We introduce for the first time the appurtenance equation and inclusion equation, which help in understanding the operations with neutrosophic numbers within the frame of neutrosophic statistics. The way of solving them resembles the equations whose coefficients are sets (not single numbers).

In this paper, a modified form of Collatz conjecture for neutrosophic numbers n ∈ (Z U I) is defined. We see for any n ∈ (Z U I) the related sequence using the formula (3a + 1) + (3b + 1)I converges to any one of the 55 elements mentioned in this paper. Using the akin formula of Collatz conjecture viz. (3a- 1) + (3b -1)I the neutrosophic numbers converges to any one of the 55 elements mentioned with (...) appropriate modifications. Thus, it is conjectured that every n ∈ (Z U I) has a finite sequence which converges to any one of the 55 elements. (shrink)

This study focuses on implementing electric vehicles (EVs) in developing countries where energy production is mainly based on fossil fuels. Although for these countries the environmental short-run benefits of the EVs cannot offset the short-run costs, it may still be the best option to implement the EVs as soon as possible. Hence, it is necessary to evaluate the alternatives to introducing EVs to the market due to the environmental concerns that created an opportunity for some developing countries to catch up (...) with the international competition. Therefore, we develop a case scenario to explore the decision-making process in implementing the EVs with three alter natives and twelve criteria. We solve the decision-making problem by using Type-2 neutrosophic numbers (T2NNs) based on the RAFSI (Ranking of Alternatives through Functional mapping of criterion sub-intervals into a Single Interval) method. The proposed model combines the advantages of the RAFSI technique, and it applies T2NNs to address the uncertainties. The results show that the alternatives that may suspend the implementation of the EVs are inferior. Direct implementation of EVs is prioritized. The policy implications of the results are discussed in the study. (shrink)

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

In this paper, we introduce for the first time the neutrosophic quadruple numbers (of the form a + bT + cI + dF) and the refined neutrosophic quadruple numbers. Then we define an absorbance law, based on a prevalence order, both of them in order to multiply the neutrosophic components T, I, F, or their sub-components tJ, Ik, Fl and thus to construct the multiplication of neutrosophic quadruple numbers.

In this paper, we propose a new hybrid neutrosophic multi criteria decision making (MCDM) framework that employs a collection of neutrosophic analytical network process (ANP), and order preference by similarity to ideal solution (TOPSIS) under bipolar neutrosophic numbers. The MCDM framework is applied for chief executive officer (CEO) selection in a case study at the Elsewedy Electric Group, Egypt. The proposed approach allows us to assemble individual evaluations of the decision makers and therefore perform accurate personnel selection. (...) The outcomes of the proposed method are compared with those of the related works such as weight sum model (WSM), weight product model (WPM), analytical hierarchy process (AHP), multiobjective optimization based on simple ratio analysis (MOORA) and ANP methods to prove and validate the results. (shrink)

In this book the notion of semigroups under + is constructed using: the MOD natural neutrosophic integers, or MOD natural neutrosophic-neutrosophic numbers, or MOD natural neutrosophic finite complex modulo integer, or MOD natural neutrosophic dual number integers, or MOD natural neutrosophic special dual like number, or MOD natural neutrosophic special quasi dual numbers.

We introduce Pura Vida Neutrosophic Algebra, an algebraic structure consisting of neutrosophic numbers equipped with two binary operations namely addition and multiplication. The addition can be calculated sometimes with the function min and other times with the max function. The multiplication operation is the usual sum between numbers. Pura Vida Neutrosophic Algebra is an extension of both Tropical Algebra (also known as Min-Plus, or Min-Algebra) and Max-Plus Algebra (also known as Max-algebra). Tropical and Max-Plus algebras are algebraic (...) structures included in semirings and their operations can be used in matrices and vectors. Pura Vida Neutrosophic Algebra is included in Neutrosophic semirings and can be used in Neutrosophic matrices and vectors. (shrink)

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

In this article, the main definitions and differentiation concepts of neutrosophic fuzzy environment will be reviewed. This article will introduce an analytical methodology for solving the second-order linear ordinary differential problem with neutrosophic fuzzy boundary values, this analysis will be under generalized Hukuhara differentiability to show the analytical solutions from a different point of view for the uncertain system, some of these solutions may be decreasing in uncertainty or maybe reflecting the behavior of some real-world systems better. Some (...) applications and numeral examples will be solved to show the behavior of the solution. (shrink)

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)

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)

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. (shrink)

The topics approached in the 52 papers included in this book are: neutrosophic sets; neutrosophic logic; generalized neutrosophic set; neutrosophic rough set; multigranulation neutrosophic rough set (MNRS); neutrosophic cubic sets; triangular fuzzy neutrosophic sets (TFNSs); probabilistic single-valued (interval) neutrosophic hesitant fuzzy set; neutro-homomorphism; neutrosophic computation; quantum computation; neutrosophic association rule; data mining; big data; oracle Turing machines; recursive enumerability; oracle computation; interval number; dependent degree; possibility degree; power aggregation operators; (...) multi-criteria group decision-making (MCGDM); expert set; soft sets; LA-semihypergroups; single valued trapezoidal neutrosophic number; inclusion relation; Q-linguistic neutrosophic variable set; vector similarity measure; fundamental neutro-homomorphism theorem; neutro-isomorphism theorem; quasi neutrosophic triplet loop; quasi neutrosophic triplet group; BE-algebra; cloud model; Maclaurin symmetric mean; pseudo-BCI algebra; hesitant fuzzy set; photovoltaic plan; decision-making trial and evaluation laboratory (DEMATEL); Choquet integral; fuzzy measure; clustering algorithm; and many more. (shrink)

The topics approached in this collection of papers are: neutrosophic sets; neutrosophic logic; generalized neutrosophic set; neutrosophic rough set; multigranulation neutrosophic rough set (MNRS); neutrosophic cubic sets; triangular fuzzy neutrosophic sets (TFNSs); probabilistic single-valued (interval) neutrosophic hesitant fuzzy set; neutro-homomorphism; neutrosophic computation; quantum computation; neutrosophic association rule; data mining; big data; oracle Turing machines; recursive enumerability; oracle computation; interval number; dependent degree; possibility degree; power aggregation operators; multi-criteria group decision-making (...) (MCGDM); expert set; soft sets; LA-semihypergroups; single valued trapezoidal neutrosophic number; inclusion relation; Q-linguistic neutrosophic variable set; vector similarity measure; fundamental neutro-homomorphism theorem; neutro-isomorphism theorem; quasi neutrosophic triplet loop; quasi neutrosophic triplet group; BE-algebra; cloud model; fuzzy measure; clustering algorithm; and many more. (shrink)

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)

In this paper we present the concept of neutrosophic quadruple algebraic structures. Specially, we study neutrosophic quadruple rings and we present their elementary properties.

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

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

We all know that problems of transportation and allocation appear frequently in practical life. We need to transfer materials from production centers to consumption centers to secure the areas’ need for the transported material or allocate machines or people to do a specific job at the lowest cost, or in the shortest time. We know that the cost factors Time is one of the most important factors that decision-makers care about because it plays an “important” role in many of the (...) practical and scientific issues that we face in our daily lives, and we need careful study to enable us to avoid losses. For this, the linear programming method was used, which is one of the research methods. Processes, where the problem data is converted into a linear mathematical model for which the optimal solution achieves the desired goal. Since these models are linear models, we can solve them using the direct simplex method and its modifications, but the specificity that these models enjoy has enabled scholars and researchers to find special methods that help us in obtaining the optimal solution. Whatever the method used, the goal is to determine the number of units transferred from any material from production centers to consumption centers, or to allocate a machine or person to do a job so that the cost or time is as short as possible. These issues were addressed according to classical logic, but the ideal solution was a specific value appropriate to the conditions in which the data was collected and does not take into account the changes that may occur in the work environment. In order to obtain results that are more accurate and enjoy a margin of freedom, we present in this book a study of transport issues and neutrosophic allocation issues and some methods for solving them. By neutrosophic issues we mean These are the problems in which the data are neutrosophic values, i.e. the required quantities and the available quantities. (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, law, medicine, social science, and some more. -/- This book explores the emerging field of Neutrosophic Algebraic Structures, focusing on both their theoretical foundations and practical applications. We apply innovative algorithmic methods to investigate the complex interactions of neutrosophic elements, such as neutrosophic numbers, sets, and functions, within algebraic systems. Our goal is to show how neutrosophic structures challenge and expand traditional algebraic approaches, offering solutions to problems across diverse fields like computer science, engineering, artificial intelligence, and decision-making. (shrink)

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

India dropped its target of 500 GW of renewable energy capacity fossil fuel sources by 2030. Its responsibilities the United Nations Framework Convention Climate Change [UNFCCC],and reducing radiations by one billion tonnes by the end of the decade at the COP26 conference, held in Glasgow in November 2022. Researchers are continually searching for inexhaustible and reasonable energy sources. Solar energy is one of the greenest sources of energy and is also one of the cleanest. The most important factor in using (...) solar energy is the location of the solar power plant. The main objective of this study is to find the best location for a new solar power plant in a specific region called Bundelkhand region of Uttar Pradesh in India. Here we offer an extension of ELECTRE III method as two-phase Pythagorean neutrosophic elimination and choice translating reality PN-ELECTRE-III) method to adapt with fuzzy, ambiguous, unsure, and indeterminate criteria. The Pythagorean neutrosophic numbers [PNNs] used by the group decision support system of PN-ELECTRE III to measure performance of the alternatives. The options are entirely outclassed in the subsequent stage in view of the past stage's evaluations of them. By defining PNN we describe the technique of indifference threshold functions, preference treshold and veto threshold functions, which provide a more stable basis to drop outranking relations. By calculating the concordance credibility, discordance credibility and net credibility degrees of each alternative, the ranking module of the PN-ELECTRE III approach is made simpler. In order to confirm the applicability of the strategy suggested in this paper, the location selection problem for solar plants is finaly solved. (shrink)

The neutrosophic cubic hesitant fuzzy set can efficiently handle the complex information in a decision-making problem because it combines the advantages of the neutrosophic cubic set and the hesitant fuzzy set. The algebraic operations based on Dombi norms and co-norms are more flexible than the usual algebraic operations as they involve an operational parameter. First, this paper establishes Dombi algebraic operational laws, score functions, and similarity measures in neutrosophic cubic hesitant fuzzy sets. Then, we proposed Dombi exponential (...) operational laws in which the exponents are neutrosophic cubic hesitant fuzzy values and bases are positive real numbers. To use neutrosophic cubic hesitant fuzzy sets in decision-making, we are developing Dombi exponential aggregation operators in the current study. In the end, we present applications of exponential aggregation operators in multiattribute decision-making problems. (shrink)

Fuzzy sets and fuzzy logics are used to model events with imprecise, incomplete, and uncertain information. Researchers have developed numerous methods and techniques to cope with fuzziness or uncertainty. This research intends to introduce the novel concepts of complex neutrosophic relations and its types based on the idea of complex neutrosophic sets. In addition, these concepts are supported by suitable examples. A CNR discusses the quality of a relationship using the degree of membership, the degree of abstinence, and (...) the degree of nonmembership. Each of these degrees is a complex number from the unit circle in a complex plane. The real part of complex-valued degrees represents the amplitude term, while the imaginary part represents the phase term. This property empowers CNRs to model multidimensional variables. Moreover, some interesting properties and useful results have also been proved. Furthermore, the practicality of the proposed concepts is verified by an application, which discusses the use of the proposed concepts in statistical decision-making. Additionally, a comparative analysis between the novel concepts of CNRs and the existing methods is carried out. (shrink)

A well known and referenced global result is the nilpotent characterisation of the finite p-groups. This un doubtedly transends into neutrosophy. Hence, this fact of the neutrosophic nilpotent p-groups is worth critical studying and comprehensive analysis. The nilpotent characterisation depicts that there exists a derived series (Lower Central) which must terminate at {ϵ} (an identity), after a finite number of steps. Now, Suppose that G(I) is a neutrosophic p-group of class at least m ≥ 3. We show (...) in this paper that Lm−1(G(I)) is abelian and hence G(I) possesses a characteristic abelian neutrosophic subgroup which is not supposed to be contained in Z(G(I)). Furthermore, If L3(G(I)) = 1 such that pm is the highest order of an element of G(I)/L2(G(I)) (where G(I) is any neutrosophic p-group) then no element of L2(G(I)) has an order higher than pm. (shrink)

In graph theory, the concept of domination is essential in a variety of domains. It has broad applications in diverse fields such as coding theory, computer net work models, and school bus routing and facility lo cation problems. If a fuzzy graph fails to obtain acceptable results, neutrosophic sets and neutrosophic graphs can be used to model uncertainty correlated with indeterminate and inconsistent information in arbitrary real-world scenario. In this study, we consider the concept of domination as it (...) relates to single-valued neutrosophic incidence graphs (SVNIGs). Given the importance of domination and its utilization in numerous fields, we propose the application of dominating sets in SVNIG with valid edges. We present some relevant definitions such as those of valid edges, cardinality, and isolated vertices in SVNIG along with some examples. Furthermore, we also show a few significant sets connected to the dominating set in an SVNIG such as independent and irredundant sets. We also investigate a relationship between the concepts of dominating sets and domination numbers as well as irredundant and independence sets in SVNIGs. Finally, a real-life deployment of domination in SVNIGsis investigated in relation to COVID-19 vaccination locations as a practical application. (shrink)

In this volume, we present a set of research that was published in cooperation with a number of researchers and those interested in keeping pace with the great scientific development that our contemporary world is witnessing, and one of its products was neutrosophic science, which was founded by the American scientist and mathematical philosopher Florentin Smarandache in 1995. Through it, we present a new vision for some research methods. Operations research to the concepts of this science.

As part of our small contribution in dialogue toward better peace development and reconciliation studies, and following Toffler & Toffler’s War and Antiwar (1993), the present article delves into a realm of logic beyond the traditional confines of negation and the excluded middle principle, exploring the nuances of "Otherness" that transcend classical and Nagatomo logics. Departing from the foundational premises of classical Aristotelian logic systems, this exploration ventures into alternative realms of reasoning, specifically examining Neutrosophic Logic and Klein bottle (...) logic (cf. Smarandache, 2005). The study challenges conventional boundaries and explores the implications of embracing paradoxes and self-reference in logic systems, aiming to redefine approaches to understanding truth and reasoning. The paper investigates how these alternative logics open avenues for philosophical inquiry, redefining entropy, and potentially influencing innovative perspectives in free energy systems. Through this exploration, it seeks to expand the discourse on logic, welcoming a broader spectrum of thought beyond established frameworks; and we also discuss shortly a number of possible implementations including in risk management and also Klein bottle entropy redefinition (Tang et al, 2018). (shrink)

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

This book is the fifth volume in the series of Collected Papers on Advancing Uncertain Combinatorics through Graphization, Hyperization, and Uncertainization: Fuzzy, Neutrosophic, Soft, Rough, and Beyond. This volume specifically delves into the concept of Various SuperHyperConcepts, building on the foundational advancements introduced in previous volumes. The series aims to explore the ongoing evolution of uncertain combinatorics through innovative methodologies such as graphization, hyperization, and uncertainization. These approaches integrate and extend core concepts from fuzzy, neutrosophic, soft, and rough (...) set theories, providing robust frameworks to model and analyze the inherent complexity of real-world uncertainties. At the heart of this series lies combinatorics and set theory—cornerstones of mathematics that address the study of counting, arrangements, and the relationships between collections under defined rules. Traditionally, combinatorics has excelled in solving problems involving uncertainty, while advancements in set theory have expanded its scope to include powerful constructs like fuzzy and neutrosophic sets. These advanced sets bring new dimensions to uncertainty modeling by capturing not just binary truth but also indeterminacy and falsity. In this fifth volume, the exploration of Various SuperHyperConcepts provides an innovative lens to address uncertainty, complexity, and hierarchical relationships. It synthesizes key methodologies introduced in earlier volumes, such as hyperization and neutrosophic extensions, while advancing new theories and applications. From pioneering hyperstructures to applications in advanced decision-making, language modeling, and neural networks, this book represents a significant leap forward in uncertain combinatorics and its practical implications across disciplines. -/- The book is structured into 17 chapters, each contributing unique perspectives and advancements in the realm of Various SuperHyperConcepts and their related frameworks: Chapter 1 introduces the concept of Body-Mind-Soul-Spirit Fluidity within psychology and phenomenology, while examining established social science frameworks like PDCA and DMAIC. It extends these frameworks using Neutrosophic Sets, a flexible extension of Fuzzy Sets, to improve their adaptability for mathematical and programming applications. The chapter emphasizes the potential of Neutrosophic theory to address multi-dimensional challenges in social sciences. Chapter 2 delves into the theoretical foundation of Hyperfunctions and their generalizations, such as Hyperrandomness and Hyperdecision-Making. It explores higher-order frameworks like Weak Hyperstructures, Hypergraphs, and Cognitive Hypermaps, aiming to establish their versatility in addressing multi-layered problems and setting a foundation for further studies. Chapter 3 extends traditional decision-making methodologies into HyperDecision-Making and n-SuperHyperDecision-Making. By building on approaches like MCDM and TOPSIS, this chapter develops frameworks capable of addressing complex decision-making scenarios, emphasizing their applicability in dynamic, multi-objective contexts. Chapter 4 explores integrating uncertainty frameworks, including Fuzzy, Neutrosophic, and Plithogenic Sets, into Large Language Models (LLMs). It proposes innovative models like Large Uncertain Language Models and Natural Uncertain Language Processing, integrating hierarchical and generalized structures to advance the handling of uncertainty in linguistic representation and processing. Chapter 5 introduces the Natural n-Superhyper Plithogenic Language by synthesizing natural language, plithogenic frameworks, and superhyperstructures. This innovative construct seeks to address challenges in advanced linguistic and structural modeling, blending attributes of uncertainty, complexity, and hierarchical abstraction. Chapter 6 defines mathematical extensions such as NeutroHyperstructures and AntiHyperstructures using the Neutrosophic Triplet framework. It formalizes structures like neutro-superhyperstructures, advancing classical frameworks into higher-dimensional realms. Chapter 7 explores the extension of Binary Code, Gray Code, and Floorplans through hyperstructures and superhyperstructures. It highlights their iterative and hierarchical applications, demonstrating their adaptability for complex data encoding and geometric arrangement challenges. Chapter 8 investigates the Neutrosophic TwoFold SuperhyperAlgebra, combining classical algebraic operations with neutrosophic components. This chapter expands upon existing algebraic structures like Hyperalgebra and AntiAlgebra, exploring hybrid frameworks for advanced mathematical modeling. Chapter 9 introduces Hyper Z-Numbers and SuperHyper Z-Numbers by extending the traditional Z-Number framework with hyperstructures. These extensions aim to represent uncertain information in more complex and multidimensional contexts. Chapter 10 revisits category theory through the lens of hypercategories and superhypercategories. By incorporating hierarchical and iterative abstractions, this chapter extends the foundational principles of category theory to more complex and layered structures. Chapter 11 formalizes the concept of n-SuperHyperBranch-width and its theoretical properties. By extending hypergraphs into superhypergraphs, the chapter explores recursive structures and their potential for representing intricate hierarchical relationships. Chapter 12 examines superhyperstructures of partitions, integrals, and spaces, proposing a framework for advancing mathematical abstraction. It highlights the potential applications of these generalizations in addressing hierarchical and multi-layered problems. Chapter 13 revisits Rough, HyperRough, and SuperHyperRough Sets, introducing new concepts like Tree-HyperRough Sets. The chapter connects these frameworks to advanced approaches for modeling uncertainty and complex relationships. Chapter 14 explores Plithogenic SuperHyperStructures and their applications in decision-making, control, and neuro systems. By integrating these advanced frameworks, the chapter proposes innovative directions for extending existing systems to handle multi-attribute and contradictory properties. Chapter 15 focuses on superhypergraphs, expanding hypergraph concepts to model complex structural types like arboreal and molecular superhypergraphs. It introduces Generalized n-th Powersets as a unifying framework for broader mathematical applications, while also touching on hyperlanguage processing. Chapter 16 defines NeutroHypergeometry and AntiHypergeometry as extensions of classical geometric structures. Using the Geometric Neutrosophic Triplet, the chapter demonstrates the flexibility of these frameworks in representing multi-dimensional and uncertain relationships. Chapter 17 establishes the theoretical groundwork for SuperHyperGraph Neural Networks and Plithogenic Graph Neural Networks. By integrating advanced graph structures, this chapter opens pathways for applying neural networks to more intricate and uncertain data representations. (shrink)

In this ninth book of scilogs collected from my nest of ideas, one may find new and old questions and solutions, – in email messages to research colleagues, or replies, and personal notes, some handwritten on the planes to, and from international conferences, about topics on Neutrosophy and its applications, such as: Neutrosophic Bipolar Set, Linguistic Neutrosophic Set, Neutrosophic Resonance Frequency, n-ary HyperAlgebra, n-ary NeutroHyperAlgebra, n-ary AntiHyperAlgebra, Plithogenic Crisp Graph, Plithogenic Fuzzy Graph, Plithogenic Intuitionistic Fuzzy Graph, Plithogenic (...) Neutrosophic Graph, Plithogenic Real Number Graph, Plithogenic Complex Number Graph, Plithogenic Neutrosophic Number Graph, and many more. -/- Exchanging ideas with: Tareq Al-Shami, Riad Khidr Al-hamido, Mohammad Akram, B. De Baets, Robert Neil Boyd, Said Broumi, Terman Frometa-Castillo, Yilmaz Ceven, D. Dubois, Harish Garg, L. Godo, Erik Gonzalez, Yanhui Guo, Mohammad Hamidi, E. Hüllermeier, W. B. Vasantha Kandasamy, Mary Jansi, Nivetha Martin, Mani Parimala, Akbar Rezaei, Bouzina Salah, Christy Vincent, Jun Ye.‬‬‬. (shrink)

In this paper, we make a short history about: the neutrosophic set, neutrosophic numerical components and neutrosophic literal components, neutrosophic numbers, neutrosophic intervals, neutrosophic hypercomplex numbers of dimension n, and elementary neutrosophic algebraic structures. Afterwards, their generalizations to refined neutrosophic set, respectively refined neutrosophic numerical and literal components, then refined neutrosophic numbers and refined neutrosophic algebraic structures.

In this eighth book of scilogs collected from my nest of ideas, one may find new and old questions and solutions, – in email messages to research colleagues, or replies, and personal notes handwritten on the planes to, and from international conferences, about all kind of topics, centered mostly on Paradoxism and Neutrosophy. -/- Exchanging ideas with: Robert Neil Boyd, Joseph Brenner, Ahmed Cevik, Victor Christianto, Adrian Curaj, Jean Dezert, Andrei-Lucian Drăgoi, Ervin Goldfain, Young Bae Jun, Yale Landsberg, Radu Munteanu, (...) Paul Piștea, Viorel Roman, Ridvan Sahin, Said Broumi, Selcuk Topal, Eric W. Weisstein, Xiaohing Zhang.‬‬‬ . (shrink)

This collection of articles explores a wide range of subject, from Godel’s incompleteness theorem, to possible technocalypse and neutrofuturology. Articles on historical debates on irrational number to electroculture, on vortex particle, or on different Neutrosophic applications are included.

This thirteenth volume of Collected Papers is an eclectic tome of 88 papers in various fields of sciences, such as astronomy, biology, calculus, economics, education and administration, game theory, geometry, graph theory, information fusion, decision making, instantaneous physics, quantum physics, neutrosophic logic and set, non-Euclidean geometry, number theory, paradoxes, philosophy of science, scientific research methods, statistics, and others, structured in 17 chapters (Neutrosophic Theory and Applications; Neutrosophic Algebra; Fuzzy Soft Sets; Neutrosophic Sets; Hypersoft Sets; (...) class='Hi'>Neutrosophic Semigroups; Neutrosophic Graphs; Superhypergraphs; Plithogeny; Information Fusion; Statistics; Decision Making; Extenics; Instantaneous Physics; Paradoxism; Mathematica; Miscellanea), comprising 965 pages, published between 2005-2022 in different scientific journals, by the author alone or in collaboration with the following 110 co-authors (alphabetically ordered) from 26 countries: Abduallah Gamal, Sania Afzal, Firoz Ahmad, Muhammad Akram, Sheriful Alam, Ali Hamza, Ali H. M. Al-Obaidi, Madeleine Al-Tahan, Assia Bakali, Atiqe Ur Rahman, Sukanto Bhattacharya, Bilal Hadjadji, Robert N. Boyd, Willem K.M. Brauers, Umit Cali, Youcef Chibani, Victor Christianto, Chunxin Bo, Shyamal Dalapati, Mario Dalcín, Arup Kumar Das, Elham Davneshvar, Bijan Davvaz, Irfan Deli, Muhammet Deveci, Mamouni Dhar, R. Dhavaseelan, Balasubramanian Elavarasan, Sara Farooq, Haipeng Wang, Ugur Halden, Le Hoang Son, Hongnian Yu, Qays Hatem Imran, Mayas Ismail, Saeid Jafari, Jun Ye, Ilanthenral Kandasamy, W.B. Vasantha Kandasamy, Darjan Karabašević, Abdullah Kargın, Vasilios N. Katsikis, Nour Eldeen M. Khalifa, Madad Khan, M. Khoshnevisan, Tapan Kumar Roy, Pinaki Majumdar, Sreepurna Malakar, Masoud Ghods, Minghao Hu, Mingming Chen, Mohamed Abdel-Basset, Mohamed Talea, Mohammad Hamidi, Mohamed Loey, Mihnea Alexandru Moisescu, Muhammad Ihsan, Muhammad Saeed, Muhammad Shabir, Mumtaz Ali, Muzzamal Sitara, Nassim Abbas, Munazza Naz, Giorgio Nordo, Mani Parimala, Ion Pătrașcu, Gabrijela Popović, K. Porselvi, Surapati Pramanik, D. Preethi, Qiang Guo, Riad K. Al-Hamido, Zahra Rostami, Said Broumi, Saima Anis, Muzafer Saračević, Ganeshsree Selvachandran, Selvaraj Ganesan, Shammya Shananda Saha, Marayanagaraj Shanmugapriya, Songtao Shao, Sori Tjandrah Simbolon, Florentin Smarandache, Predrag S. Stanimirović, Dragiša Stanujkić, Raman Sundareswaran, Mehmet Șahin, Ovidiu-Ilie Șandru, Abdulkadir Șengür, Mohamed Talea, Ferhat Taș, Selçuk Topal, Alptekin Ulutaș, Ramalingam Udhayakumar, Yunita Umniyati, J. Vimala, Luige Vlădăreanu, Ştefan Vlăduţescu, Yaman Akbulut, Yanhui Guo, Yong Deng, You He, Young Bae Jun, Wangtao Yuan, Rong Xia, Xiaohong Zhang, Edmundas Kazimieras Zavadskas, Zayen Azzouz Omar, Xiaohong Zhang, Zhirou Ma.‬‬‬‬‬‬‬‬. (shrink)

Cloud computing (ClC) has become a more popular computer paradigm in the preceding few years. Quality of Service (QoS) is becoming a crucial issue in service alteration because of the rapid growth in the number of cloud services. When evaluating cloud service functioning using several performance measures, the issue becomes more complex and non-trivial. It is therefore quite difficult and crucial for consumers to choose the best cloud service. The user's choices are provided in a quantifiable manner in the (...) current methods for choosing cloud services. Hence, this study attempts to achieve this objective through construction. decision-making framework so-called cloud services evaluator (CSsEv). The main indicator and its sub-indicators are formed as nodes at levels(n) in tree soft sets (TSSs). Thereafter Single Value Neutrosophic Sets (SVNSs) as branch of neutrosophic sets which conjunction with the Multi-Criteria Decision Making (MCDM) technique to facilitate analysis and evaluation process for the available Cloud services providers. Hence, entropy is employed to obtain indicators and sub_indicators weights and Complex Proportional Assessment utilizes these weights to facilitate the decision process of selecting optimal ClSPs. (shrink)

Beyond the predominant paradigm of an essentially rational human cognition, based on the classical binary logic, we want to propose some reflections that are organized around the intuition that the representations we have of the world are weighted with appreciations, for example affective ones. resulting from our integration into a social environment. We see these connotations as essentially ternary in nature, depending on the concepts underlying neutrosophy: either positive, negative or neutral. This form of representation would then influence the very (...) nature of the cognitive process, which in complex real-world situations, has to deal with problems of a combinatorial nature leading to a number of cases too large for our abilities. Forced to proceed by shortcuts on the basis of heuristics, cognition would use these assessments of the representations it manipulates to decide whether partial solutions are attractive for solving the problem or on the contrary are judged negative and are then quickly rejected. There is still the case of a neutral weighting that allows processing to continue. Thus a neutrosophical conception of our representations of the world explains how our cognition functions in its treatment of combinatorial problems in the form of producing processing accelerating heuristics, both in terms of partial solutions selection and processing optimization. (shrink)

In this paper, we recall and study the new type of algebraic structures called Symbolic Plithogenic Algebraic Structures. Their operations are given under the Absorbance Law and the Prevalence Order.