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NeutroGeometry is one of the most recent approaches to geometry. In NeutroGeometry models, the main condition is to satisfy an axiom, definition, property, operator and so on, that is neither entirely true nor entirely false. When one of these concepts is not satisfied at all it is called AntiGeometry. One of the problems that this new theory has had is the scarcity of models. Another open problem is the definition of angle and distance measurements within the framework of NeutroGeometry. This paper aims to introduce a general theory of distance measures in any NeutroGeometry. We also present an algorithm for distance measurement in real-life problems. [ABSTRACT FROM AUTHOR]

If we know the determinate and sub-indeterminate part(s) of an argument a0 + a1I1 + a2I2 + ... +anIn how do we similarly find the determinate and sub-indeterminate part(s) of a function (or operation) of this argument f (a0 + a1I1 + a2I2 + ... +anIn)? The AH-Isometry is designed to do just that. The function f may be of one or more variables (arguments), it may also be some unary or n-ary operation. Real examples are presented in this paper of such arguments and functions. We now present for the first time the most general form, the n-Refined AH-Isometry. [ABSTRACT FROM AUTHOR]

In this paper, we presented the indefinite integrals of refined neutrosophic trigonometric functions, where we discussed the integrals by distinguishing several cases of these types of refined neutrosophic integrals. Also, we introduced the refined neutrosophic trigonometric identities, which makes it easier for us to find these types of integrals. In addition, to supporting this with appropriate examples. [ABSTRACT FROM AUTHOR]

Plant diseases are one of the factors that lead to yield and economic losses, which have a direct effect on national and international food production systems. One of the most essential ways to avoid agricultural product loss or reduction in amount is to diagnose plant diseases promptly and accurately. Hence, the diagnosis process for plants is crucial and should be conducted accurately. Moreover, this study focuses on this process by constructing an Artificiality Diagnostics Framework (ADF) to serve the study's objectives which entailed conducting diagnosis for plants in a professional and precise manner over uncertain environments. Thus, neutrosophic theory is considered the principal ingredient in our ADF. Due to the ability of neutrosophic to divide images into Truth (T), Falsity(F), and Indeterminacy (I). Also, deep learning (DL) is considered another principal ingredient in treating vast samples of datasets. Our comparative analysis of the leaves of potatoes is conducted whether leveraging neutrosophic and without utilizing Neutrosophic. ResNet50, ResNet152, and Mobile Net are the principal ingredients for the training dataset. The findings of implementing these networks indicated that ResNet50 achieved the highest accuracy of 0.915 in the T domain, ResNet152 achieved the highest accuracy of 0.905 in the True(T) domain, and Mobile Net achieved the highest accuracy of 0.915 in Truth(T) domain. Accuracy of 0.863 in Indeterminate(I). [ABSTRACT FROM AUTHOR]

Due to data privacy concerns and a lack of broadly applicable modelling approaches, mental health prediction encounters substantial challenges. This research introduces a pioneering decentralized framework integrating federated learning with Neutrosophic Cognitive Maps (NCMs) to facilitate secure and accurate mental health predictions while preserving data privacy. This innovative approach allows collaborative NCMs training on sensitive patient data across diverse sites without centralizing or transferring the data. The NCMs incorporated into the framework effectively model relationships between various symptoms and mental health states, offering interpretable insights into the complex dynamics of mental health. To address the limitations of local data availability, a multi-task learning methodology is employed, leveraging commonalities between related mental health prediction tasks to enhance modelling. Experiments are done on a synthetic mental health dataset to validate the proposed approach, demonstrating significant improvements. The decentralized nature of the approach ensures robust privacy guarantees by preventing direct access to patient data. The proposed framework contributes to the responsible application of soft computing and AI in the sensitive mental health domain. Furthermore, the interpretability of NCM models facilitates a nuanced analysis of indeterminate interrelationships between various psychological concepts, offering valuable support for data-driven decision-making in mental health contexts. [ABSTRACT FROM AUTHOR]

One of the major challenges facing decision-makers at the present time is obtaining complete information about the issue under study, due to the unstable conditions of the work environment that are beyond the control of decision-makers, which requires them to reach an optimal decision in light of these circumstances and fluctuations and to benefit from the data that is collected. Collected by specialists to determine the appropriate probability distribution corresponding to random cases of nature, here we are faced with the issue of making a decision in the event of risk because the probability distribution is a distribution linked to the data controlled by the conditions of the work environment, which entails a great risk. Decision makers bear the responsibility of choosing the optimal decision that reduces This risk is achieved and the greatest possible profit and the least possible loss are achieved. The issue of decision-making becomes more complex as the number of events increases, and we are in dire need of an ideal study of the issue that takes into account all the circumstances of the work environment. The concept of missed opportunity is very useful in analyzing the decision making under risk, after making the decision and the occurrence of events, the decision makers may regret and wish they had chosen actions different from those they chose at the beginning. To reduce the regret of the decision makers and minimize the expected lost opportunity, researchers in the field of classical operations research presented the criterion of the expected lost opportunity through which the decision can be determined. The ideal with the least percentage of regret. In this research, we present a neutrosophic vision of the expected opportunity loss criterion by taking the data of the issue under study. neutrosophic values are ranges whose lowest limit expresses profit in the worst conditions, and only the highest represents profit in the best conditions. [ABSTRACT FROM AUTHOR]

This article is a brief foray into the intricate realm of mythological and folkloric composite entities. Our analysis posits that these hybrid and superhybrid creatures serve as compelling evidence that the human psyche consistently transcends binary oppositions as in neutrosophy. Across diverse cultures and epochs, the human mind exhibits a propensity for nuanced and neutrosophic vantage points, defying simplistic categorizations. Additionally, we make some remarks pertaining to the subject matter, and open up a few less conventional questions. [ABSTRACT FROM AUTHOR]

Because of the advancements in technology, classification learning has become an essential activity in today's environment. Unfortunately, through the classification process, we noticed that the classifiers are unable to deal with the imbalanced data, which indicates there are many more instances (majority instances) in one class than in another. Identifying an appropriate classifier among the various candidates is a time-consuming and complex effort. Improper selection can hinder the classification model's ability to provide the right outcomes. Also, this operation requires preference among a set of alternatives by a set of criteria. Hence, multi-criteria decisionmaking (MCDM) methodology is the appropriate methodology can deploy in this problem. Accordingly, we applied MCDM and supported it through harnessing neurotrophic theory as motivators in uncertainty circumstances. Single value Neutrosophic sets (SVNSs) are applied as branch of Neutrosophic theory for evaluating and ranks classifiers and allows experts to select the best classifier So, to select the best classifier (alternative), we use MCDM method called Multi-Attributive Ideal-Real Comparative Analysis (MAIRAC) and the criteria weight calculation method called Stepwise Weight Assessment Ratio Analysis (SWARA) where these methods consider singlevalue neutrosophic sets (SVNSs) to improve and boost these techniques in uncertain scenarios. All these methods are applied after modeling criteria and its sub-criteria through a novel technique is Tree Soft Sets (TrSS). Ultimately, the findings of leveraging these techniques indicated that the hybrid multi-criteria meta-learner (HML)-based classifier is the best classifier compared to the other compared models. [ABSTRACT FROM AUTHOR]

Simulation is a numerical technique used to perform tests on a numerical computer, and involves logical and mathematical relationships interacting with each other to describe the behavior and structure of a complex system in the real world over a period of time. Analysis using simulation is a "natural" and logical extension of the mathematical analytical models inherent in operations research, because most operations research methods depend on building mathematical models that closely approximate the real-world environment and we obtain the optimal solution for them using algorithms appropriate to the type of these models. The importance of the simulation process comes In all branches of science, there are many systems that cannot be studied directly, due to the great difficulty that we may encounter when studying, and the high cost, in addition to the fact that some systems cannot be studied directly. The simulation process depends on generating a series of numbers. Randomness subject to a uniform probability distribution over the domain [0,1], then converting these numbers into random variables subject to the law of probability distribution by which the system to be simulated works, using known transformation methods. In previous research, we presented a neutrosophical vision of the reverse transformation method and the method of rejection and acceptance. Which are used to transform random numbers into random variables that follow probability distributions such as: uniform distribution, exponential distribution, beta distribution..., In this research, we present a neutrosophical vision of the Composition method)the mixed method of inverse transformation method and rejection method), used to generate random variables that follow... To some Poisson distribution, the aim is to obtain neutrosophic random variables that we use when simulating systems that operate according to this distribution in order to obtain more accurate simulation results. [ABSTRACT FROM AUTHOR]

In this paper we extend the SuperHyperAlgebra, SuperHyperGraph, SuperHyperTopology, SuperHyperSoft Set, endowed with SuperHyperOperations, SuperHyperAxioms, and SuperHyperFunctions, to the most general form of structure, from our real world, called SuperHyperStructure in any field of knowledge. A practical application of the SuperHyperStructure is presented at the end. The prefix "Hyper" [Marty, 1934] stand for the codomain of the functions and operations to be P(H), or the powerset of the set H. While the prefix "Super" [Smarandache, 2016] stands for using the Pn(H), n = 2, or the n-th PowerSet of the Set H {because the set (or system) H (that may be a set of items, a company, institution, country, region, etc.) is organized in sub-systems, which in their turn are organized in sub-subsystems, and so on} in the domain and/or codomain of the functions and operations. [ABSTRACT FROM AUTHOR]

The goal of this short note is to expand the concepts of 'pluralism', 'neutrosophy', 'refined neutrosophy', 'refined neutrosophic set', 'multineutrosophic set', and 'plithogeny' (Smarandache 2002, 2013, 2017, 2019, 2021, 2023a, 2023b, 2023c), into a larger category that I will refer to as MultiAlism (or MultiPolar). As a straightforward generalization, I propose the conceptualization of a MultiPolar System (different from a PluriPolar System), which is formed not only by multiple elements that might be random, or contradictory, or adjuvant, but also by accepting features from more than one basic system (UniPolar, BiPolar, TriPolar, or PluriPolar systems). PluriAlism is a closed dynamic system without neutralities nor indeterminacies, while MultiAlism is an open dynamic system with neutralities and indeterminacies. PluriAlism is a uni-system (formed by elements from a single system), while MultiAlism is a MultiSystem (formed by elements from many systems). [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this article, the concept of system of symbolic 2-plithogenic linear equations and its solutions are introduced and studied. The Cramer’s rule was applied to solve the system of symbolic 2-plithogenic linear equations. Also, provided enough examples for each case to enhance understanding. [ABSTRACT FROM AUTHOR]

A ring is said to be clean if every element of the ring can be written as a sum of an idempotent element and a unit element of the ring and a ring is said to be nil-clean if every element of the ring can be written as a sum of an idempotent element and a nilpotent element of the ring. In this paper, we generalize these arguments to symbolic 2-plithogenic structure. We introduce the structure of clean and nil-clean symbolic 2-plithogenic rings and some of its elementary properties are presented. Also, we have found the equivalence between classical clean(nil-clean) ring R and the corresponding symbolic 2-plithogenic ring 2 − SPR. [ABSTRACT FROM AUTHOR]

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). [ABSTRACT FROM AUTHOR]

The linear programming method is one of the important methods of operations research that has been used to address many practical issues and provided optimal solutions for many institutions and companies, which helped decision makers make ideal decisions through which companies and institutions achieved maximum profit, but these solutions remain ideal and appropriate in If the conditions surrounding the work environment are stable, because any change in the data provided will affect the optimal solution and to avoid losses and achieve maximum profit, we have, in previous research, reformulated the linear models using the concepts of neutrosophic science, the science that takes into account the instability of conditions and fluctuations in the work environment and leaves nothing to chance. While taking data, neutrosophic values carry some indeterminacy, giving a margin of freedom to decision makers. In another research, we reformulated one of the most important methods used to solve linear models, which is the simplex method, using the concepts of this science, and as a continuation of what we did in the previous two researches, we will reformulate in this research. The graphical method for solving linear models using the concepts of neutrosophics. We will also shed light on a case that is rarely mentioned in most operations research references, which is that when the difference between the number of unknowns and the number of constraints is equal to one, two, or three, we can also find the optimal solution graphically for some linear models. This is done by taking advantage of the conditions of non-negativity that linear models have, and we will explain this through an example in which the difference is equal to two. Also, through examples, we will explain the difference between using classical values and neutrosophic values and the extent of this’s impact on the optimal solution. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper we recall the six new types of topologies, and their corresponding topological spaces, that we introduced in the last years (2019-20223), such as: NeutroTopology, AntiTopology, Refined Neutrosophic Topology, Refined Neutrosophic Crisp Topology, SuperHyperTopology, and Neutrosophic SuperHyperTopology. The nth-PowerSets Pn(H) and Pn(H), that the SuperHyperTopology and respectively Neutrosophic SuperHyperTopology are based on, better describe our real world, since a system H (that may be a set, company, institution, country, region, etc.) is organized in sub-systems, which in their turn are organized each in sub-sub-systems, and so on. [ABSTRACT FROM AUTHOR]

Operations research methods focus on formulating decision models, models that approximate the real work environment to mathematical models, and their optimal solution helps the decision maker to make optimal decisions that guarantee the greatest profit or the lowest cost. These models were built based on the data collected on the practical issue under study, and these data are classic values, specific values that are correct and accurate during the period in which these data are collected, which means that they remain ideal if the surrounding conditions remain the same as those in which they were collected. Data, and since the aim of this study is to make ideal decisions and to develop future plans through which we achieve the greatest profit or the least cost, It was necessary to search for a study that fits all the conditions in which the work environment passes in the present and the future. If we take data, neural values, indeterminate values - uncertain - leave nothing to chance, and take into account all conditions from worst to best. In this paper, we will present a complementary study to what we have done in previous research, the purpose of which was to reformulate some operational research methods using neutrosophic data. We will reformulate the assignment problem and one of the methods for its solution, which is the Hungarian method, using neutrosophic concepts, and we will explain the difference between using classical and neutrosophic data through examples. [ABSTRACT FROM AUTHOR]

In this paper, we extend, for the first time, the Law of Included Multiple-Middles to the Law of Infinitely-Many-Middles. And we present several practical applications. Also, we discuss Aristotle's Syllogism, Principle of Identity, and Principle of NonContradiction. [ABSTRACT FROM AUTHOR]

Plithogenic sets coined by Smarandache in the year 2018 has unveiled new research opportunities in the field of Multi criteria decision making (MCDM). The contributions and developments of new decision making approaches based on plithogeny is gaining high momentum presently. The theoretical conceptualization of different phenomenon with plithogenic sets are also applied in designing optimal solutions to the decision making problems. This review paper presents the applications of plithogenic MCDM from the year 2018 to till date in almost all the spheres of decision making scenario. The literature works of the researchers presented in this paper will certainly portray the compatibility and flexibility of plithogenic sets, operators and other decision making tools. Though the time span considered for counting on the plithogeny based works is short, the applications of plithogenic sets are growing many in number and also plithogeny based theories are amplifying in a speedy manner. This has motivated the authors to investigate on the proliferation of plithogeny applications in decision making. This review paper has focused on the dimensions of different fields to which plithogeny is applied, new plithogeny based theories, extension of plithogeny, plithogenic based operators and measures. In addition to it the data on the publications of plithogeny based articles and interests of researchers are also presented as a part of this review work. The overall impact of plithogeny in the arena of decision making science and on the researchers of the same field is well sketched in this paper with the intention and hope of inspiring plithogenic researchers. [ABSTRACT FROM AUTHOR]

The softmax function is a well-known generalization of the logistic function. It has been extensively applied in various probabilistic classification methods such as softmax regression, linear discriminant analysis, naive Bayes classifiers, and artificial neural networks. Inspired by the advantages of the softmax function, we have developed the softmax function-based single-valued neutrosophic aggregation operators. Then we have established some essential properties of aggregation operators based on the softmax function with the neutrosophic set. Additionally, we have defined a multi-attribute decision-making method based on the proposed aggregation operators. Using the proposed MCDM method, we have developed a novel algorithm. This algorithm helps to examine FD-risk assessment problems. Also, the proposed algorithm process is a reasonable strategy for the decision-making problem. It is easy to recognize when choosing a neutrosophic set of information for a practical decision problem. We used this proposed MADM method to exercise a realistic MADM problem with neutrosophic information. Finally, we have considered one numerical illustration to show the validity and reliability of the proposed methods. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The IndetermSoft Set is as an extension of the Soft Set, because the data, or the function, or the sets involved in the definition of the soft set have indeterminacy - as in our everyday life, and we still need to deal with such situations. And similarly, IndetermHyperSoft Set as extension of the HyperSoft Set, when there is indeterminate data, or indeterminate functions, or indeterminate sets. Herein, 'Indeterm' stands for 'Indeterminate' (uncertain, conflicting, incomplete, not unique outcome). We now introduce for the first time the TreeSoft Set as extension of the MultiSoft Set. Several applications are presented for each type of soft set. [ABSTRACT FROM AUTHOR]

In the fifth version of our response-paper [26] to Imamura's criticism, we recall that NonStandard Neutrosophic Logic was never used by neutrosophic community in no application, that the quarter of century old neutrosophic operators (1995-1998) criticized by Imamura were never utilized since they were improved shortly after but he omits to tell their development, and that in real world applications we need to convert/approximate the NonStandard Analysis hyperreals, monads and binads to tiny intervals with the desired accuracy - otherwise they would be inapplicable. We point out several errors and false statements by Imamura [21] with respect to the inf/sup of nonstandard subsets, also Imamura's "rigorous definition of neutrosophic logic" is wrong and the same for his definition of nonstandard unit interval, and we prove that there is not a total order on the set of hyperreals (because of the newly introduced Neutrosophic Hyperreals that are indeterminate), whence the Transfer Principle from R to R* is questionable. After his criticism, several response publications on theoretical nonstandard neutrosophics followed in the period 2018-2022. As such, I extended the NonStandard Analysis by adding the left monad closed to the right, right monad closed to the left, pierced binad (we introduced in 1998), and unpierced binad - all these in order to close the newly extended nonstandard space (R*) under nonstandard addition, nonstandard subtraction, nonstandard multiplication, nonstandard division, and nonstandard power operations [23, 24]. Improved definitions of NonStandard Unit Interval and NonStandard Neutrosophic Logic, together with NonStandard Neutrosophic Operators are presented. [ABSTRACT FROM AUTHOR]

There are several decision-making based situations in which it is necessary to categorize the evaluating parameters into their respective sub-parametric values based non-overlapping sets. The existing soft set model is not compatible with such situations therefore hypersoft set (Hs-set) is developed which manages such situations by utilizing a novel mapping called multi-argument approximate mapping which broadens the domain of soft approximate mapping. This research presents the characterization of several essential axiomatic properties and set-based operations of Hs-set which will help the researchers to implement this emerging theory to other fields of study. The brief discussion on some hybridized structures of Hs-set with fuzzy set-like models is also provided. [ABSTRACT FROM AUTHOR]

Clustering research is an important field in machine learning, pattern recognition and other fields. The neutrosophic set characterizes the data through true membership functions, indeterminate membership functions and false membership functions. Data clustering using neutrosophic set has become one of the current research hotspots. In this paper, first, a new definition of data uncertainty in a neutrosophic set is proposed in this paper based on the density of data. Next, a clustering model based on the uncertainty value of neutrosophic set data is proposed by considering the main cluster (true membership) and the noise cluster (false membership) in the data set. The model takes into account the distance of the data points to the cluster centers and the indeterminacy value of each data point, and then minimizes the proposed cost function by the method of Lagrangian multipliers. The true membership value and false membership value of each data point can be obtained. Finally, the effectiveness of the method is demonstrated by experiments on the various datasets. Experimental results show that the cost function has more accurate membership degree when dealing with boundary points and outliers, and outperforms existing clustering methods on datasets. [ABSTRACT FROM AUTHOR]

In this paper, we shall study some new concepts of weakly neutrosophic crisp separation axioms, which are called "neutrosophic crisp α-separation and neutrosophic crisp semi-α-separation axioms" such as neutrosophic crisp α-Ti and neutrosophic crisp semi-α-Ti, ∀ i= 0, 1,..., 4. Moreover, we shall study the relationship between usual neutrosophic crisp separation axioms and these kinds of weakly neutrosophic crisp separation axioms. [ABSTRACT FROM AUTHOR]

Aggregation operators can be used to combine and synthesise a finite number of numerical values into a single numerical value. Many areas, including decision-making, expert systems, risk analysis, and image processing, rely heavily on aggregating functions. In real-world situations, the neutrosophic set can manage the uncertainties associated with information from any decision-making challenge, whereas the fuzzy set and intuitionistic set cannot. The term "bipolarity" refers to the propensity of the human mind to weigh pros and drawbacks when thinking through decisions. Triangular norms are aggregation operators in a variety of fields, including fuzzy set theory, probability and statistics, and decision sciences. Thus, the individual assessments in this paper's study of and approach to multi-criteria decision-making (MCDM) problems that use bipolar neutrosophic numbers as the individual evaluations. Frank operational laws of bipolar neutrosophic numbers, bipolar neutrosophic Frank weighted geometric aggregation (BNFWGA) and the bipolar neutrosophic frank ordered weighted geometric aggregation (BNFOWGA) operators have been developed with its desirable properties. Additionally, the suggested aggregation operators have been used in the selection of bridges. The outcomes demonstrate the applicability and validity of the suggested approach. Comparative analysis has been performed using the current approach. [ABSTRACT FROM AUTHOR]

This article aims to introduce some modern algebraic structures as hyper super matrices. The classical algebra and matrices cannot process higher-dimensional information with several levels of ambiguity and uncertainty. Hence, it is necessary to establish such superalgebraic structures that can organize and classify the uncertain and incomplete information floating in parallel higher dimensions as facts, events, or realities. To achieve the desired goal, a particular construction of Hypersoft Matrix (HS-Matrix) and Subjectively Whole Hyper-SuperSoft Matrix (SWHSS-Matrix) is offered in a plithogenic Fuzzy environment initially, and some aggregation operators are formulated. A Local-Global-Universal Combined Consciousness State Ranking Model is formulated as an application. As the classification of non-physical phenomena like state of physical health or Consciousness has not yet been addressed in the area of decision making therefor the proposed model will open a new dimension of classification of the non-physical part of the universe in which one can select the most suitable possible reality from several parallel realities which would be useful in the field of artificial intelligence. This model classifies the accumulated states of matter bodies (subjects). And gives a possible description of the Combined-Consciousness State of a Universe. In addition, it offers a local ranking by observing the information through several angles of vision, just like a human mind does, and a universal ranking by classifying the accumulated states. Furthermore, the final Global Ranking is achieved by constructing a percentage frequency-matrix and an authenticity measure of the order is offered. A numerical example is constructed to describe SWHSS-Matrix and LGU-Ranking Model. Some pie graphs are used to describe the individual states, accumulated states, and the ultimate accumulated universal state of all given subjects (a Combined Conscious State of Universe). [ABSTRACT FROM AUTHOR]

In this paper one introduces for the first time the IndetermSoft Set, as extension of the classical (determinate) Soft Set, that deals with indeterminate data, and similarly the HyperSoft Set extended to IndetermHyperSoft Set, where 'Indeterm' stands for 'Indeterminate' (uncertain, conflicting, not unique outcome). They are built on an IndetermSoft Algebra that is an algebra dealing with IndetermSoft Operators resulted from our real world. Afterwards, the corresponding Fuzzy / Intuitionistic Fuzzy / Neutrosophic / and other fuzzy-extension IndetermSoft Set & IndetermHyperSoft Set are presented together with their applications. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, one recalls the general definition of the SuperHyperAlgebra with its SuperHyperOperations and SuperHyperAxioms [2, 6]. Then one introduces for the first time the SuperHyperTopology and especially the SuperHyperFunction and Neutrosophic SuperHyperFunction. One gives a numerical example of a SuperHyperGroup. [ABSTRACT FROM AUTHOR]

The newly introduced theories, proposed as extensions of the fuzzy theory, such as the Neutrosophic, Pythagorean, Spherical, Picture, Cubic theories, and their numerous hybrid forms, are criticized by the authors of [1]. In this paper we respond to their critics with respect to the neutrosophic theories and show that the DST, that they want to replace the A-IFS with, has many flaws. Their misunderstanding, with respect to the partial and total independence of the neutrosophic components, is that in the framework of the neutrosophic theories we deal with a MultiVariate Truth-Value (truth upon many independent random variables) as in our real-life world, not with a UniVariate Truth-Value (truth upon only one random variable) as they believe. About the membership degrees outside of the interval [0, 1], which are now in the arXiv and HAL mainstream, it is normal that somebody who over-works (works overtime) to have an over-membership (i.e., membership degree above 1) to be distinguished from those who do not work overtime (whose membership degree is between 0 and 1). And, similarly, a negative employee (that who does only damages to the company) to have a negative membership (i.e., membership degree below 0) in order to distinguish him from the positive employees (those whose membership degree is above 0). There are elementary practical applications in this paper that allow us to think out of box (in this case the box is the interval [0, 1]). [ABSTRACT FROM AUTHOR]

In this paper we extend the NeutroAlgebra & AntiAlgebra to the geometric space, by founding the NeutroGeometry & AntiGeometry. While the Non-Euclidean Geometries resulted from the total negation of only one specific axiom (Euclid’s Fifth Postulate), the AntiGeometry results from the total negation of any axiom and even of more axioms from any geometric axiomatic system (Euclid’s, Hilbert’s, etc.), and the NeutroAxiom results from the partial negation of one or more axioms [and no total negation of no axiom] from any geometric axiomatic system. Therefore, the NeutroGeometry and AntiGeometry are respectively alternatives and generalizations of the Non-Euclidean Geometries. In the second part, we recall the evolution from Paradoxism to Neutrosophy, then to NeutroAlgebra & AntiAlgebra, afterwards to NeutroGeometry & AntiGeometry, and in general to NeutroStructure & AntiStructure that naturally arise in any field of knowledge. At the end, we present applications of many NeutroStructures in our real world. [ABSTRACT FROM AUTHOR]

A single-valued neutrosophic set is one of the advanced fuzzy sets that is capable of handling complex real-world information satisfactorily. A development of single-valued neutrosophic set and fuzzy bipolar set, called a bipolar neutrosophic set, was introduced. Distance measures between fuzzy sets and advanced fuzzy sets are important tools in diagnostics and prediction problems. Sometimes they are defined without considering the condition of the inclusion relation on sets. In decision-making applications, this condition should be required (here it is called the inference of the measure). Moreover, in many cases, a distance measure capable of discriminating between two nearly identical objects is considered an effective measure. Motivated by these observations, in this paper, a new distance measure is proposed in a bipolar neutrosophic environment. Furthermore, an entropy measure is also developed by the similarity between two sets of mutual negation. Finally, an application to medical diagnosis is presented to illustrate the effective applicability of the proposed distance measure, where entropy values are used to characterize noises of different attributes. [ABSTRACT FROM AUTHOR]

A Plithogenic Logical proposition P is a proposition that is characterized by many degrees of truth-values with respect to many corresponding attribute-values (or random variables) that characterize P. Each degree of truth-value may be classical, fuzzy, intuitionistic fuzzy, neutrosophic, or other fuzzy extension type logic. At the end, a cumulative truth of P is computed. [ABSTRACT FROM AUTHOR]

Uninorm generalizes the notion of t-norm and t-conorm in fuzzy logic theory. They are three increasing, commutative and associate operators having one neutral element. However, such specific value identifies the kind of operator it is; t-norms have the 1 as neutral element, t-conorms have the 0 and uninorms have every number lying between 0 and 1. Uninorms have been applied as aggregators in many fields of Artificial Intelligence and Decision Making. This theory has also been extended to the framework of interval-valued fuzzy sets, intuitionistic fuzzy sets, interval-valued intuitionistic fuzzy sets and L-fuzzy sets. This paper aims to explore neutrosophic uninorms. We demonstrate that it is possible to define uninorms operators from neutrosophic logic. Additionally, we define neutrosophic implicators induced by neutrosophic uninorms. The combination of both, Neutrosophy and uninorms, enriches the applicability of uninorms operators due to the possibility of incorporating indeterminancy as part of the Neutrosophy contribution. [ABSTRACT FROM AUTHOR]

In this paper we exemplify the types of Plithogenic Probability and respectively Plithogenic Statistics. Several applications are given. The Plithogenic Probability of an event to occur is composed from the chances that the event occurs with respect to all random variables (parameters) that determine it. Each such a variable is described by a Probability Distribution (Density) Function, which may be a classical, (T,I,F)-neutrosophic, I-neutrosophic, (T,F)-intuitionistic fuzzy, (T,N,F)-picture fuzzy, (T,N,F)-spherical fuzzy, or (other fuzzy extension) distribution function. The Plithogenic Probability is a generalization of the classical MultiVariate Probability. The analysis of the events described by the plithogenic probability is the Plithogenic Statistics. [ABSTRACT FROM AUTHOR]

The introducing of NeutroAlgebra by Smarandache opened the door for researchers to define many related new concepts. NeutroOrderedAlgebra was one of these new related definitions. The aim of this paper is to study productional NeutroOrderedSemigroup. In this regard, we firstly present many examples and study subsets of productional NeutroOrderedSemigroups. Then, we find sufficient conditions for the productional NeutroSemigroup to be a NeutroOrderedSemigroup. Finally, we find sufficient conditions for subsets of the productional NeutroOrderedSemigroup to be NeutroOrderedSubSemigroups, NeutroOrderedIdeals, and NeutroOrderedFilters. [ABSTRACT FROM AUTHOR]

Neutrosophic sets deals with inconsistent, indeterminate and imprecise datas. The concept of Neutrosophic Metric Space (NMS) uses the idea of continuous t- norm and continuous t - conorm in intuitionistic fuzzy metric spaces. In this paper, we introduce the definition of subcompatible maps of types (J-1 and J-2). We extend the structure of weak non-Archimedian with the help of sub compatible maps of types (J-1 and J-2) in NMS. Finally, we obtain common fixed point theorems for four sub compatible maps of type (J-1) in weak non-Archimedean NMS. [ABSTRACT FROM AUTHOR]

In the current paper, particular acheivments of single-valued neutrosophic continuity on a singlevalued neutrosophic topological space (𝔗 ̃, 𝜏̃𝛾̃, 𝜏̃𝜂̃, 𝜏̃𝜇̃) are introduced. Some necessary implications between them are illustrated. The theories of r-single-valued neutrosophic compact, r-single-valued neutrosophic ideal compact, r-single-valued neutrosophic quasi H-closed and r-single-valued neutrosophic compact modulo an single-valued neutrosophic ideal I~ are presented and investigated. [ABSTRACT FROM AUTHOR]

Multi-criteria decision making (MCDM) is the technique of selecting the best alternative from multiple alternatives and multiple conditions. The technique for order preference by similarity to an ideal solution (TOPSIS) is a crucial practical technique for ranking and selecting different options by using a distance measure. In this article, we protract the fuzzy TOPSIS technique to neutrosophic fuzzy TOPSIS and prove the accuracy of the method by explaining the MCDM problem with single-valued neutrosophic information and use the method for supplier selection in the production industry. We hope that this article will promote future scientific research on numerous existing issues based on multi-criteria decision making. [ABSTRACT FROM AUTHOR]

The notions of an implicative (∈, ∈)-neutrosophic ideal and an implicative falling neutrosophic ideal are introduced, and several properties are investigated. Characterizations of an implicative (∈, ∈)-neutrosophic ideal are considered, and relations between an implicative (∈, ∈)-neutrosophic ideal and an (∈, ∈)-neutrosophic ideal are discussed. Conditions for an (∈, ∈)-neutrosophic ideal to be an implicative (∈, ∈)-neutrosophic ideal are provided, and relations between an implicative (∈, ∈)-neutrosophic ideal, a falling neutrosophic ideal and an implicative falling neutrosophic ideal are studied. Conditions for a falling neutrosophic ideal to be implicative are provided. Relations between implicative falling neutrosophic ideal, commutative falling neutrosophic ideal and positive implicative falling neutrosophic ideal are discussed. [ABSTRACT FROM AUTHOR]

Multi-criteria decision-making (MCDM) focuses on coordination, choice and planning issues, including multi-criteria. the neutrosophic soft set cannot handle environments involving multiple attributes. In order to overcome these obstacles, the neutrosophic hypersoft set (NHSS) and Interval Value neutrosophic hypersoft set (IVNHSS) are defined. In this paper, we extend the concept of IVNHSS with basic properties. We also developed some basic operations on IVNHSS such as union, intersection, addition, difference, Truth-favorite, and False-favorite, etc. with their desirable properties. Finally, the necessity and possibility operations on IVNHSS with properties are presented in the following research. [ABSTRACT FROM AUTHOR]

In this paper we prove that the Single-Valued (and respectively Interval-Valued, as well as Subset-Valued) Score, Accuracy, and Certainty Functions determine a total order on the set of neutrosophic triplets (T, I, F). This total order is needed in the neutrosophic decision-making applications. [ABSTRACT FROM AUTHOR]

Multi-criteria decision making (MCDM) is the technique of selecting the best alternative from multiple alternatives and multiple conditions. The technique for order preference by similarity to an ideal solution (TOPSIS) is a crucial practical technique for ranking and selecting different options by using a distance measure. In this article, we protract the fuzzy TOPSIS technique to neutrosophic fuzzy TOPSIS, and prove the accuracy of the method by explaining the MCDM problem with single-value neutrosophic information, and use the method for supplier selection in the production industry. We hope that this article will promote future scientific research on numerous existence issues based on multi-criteria decision making. [ABSTRACT FROM AUTHOR]

In this age of information, the industrial sectors are embedding its functioning principles with the components of Industry 4.0. This article proposes a production inventory model discussing the paradigm shift towards smart production process involving many new cost parameters in addition to the conventional inventory costs. The proposed Industry 4.0 production inventory model is discoursed and compared in both deterministic and neutrosophic environments. The trapezoidal neutrosophic number representation of the parameters enhances the efficiency of the model in determining the optimal order time that minimizes the total costs. The model is highly comprehensive in nature and it is validated with a numerical example. [ABSTRACT FROM AUTHOR]

Smarandache in 2019 has generalizes the algebraic structures to NeutroAlgebraic structures and AntiAlgebraic structures. In this paper, authors, for the first time, define the NeutroAlgebraic structures and AntiAlgebra of neutrosophic triplets group usual + and x, built using {Zn, ×}, n a composite number, 5< n< ∞, 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 + 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 associated with neutrosophic triplet groups in {Zn, ×} 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. [ABSTRACT FROM AUTHOR]