Florentin Smarandache

The interval neutrosophic uncertain linguistic variables can easily express the indeterminate and inconsistent information in real world, and TOPSIS is a very effective decision making method more and more extensive applications. In this paper, we will extend the TOPSIS method to deal with the interval neutrosophic uncertain linguistic information, and propose an extended TOPSIS method to solve the multiple attribute decision making problems in which the attribute value takes the form of the interval neutrosophic uncertain linguistic variables and attribute weight is unknown. Firstly, the operational rules and properties for the interval neutrosophic variables are introduced. Then the distance between two interval neutrosophic uncertain linguistic variables is proposed and the attribute weight is calculated by the maximizing deviation method, and the closeness coefficients to the ideal solution for each alternatives. Finally, an illustrative example is given to illustrate the decision making steps and the effectiveness of the proposed method.

In this paper we have defined neutrosophic ideals, neutrosophic interior ideals, netrosophic quasi-ideals and neutrosophic bi-ideals (neutrosophic generalized bi-ideals) and proved some results related to them. Furthermore, we have done some characterization of a neutrosophic LA-semigroup by the properties of its neutrosophic ideals. It has been proved that in a neutrosophic intra-regular LA-semigroup neutrosophic left, right, two-sided, interior, bi-ideal, generalized bi-ideal and quasi-ideals coincide and we have also proved that the set of neutrosophic ideals of a neutrosophic intra-regular LA-semigroup forms a semilattice structure.

In this paper we extend Neutro-Algebra and Anti-Algebra to geometric spaces, founding Neutro/Geometry and AntiGeometry. While Non-Euclidean Geometries resulted from the total negation of a specific axiom (Euclid's Fifth Postulate), AntiGeometry results from the total negation of any axiom or even more axioms of any geometric axiomatic system (Euclidean, Hilbert, etc. ) and of any type of geometry such as Geometry (Euclidean, Projective, Finite, Differential, Algebraic, Complex, Discrete, Computational, Molecular, Convex, etc.), and Neutro-Geometry results from the partial negation of one or more axioms [and without total negation of any axiom] of any geometric axiomatic system and of any type of geometry. Generally, instead of a classical geometric Axiom, one can take any classical geometric Theorem of any axiomatic system and of any type of geometry, and transform it by Neutrosophication or Antisofication into a Neutro-Theorem or Anti-Theorem respectively to construct a Neutro-Geometry or Anti-Geometry. Therefore, Neutro-Geometry and Anti-Geometry are respectively alternatives and generalizations of Non-Euclidean Geometries. In the second part, the evolution from Paradoxism to Neutrosophy, then to NeutroAlgebra and Anti-Algebra, then to Neutro-Geometry and Anti-Geometry, and in general to Neutro-Structure and Anti-Structure that arise naturally in any field of knowledge is recalled. At the end, applications of many Neutro-Structures in our real world are presented.

In this book, the authors introduce the linguistic set associated with a linguistic variable and the structure of matrices, which they define as linguistic matrices. The authors build linguistic matrices only for those linguistic variables which yield a linguistic continuum or an ordered linguistic set. This book is organised into three chapters. The first chapter is introductory, in which we introduce all the basic concepts of linguistic variables and the associated linguistic set to make this book self-contained. Chapter two introduces linguistic matrices and develops basic properties associated with them, like types of matrices, transpose of matrices and diagonal matrices. Most of the properties enjoyed by real or complex matrices are satisfied by these linguistic matrices. Chapter three deals with operations on the linguistic matrices.

This paper presents for the first time the IndetermSoft Set, as an extension of the classical (determinate) Soft Set, which operates on indeterminate data, and similarly the HyperSoft Set extended to the IndetermHyperSoft Set, where 'Indeterm' means 'Indeterminate' (uncertain, conflicting, non-unique result). They are built on an IndetermSoft Algebra which is an algebra dealing with IndetermSoft Operators resulting from our real world. Subsequently, the IndetermSoft and IndetermHyperSoft Sets and their Fuzzy/Fuzzy Intuitionistic/Neutrosophic and other fuzzy extensions and their applications are presented.

In this paper we extend the NeutroAlgebra & AntiAlgebra to the geometric spaces, by founding the NeutroGeometry & AntiGeometry. While the Non-Euclidean Geometries resulted from the total negation of one specific axiom (Euclid’s Fifth Postulate), the AntiGeometry results from the total negation of any axiom or even of more axioms from any geometric axiomatic system (Euclid’s, Hilbert’s, etc.) and from any type of geometry such as (Euclidean, Projective, Finite, Affine, Differential, Algebraic, Complex, Discrete, Computational, Molecular, Convex, etc.) Geometry, and the NeutroGeometry results from the partial negation of one or more axioms [and no total negation of no axiom] from any geometric axiomatic system and from any type of geometry. Generally, instead of a classical geometric Axiom, one may take any classical geometric Theorem from any axiomatic system and from any type of geometry, and transform it by NeutroSophication or AntiSophication into a NeutroTheorem or AntiTheorem respectively in order to construct a NeutroGeometry or AntiGeometry. 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.

In this paper, we have developed an Excel package to be utilized for calculating neutrosophic data and analyze them. The use of object oriented programming techniques and concepts as they may apply to the design and development a new framework to implement neutrosophic data operations, the c# programming language, NET Framework and Microsoft Visual Studio are used to implement the neutrosophic classes. We have used Excel as it is a powerful tool that is widely accepted and used for statistical analysis. Figure 1 shows Class Diagram of the implemented package. Figure 2 presents a working example of the package interface calculating the complement. Our implemented Neutrosophic package can calculate Intersection, Union, and Complement of the nuetrosophic set. Figure 3 presents our neutrosphic package capability to draw figures of presented neutrosphic set. Figure 4 presents charting of Union operation calculation, and figure 5 Intersection Operation. nuetrosophic set are characterized by its efficiency as it takes into consideration the three data items: True, Intermediate, and False.

In this paper, we define projection and bidirectional projection measures between interval rough neutrosophic sets and prove their basic properties. Then two new multi attribute decision making strategies are proposed based on interval rough neutrosophic projection and bidirectional projection measures respectively. Then the proposed methods are applied for solving multi attribute decision making problems. Finally, a numerical example is solved to show the feasibility, applicability and effectiveness of the proposed strategies.

Este es el cuarto volumen de la Enciclopedia de Investigadores Neutróficos, editados a partir de materiales ofrecidos por los autores que respondieron a la invitación del editor. Los autores se enumeran alfabéticamente. La introducción contiene una breve historia de la neutrosófica, y en especial de su impacto en Latinoamérica junto con enlaces a los principales artículos y libros. Conjunto neutrosófico, lógica neutrosófica, probabilidad neutrosófica, estadística neutrosófica, medida neutrosófica, precálculo neutrosófico, cálculo neutrosófico, psicología neutrosófica, sociología neutrosófica etc., están ganando una atención significativa en resolver muchos problemas de la vida real que implican incertidumbre, imprecisión, vaguedad, incompletitud, inconsistencia e indeterminación.

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.

Recently, a friend of one of these writers told her story of using one of a healthcare product to activate her stem cells as part of regenerative medicine. Regenerative medicine is a field of medicine that seeks to repair or replace damaged or diseased tissues and organs. This can be done through a variety of methods, including stem cell therapy, tissue engineering, and gene therapy. This is a short review article on this rapid field called regenerative medicine, in particular via a new method called photobiomodulation, especially by virtue of low-intensity laser pen treatment. Hopefully it will attract more research in this interesting direction, both for laser/photobiomodulation to boost plant growth as well as in healthcare (see also: Christianto & Smarandache, 2022; Christianto, 2023).

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

Algebraic structures on linguistic sets associated with a linguistic variable are introduced. The linguistics with single closed binary operations are only semigroups and monoids. We describe the new notion of linguistic semirings, linguistic semifields, linguistic semivector spaces and linguistic semilinear algebras defined over linguistic semifields. We also define algebraic structures on linguistic subsets of a linguistic set associated with a linguistic variable. We define the notion of linguistic subset semigroups, linguistic subset monoids and their respective substructures. We also define as in case of deals in classical semigroups, linguistic ideals in linguistic semigroups and linguistic monoids. This concept of linguistic ideals is extended to the case of linguistic subset semigroups and linguistic subset monoids. We also define linguistic substructures.

In the ancient world, the Greeks believed that all great insights came from one of nine muses, divine sisters who brought inspiration to mere mortals. In the modern world, few people still believe in the muses, but we all still love to hear stories of sudden inspiration. Like Newton and the apple, or Archimedes and the bathtub (both another type of myth), we’re eager to hear and to share stories about flashes of insight. But what does it take to be actually creative? How to have such a flash insight? Turns out, there is real science behind "aha moments." We prefer to call it “intuilytics.”

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.

In this paper, we define a new cosine similarity between two interval valued neutrosophic sets based on Bhattacharya’s distance [19]. The notions of interval valued neutrosophic sets (IVNS, for short) will be used as vector representations in 3D-vector space. Based on the comparative analysis of the existing similarity measures for IVNS, we find that our proposed similarity measure is better and more robust. An illustrative example of the pattern recognition shows that the proposed method is simple and effective.

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

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