper is to introduce the new concept of Quadripartitioned Neutrosophic Pythagorean soft set with T, C, U, F as dependent neutrosophic components and have also discussed some of its properties. [ABSTRACT FROM AUTHOR]

The postulation of neutrosophic numbers has been analyzed from different angles in this paper. In this current era, our main purpose is to mention Decagonal Neutrosophic numbers. The types of linear and non-linear generalized decagonal neutrosophic numbers play a very critical role in the theory related to uncertainty This approach is helpful in getting a crisp number from a neutrosophic number. The definitions regarding Linear, Non-Linear, symmetry, Asymmetry, alpha cuts have been introduced and large decision-making problems using fuzzy TOPSIS have been solved. [ABSTRACT FROM AUTHOR]

Neutrophic offsets are neutrosophic sets whose truth-values lie outside the interval [0, 1].Uninorms are aggregation operators defined in fuzzy logic to generalize t-norms and t-conorms. They satisfy the axioms of symmetry, associativity, monotony and the existence of a neutral element. Fuzzy uninorms have been generalized to intuitionistic fuzzy sets, neutrosophic sets, and neutrosophic offsets, which are called offuninorms in the latter case. This paper aims to demonstrate that offsets and offuninorms can be used in digital image processing, especially for image segmentation and edge detection, moreover algorithms and examples are also provided. [ABSTRACT FROM AUTHOR]

'Angel particle' bearing properties of both particles and anti-particles, which was recently discovered by the Stanford team of experimental physicists, is usually associated with Majorana fermions (predicted in 1937 by Ettore Majorana). In this message we point out that particles bearing properties of both matter and anti-matter were as well predicted without any connexion with particle physics, but on the basis of pure mathematics, namely--neutrosophic logic which is a generalization of fuzzy and intuitionistic fuzzy logics in mathematics., Recently, a group of experimental physicists conducted by Prof. Shoucheng Zhang, in Stanford University, claimed about discovery of the particles that bear properties of both particles and anti-particles. The press-release [...]

This paper introduces the concept of single–valued neutrosophic EQ–subalgebras, single–valued neutrosophic EQ–prefilters and single–valued neutrosophic EQ–filters. We study some properties of single–valued neutrosophic EQ–prefilters and show how to construct single–valued neutrosophic EQ–filters. Finally, the relationship between single–valued neutrosophic EQ–filters and EQ–filters are studied. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new correlation coefficient between intuitionistic fuzzy sets. We then use this new result to compute some examples through which we find that it benefits from such an outcome with some well-known results in the literature. As in statistics with real variables, we refer to variance and covariance between two intuitionistic fuzzy sets. Then, we determined the formula for calculating the correlation coefficient based on the variance and covariance of the intuitionistic fuzzy set, the value of this correlation coefficient is in [–1,1]. Then, we develop this direction to build correlation coefficients between the interval-valued intuitionistic fuzzy sets and apply it in the pattern recognition problem. Finally, we apply this correlation coefficient in clustering problem with intuitionistic fuzzy information. [ABSTRACT FROM AUTHOR]

In many real-life situations, it is often observed that the degree of indeterminacy (neutrality) plays an important role along with the satisfaction and dissatisfaction levels of the decision maker(s) (DM(s)) in any decision making process. Due to some doubt or hesitation, it may necessary for DM(s) to take opinions from experts which leads towards a set of conflicting values regarding satisfaction, indeterminacy and dis-satisfaction level of DM(s). In order to highlight the above-mentioned insight, we have developed an effective framework which reflects the reality involved in any decision-making process. In this study, a multiobjective nonlinear programming problem (MO-NLPP) has been formulated in the manufacturing system. A new algorithm, neutrosophic hesitant fuzzy programming approach (NHFPA), based on singlevalued neutrosophic hesitant fuzzy decision set has been proposed which contains the concept of indeterminacy hesitant degree along with truth and falsity hesitant degrees of different objectives. In order to show the validity and applicability of the proposed approach, a numerical example has been presented. The superiority of the proposed approach has been shown by comparing with other existing approaches. Based on the present work, conclusions and future scope have been presented. [ABSTRACT FROM AUTHOR]

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' 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 between 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. [ABSTRACT FROM AUTHOR]

The notion of a neutrosophic positive implicative N-ideal in BCK-algebras is introduced, and several properties are investigated. Relations between a neutrosophic N-ideal and a neutrosophic positive implicative N-ideal are discussed. Characterizations of a neutrosophic positive implicative N-ideal are considered. Conditions for a neutrosophic N-ideal to be a neutrosophic positive implicative N-ideal are provided. An extension property of a neutrosophic positive implicative N-ideal based on the negative indeterminacy membership function is discussed. [ABSTRACT FROM AUTHOR]

The notions of the neutrosophic triplet and neutrosophic duplet were introduced by Florentin Smarandache. From the existing research results, the neutrosophic triplets and neutrosophic duplets are completely different from the classical algebra structures. In this paper, we further study neutrosophic duplet sets, neutrosophic duplet semi-groups, and cancellable neutrosophic triplet groups. First, some new properties of neutrosophic duplet semi-groups are funded, and the following important result is proven: there is no finite neutrosophic duplet semi-group. Second, the new concepts of weak neutrosophic duplet, weak neutrosophic duplet set, and weak neutrosophic duplet semi-group are introduced, some examples are given by using the mathematical software MATLAB (MathWorks, Inc., Natick, MA, USA), and the characterizations of cancellable weak neutrosophic duplet semi-groups are established. Third, the cancellable neutrosophic triplet groups are investigated, and the following important result is proven: the concept of cancellable neutrosophic triplet group and group coincide. Finally, the neutrosophic triplets and weak neutrosophic duplets in BCI-algebras are discussed. [ABSTRACT FROM AUTHOR]

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 (INS) environment, which we call IN-cross entropy measure and prove its basic properties. We also develop weighted IN-cross entropy measure and investigats its basic properties. Based on the weighted IN-cross entropy measure, we develop a novel strategy for multi attribute group decision making (MAGDM) strategy under interval neutrosophic environment. The proposed multi attribute group decision making strategy is compared with the existing cross entropy measure based strategy in the literature under interval neutrosophic set environment. Finally, an illustratative example of multi attribute group decision making problem is solved to show the feasibility, validity and efficiency of the proposed MAGDM strategy. [ABSTRACT FROM AUTHOR]

Neutrosophic set is a generalization of classical set, fuzzy set, and intuitionistic fuzzy set by employing a degree of truth (T), a degree of indeterminacy (I), and a degree of falsehood (F) associated with an element of the dataset. One of the most essential problems is studying set-theoretic operators in order to be applied to practical applications. Developing Matlab toolboxes for computing the operational matrices in neutrosophic environments is essential to gain more widely-used of neutrosophic algorithms. In this paper, we propose some computing procedures in Matlab for neutrosophic operational matrices, especially i) computing the single-valued neutrosophic matrix; ii) determining complement of a single-valued neutrosophic matrix; iii) computing max-min-min and min-maxmax of two single-valued neutrosophic matrices; v) computing power of a single-valued neutrosophic matrix; vi) computing additional operation and subtraction of two single-valued neutrosophic matrices; and ix) computing transpose of a singlevalued neutrosophic matrix. Numerical examples are given to illustrate their applicability. [ABSTRACT FROM AUTHOR]

For i,j,k,l,m,n ∊ {1,2,3,4}, the notion of (T(i, j),I(k, l),F(m, n))-interval neutrosophic ideals in BCK/BCI-algebras is introduced, and their properties and relations are investigated. [ABSTRACT FROM AUTHOR]

Extreme learning machine (ELM) is known as a kind of single-hidden layer feedforward network (SLFN), and has obtained considerable attention within the machine learning community and achieved various real-world applications. It has advantages such as good generalization performance, fast learning speed, and low computational cost. However, the ELM might have problems in the classification of imbalanced data sets. In this paper, we present a novel weighted ELM scheme based on neutrosophic set theory, denoted as neutrosophic weighted extreme learning machine (NWELM), in which neutrosophic c-means (NCM) clustering algorithm is used for the approximation of the output weights of the ELM. We also investigate and compare NWELM with several weighted algorithms. The proposed method demonstrates advantages to compare with the previous studies on benchmarks. [ABSTRACT FROM AUTHOR]

In this paper, we define correlation coefficient measure between any two rough neutrosophic sets. We also prove some of its basic properties. We develop a new multiple attribute group decision making method based on the proposed correlation coefficient measure. An illustrative example of medical diagnosis is solved to demonstrate the applicability and effecriveness of the proposed method. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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 support-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. [ABSTRACT FROM AUTHOR]

In this paper, we use PCR5 in order to fusion the information of two sources providing subjective probabilities of an event A to occur in the following form: chance that A occurs, indeterminate chance of occurrence of A, chance that A does not occur. [ABSTRACT FROM AUTHOR]

Conditions for the P-intersection and P-intersection of falsity-external (resp. indeterminacy-external and truth-external) neutrosophic cubic sets to be an falsity-external (resp. indeterminacy-external and truth-external) neutrosophic cubic set are provided. Conditions for the P-union and the P-intersection of two truth-external (resp. indeterminacy-external and falsity-external) neutrosophic cubic sets to be a truth-internal (resp. indeterminacy-internal and falsity-internal) neutrosophic cubic set are discussed. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of neutrosophic less than or equal to. The neutrosophy considers every idea < A > together with its opposite or negation < antiA > and with their spectrum of neutralities < neutA > in between them (i.e. notions or ideas supporting neither < A > nor < antiA >). The < neutA > and < antiA > ideas together are referred to as < nonA >. Neutrosophic Set and Neutrosophic Logic are generalizations of the fuzzy set and respectively fuzzy logic (especially of intuitionistic fuzzy set and respectively intuitionistic fuzzy logic) [5]. In neutrosophic logic, a proposition has a degree of truth (T), a degree of indeterminacy (I), and a degree of falsity (F), where T, I, F are standard or non-standard subsets of ]-0, 1+[. Another purpose of this article is to explain the mathematical theory of neutrosophic geometric programming (the unconstrained posynomial case). It is necessary to work in fuzzy neutrosophic space FNs = [0,1] ∪ [0, nI], n ∈ [0,1]. The theory stated in this article aims to be a complementary theory of neutrosophic geometric programming. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The notion of interval valued neutrosophic sets is a generalization of fuzzy sets, intuitionistic fuzzy sets, interval valued fuzzy sets, interval valued intuitionstic fuzzy sets and single valued neutrosophic sets. We apply for the first time to graph theory the concept of interval valued neutrosophic sets, an instance of neutrosophic sets. We introduce certain types of interval valued neutrosophc graphs (IVNG) and investigate some of their properties with proofs and examples. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the goal programming in neutrosophic environment. The degrees of acceptance, indeterminacy and rejection of objectives are simultaneously considered. We propose two models for solving Neutrosophic Goal Programming Problem (NGPP), at one hand aiming to minimize the sum of the deviation (the Ist model), and at the other hand, transforming NGPP into a crisp programming model, using truth membership, indeterminacy membership, and falsity membership functions (the IInd model). Finally, an industrial design problem is given to illustrate the efficiency of the proposed models. The obtained results of the Ist model and of the IInd model are compared with other methods. [ABSTRACT FROM AUTHOR]

In spite of researchers' concerns to find causalities, reviewing the literature of psychological studies one may argue that the classical statistical methods applied in order to find causalities are unable to find uncertainty and indeterminacies of the relationships between concepts. In this paper, we introduce two methods to find effective solutions by identifying "hidden" patterns in the patients' cognitive maps. Combined Overlap Block Fuzzy Cognitive Map (COBFCM) and Combined Overlap Block Neutrosophic Map (COBNCM) are effective when the number of concepts can be grouped and are large in numbers. In the first section, we introduce COBFCM, COBNCM, their applications, and the advantages of COBNCM over COBFCM in some cases. In the second section, we explain eight overlapped cognitive concepts related to ADHD in children and apply COBNCM and COBFCM to analyze the modeled data, comparing their results. Conclusions, limitations, and implications for applying COBNCM in other psychological areas are also discussed. [ABSTRACT FROM AUTHOR]

In this paper, we introduce for the first time the notions of Neutrosophic Axiom, Neutrosophic Axiomatic System, Neutrosophic Deducibility and Neutrosophic Inference, Neutrosophic Proof, Neutrosophic Tautologies, Neutrosophic Quantifiers, Neutrosophic Propositional Logic, Neutrosophic Axiomatic Space, Degree of Contradiction (Dissimilarity) of Two Neutrosophic Axioms, and Neutrosophic Model. A class of neutrosophic implications is also introduced. A comparison between these innovatory neutrosophic notions and their corresponding classical notions is made. Then, three concrete examples of neutrosophic axiomatic systems, describing the same neutrosophic geometrical model, are presented at the end of the paper. [ABSTRACT FROM AUTHOR]

In this paper a new concept is called n-valued interval neutrosophic sets is given. The basic operations are introduced on n-valued interval neutrosophic sets such as; union, intersection, addition, multiplication, scalar multiplication, scalar division, truthfavorite and false-favorite. Then, some distances between n-valued interval neutrosophic sets (NVINS) are proposed. Also, we propose an efficient approach for group multi-criteria decision making based on n-valued interval neutrosophic sets. An application of n-valued interval neutrosophic sets in medical diagnosis problem is given. [ABSTRACT FROM AUTHOR]

The idea of neutrosophic code came into my mind at that time when i was reading the literature about linear codes and i saw that, if there is data transfremation between a sender and a reciever. They want to send 11and 00 as codewords. They suppose 11 for true and 00 for false. When the sender sends the these two codewords and the error occures. As a result the reciever recieved 01 or 10 instead of 11 and 00. This story give a way to the neutrosophic codes and thus we introduced neutrosophic codes over finite field in this paper. [ABSTRACT FROM AUTHOR]

In this paper, we introduce for the first time the discounting of a neutrosophic mass in terms of reliability and respectively the importance of the source. We show that reliability and importance discounts commute when dealing with classical masses. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we extend the lower and upper soft interval valued intuitionstic fuzzy rough approximations of IVIFS -relations proposed by Anjan et al. to the case of interval valued neutrosophic soft set relation(IVNSS-relation for short) [ABSTRACT FROM AUTHOR]

In this paper we proposed a new distance and several similarity measures between interval neutrosophic sets. [ABSTRACT FROM AUTHOR]

Soft set theory is a general mathematical tool for dealing with uncertain, fuzzy, not clearly defined objects. In this paper we introduced soft neutrosophic groupoid and their generalization with the discuissionf of some of their characteristics. We also introduced a new type of soft neutrophic groupoid, the so called soft strong neutrosophic goupoid 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. [ABSTRACT FROM AUTHOR]

The aim of this presentation is to connect Extension Logic with new fields of research, i.e. Fuzzy Logic and Neutrosophic Logic. It shows herein: How Extension Logic is connected to the Three-Valud Neutrosophic Logic, How Extension Logic is connected to the Four-Valued Neutrosophic Logic, How Extension Logic is connected to the n-Valued Neutrosophic Logic, when contradictions occurs. As extension transformation one uses the normalization of the Neutrosophic Logic components. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this article, we shall define the addition and multiplication of two neutrosophic fuzzy matrices. Thereafter, some properties of addition and multiplication of these matrices are also put forward. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Neutrosophic set is a part of neutrosophy which studies the origin, nature, and scope of neutralities, as well as their interactions with different ideational spectra. Neutrosophic set is a powerful general formal framework that has been recently proposed. However, neutrosophic set needs to be specified from a technical point of view. To this effect, we define the set-theoretic operators on an instance of neutrosophic set, we call it single valued neutrosophic set (SVNS). We provide various properties of SVNS, which are connected to the operations and relations over SVNS. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]