Neutrosophic Triplets in Neutrosophic Rings
20 Jun 2019-Vol. 7, Iss: 6, pp 563
TL;DR: In this paper, the neutrosophic triplets in the ring of integers Z ∪ I 〉 and R ∪ X ∪ Y ∪ Z ⌫ are investigated and it is proved that these rings can contain only three types of neutrosphic triplet, these collections are distinct, and these collections form a torsion free abelian group as triplets under component wise product.
Abstract: The neutrosophic triplets in neutrosophic rings 〈 Q ∪ I 〉 and 〈 R ∪ I 〉 are investigated in this paper. However, non-trivial neutrosophic triplets are not found in 〈 Z ∪ I 〉 . In the neutrosophic ring of integers Z \ { 0 , 1 } , no element has inverse in Z. It is proved that these rings can contain only three types of neutrosophic triplets, these collections are distinct, and these collections form a torsion free abelian group as triplets under component wise product. However, these collections are not even closed under component wise addition.
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TL;DR: In this paper, some new properties of neutrosophic duplet semi-groups are funded, and the following important result is proven: there is no finite neutrosophile duplet Semi-group.
Abstract: The notions of the neutrosophic triplet and neutrosophic duplet were introduced by Florentin Smarandache. From the existing research results, the neutrosophic triplets and neutrosophic duplets are completely different from the classical algebra structures. In this paper, we further study neutrosophic duplet sets, neutrosophic duplet semi-groups, and cancellable neutrosophic triplet groups. First, some new properties of neutrosophic duplet semi-groups are funded, and the following important result is proven: there is no finite neutrosophic duplet semi-group.
56 citations
01 Jan 2017
TL;DR: In this paper, the authors present the last developments in the field of neutrosophic theories and their applications, starting by the author in 1998, and present various new applications in: neutroophic multi-criteria decision-making, neutrophic psychology, neutrophic geography function (the equatorial virtual line).
Abstract: This book is part of the book-series dedicated to the advances of neutrosophic theories and their applications, started by the author in 1998. Its aim is to present the last developments in the field.
This is the second extended and improved edition of Neutrosophic Perspectives (September 2017; first edition was published in June 2017).
For the first time, we now introduce:
— Neutrosophic Duplets and the Neutrosophic Duplet Structures;
— Neutrosophic Multisets (as an extension of the classical multisets);
— Neutrosophic Spherical Numbers;
— Neutrosophic Overnumbers / Undernumbers / Offnumbers;
— Neutrosophic Indeterminacy of Second Type;
— Neutrosophic Hybrid Operators (where the heterogeneous t-norms and t-conorms may be used in designing neutrosophic aggregations);
— Neutrosophic Triplet Loop;
— Neutrosophic Triplet Function;
— Neutrosophic Modal Logic;
— and Neutrosophic Hedge Algebras.
The Refined Neutrosophic Set / Logic / Probability were introduced in 2013 by F. Smarandache. Since year 2016 a new interest has been manifested by researchers for the Neutrosophic Triplets and their corresponding Neutros-ophic Triplet Algebraic Structures (introduced by F. Smarandache & M. Ali). Subtraction and Division of Neutrosophic Numbers were introduced by F. Smarandache - 2016, and Jun Ye – 2017.
We also present various new applications in: neutrosophic multi-criteria decision-making, neutrosophic psychology, neutrosophic geographical function (the equatorial virtual line), neutrosophic probability in target identification, neutrosophic dynamic systems, neutrosophic quantum computers, neutrosophic theory of evolution, and neutrosophic triplet structures in our everyday life.
41 citations
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TL;DR: The notion of neutrosophic triplet was introduced in this article, which is a group of three elements that satisfy certain properties with some binary operation. But it is not the most fundamental and rich algebraic structure in the study of algebra.
Abstract: Groups are the most fundamental and rich algebraic structure with respect to some binary operation in the study of algebra. In this paper, for the first time, we introduced the notion of neutrosophic triplet which is a group of three elements that satisfy certain properties with some binary operation.
16 citations
19 Aug 2019
TL;DR: In this article, the authors introduce the concept of Neutrosophic Quadruple (NQ) vector spaces and study their properties, and they show that all quadruple vector spaces are of dimension four.
Abstract: In this paper authors for the first time introduce the concept of Neutrosophic Quadruple (NQ) vector spaces and Neutrosophic Quadruple linear algebras and study their properties. Most of the properties of vector spaces are true in case of Neutrosophic Quadruple vector spaces. Two vital observations are, all quadruple vector spaces are of dimension four, be it defined over the field of reals R or the field of complex numbers C or the finite field of characteristic p, Z p ; p a prime. Secondly all of them are distinct and none of them satisfy the classical property of finite dimensional vector spaces. So this problem is proposed as a conjecture in the final section.
13 citations
03 Jun 2019
TL;DR: In this paper, a characterization of neutrosophic semi-idempotents in modulo integers is presented, and several interesting properties about them are also derived and some open problems are suggested.
Abstract: In complex rings or complex fields, the notion of imaginary element i with i 2 = − 1 or the complex number i is included, while, in the neutrosophic rings, the indeterminate element I where I 2 = I is included. The neutrosophic ring 〈 R ∪ I 〉 is also a ring generated by R and I under the operations of R. In this paper we obtain a characterization theorem for a semi-idempotent to be in 〈 Z p ∪ I 〉 , the neutrosophic ring of modulo integers, where p a prime. Here, we discuss only about neutrosophic semi-idempotents in these neutrosophic rings. Several interesting properties about them are also derived and some open problems are suggested.
12 citations
Cites background from "Neutrosophic Triplets in Neutrosoph..."
...As the newly introduced notions of neutrosophic triplet groups [17,18] and neutrosophic triplet rings [19], neutrosophic triplets in neutrosophic rings [20] and their relations to neutrosophic refined sets [21,22] depend on idempotents, thus the relative study of semi-idempotents will be an innovative research for any researcher interested in these fields....
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References
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TL;DR: This work defines the settheoretic operators on an instance of neutrosophic set, and provides various properties of SVNS, which are connected to the operations and relations over SVNS.
Abstract: Neutrosophic set is a part of neutrosophy which studies the origin, nature, and scope of neutralities, as well as their interactions with different ideational spectra. Neutrosophic set is a powerful general formal framework that has been recently proposed. However, neutrosophic set needs to be specified from a technical point of view. To this effect, we define the settheoretic operators on an instance of neutrosophic set, we call it single valued neutrosophic set (SVNS). We provide various properties of SVNS, which are connected to the operations and relations over SVNS.
1,408 citations
Book•
05 Aug 2020
TL;DR: In this article, Smarandache generalized the fuzzy logic and introduced two new concepts: a) "neutrosophy" -study of neutralities as an extension of dialectics; b) and its derivative, such as Neutrosophic logic, NeUTrosophistic set, Neutroscophic probability, and NEUTrosophy statistics, thus opening new ways of research in four fields: philosophy, logics, set theory, and probability/statistics.
Abstract: It was a surprise for me when in 1995 I received a manuscript from the mathematician, experimental writer and innovative painter Florentin Smarandache, especially because the treated subject was of philosophy - revealing paradoxes - and logics. He had generalized the fuzzy logic, and introduced two new concepts: a) "neutrosophy" - study of neutralities as an extension of dialectics; b) and its derivative "neutrosophic", such as "neutrosophic logic", "neutrosophic set", "neutrosophic probability", and "neutrosophic statistics" and thus opening new ways of research in four fields: philosophy, logics, set theory, and probability/statistics.
861 citations
01 Jan 2010
TL;DR: The authors generalizes the intuitionistic fuzzy set (IFSFS), paraconsistent set, and intuitionistic set to the neutrosophic set (NS), and distinguishes between NS and IFS.
Abstract: In this paper one generalizes the intuitionistic fuzzy set (IFS), paraconsistent set, and intuitionistic set to the neutrosophic set (NS). Many examples are presented. Distinctions between NS and IFS are underlined.
551 citations
10 May 2006
TL;DR: The authors generalizes the intuitionistic fuzzy set (IFSFS), paraconsistent set, and intuitionistic set to the neutrosophic set (NS), and distinguishes between NS and IFS.
Abstract: In this paper one generalizes the intuitionistic fuzzy set (IFS), paraconsistent set, and intuitionistic set to the neutrosophic set (NS). Many examples are presented. Distinctions between NS and IFS are underlined.
171 citations
Book•
31 Jan 2005TL;DR: The neutrosophic models are fuzzy models that permit the factor of indeterminacy, and it also plays a significant role, and utilizes the concept of neutrosophile graphs, which are introduced and studied for the first time.
Abstract: The involvement of uncertainty of varying degrees when the total of the membership degree exceeds one or less than one, then the newer mathematical paradigm shift, Fuzzy Theory proves appropriate. For the past two or more decades, Fuzzy Theory has become the potent tool to study and analyze uncertainty involved in all problems. But, many real-world problems also abound with the concept of indeterminacy. In this book, the new, powerful tool of neutrosophy that deals with indeterminacy is utilized. Innovative neutrosophic models are described. The theory of neutrosophic graphs is introduced and applied to fuzzy and neutrosophic models. This book is organized into four chapters. In Chapter One we introduce some of the basic neutrosophic algebraic structures essential for the further development of the other chapters. Chapter Two recalls basic graph theory definitions and results which has interested us and for which we give the neutrosophic analogues. In this chapter we give the application of graphs in fuzzy models. An entire section is devoted for this purpose. Chapter Three introduces many new neutrosophic concepts in graphs and applies it to the case of neutrosophic cognitive maps and neutrosophic relational maps. The last section of this chapter clearly illustrates how the neutrosophic graphs are utilized in the neutrosophic models. The final chapter gives some problems about neutrosophic graphs which will make one understand this new subject.
112 citations
"Neutrosophic Triplets in Neutrosoph..." refers background in this paper
...Neutrosophic algebraic structures such as neutrosophic rings, groups and semigroups are presented and analyzed and their application to fuzzy and neutrosophic models are developed in [6]....
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