Neutrosophic Triplet Group (revisited)

We have introduced for the first time the notion of neutrosophic triplet since 2014, which has the form (x, neut(x), anti(x)) with respect to a given binary well-defined law, where neut(x) is the neutral of x, and anti(x) is the opposite of x. Then we define the neutrosophic triplet group (2016), pr...

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Main Authors: Florentin Smarandache, Mumtaz Ali
Format: Article
Language:English
Published: University of New Mexico 2019-11-01
Series:Neutrosophic Sets and Systems
Online Access:http://fs.unm.edu/NSS/NeutrosophicTripletGroup-v2.pdf
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spelling doaj-e1fd33e2fefc49f98ebdd503bc0359a12020-11-24T21:27:39ZengUniversity of New MexicoNeutrosophic Sets and Systems2331-60552331-608X2019-11-0110.5281/zenodo.3244162 Neutrosophic Triplet Group (revisited)Florentin SmarandacheMumtaz AliWe have introduced for the first time the notion of neutrosophic triplet since 2014, which has the form (x, neut(x), anti(x)) with respect to a given binary well-defined law, where neut(x) is the neutral of x, and anti(x) is the opposite of x. Then we define the neutrosophic triplet group (2016), prove several theorems about it, and give some examples. This paper is an improvement and a development of our 2016 published paper. 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 collection of three elements that satisfy certain axioms with respect to a binary operation. These neutrosophic triplets highly depend on the defined binary operation. Further, in this paper, we used these neutrosophic triplets to introduce the innovative notion of neutrosophic triplet group, which is a completely different from the classical group in the structural properties. A big advantage of neutrosophic triplet is that it gives a new group (neutrosophic triplet group) structure to those algebraic structures, which are not group with respect to some binary operation in the classical group theory. In neutrosophic triplet group, we apply the fundamental law of Neutrosophy that for an idea A, we have the neutral of A denoted as neut(a) and the opposite of A dented as anti(A) to capture this beautiful picture of neutrosophic triplet group in algebraic structures. We also studied some interesting properties of this newly born structure. We further defined neutro-homomorphisms for neutrosophic triplet groups. A neutro-homomorphism is the generalization of the classical homomorphism with two extra conditions. As a further generalization, we gave rise to a new field or research called Neutrosophic Triplet Structures (such as neutrosophic triplet ring, neutrosophic triplet field, neutrosophic triplet vector space, etc.). In the end, we gave main distinctions and comparison of neutrosophic triplet group with the Molaei’s generalized group as well as the possible application areas of the neutrosophic triplet groups. In this paper we improve our [13] results on neutrosophic triplet groups.http://fs.unm.edu/NSS/NeutrosophicTripletGroup-v2.pdf
collection DOAJ
language English
format Article
sources DOAJ
author Florentin Smarandache
Mumtaz Ali
spellingShingle Florentin Smarandache
Mumtaz Ali
Neutrosophic Triplet Group (revisited)
Neutrosophic Sets and Systems
author_facet Florentin Smarandache
Mumtaz Ali
author_sort Florentin Smarandache
title Neutrosophic Triplet Group (revisited)
title_short Neutrosophic Triplet Group (revisited)
title_full Neutrosophic Triplet Group (revisited)
title_fullStr Neutrosophic Triplet Group (revisited)
title_full_unstemmed Neutrosophic Triplet Group (revisited)
title_sort neutrosophic triplet group (revisited)
publisher University of New Mexico
series Neutrosophic Sets and Systems
issn 2331-6055
2331-608X
publishDate 2019-11-01
description We have introduced for the first time the notion of neutrosophic triplet since 2014, which has the form (x, neut(x), anti(x)) with respect to a given binary well-defined law, where neut(x) is the neutral of x, and anti(x) is the opposite of x. Then we define the neutrosophic triplet group (2016), prove several theorems about it, and give some examples. This paper is an improvement and a development of our 2016 published paper. 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 collection of three elements that satisfy certain axioms with respect to a binary operation. These neutrosophic triplets highly depend on the defined binary operation. Further, in this paper, we used these neutrosophic triplets to introduce the innovative notion of neutrosophic triplet group, which is a completely different from the classical group in the structural properties. A big advantage of neutrosophic triplet is that it gives a new group (neutrosophic triplet group) structure to those algebraic structures, which are not group with respect to some binary operation in the classical group theory. In neutrosophic triplet group, we apply the fundamental law of Neutrosophy that for an idea A, we have the neutral of A denoted as neut(a) and the opposite of A dented as anti(A) to capture this beautiful picture of neutrosophic triplet group in algebraic structures. We also studied some interesting properties of this newly born structure. We further defined neutro-homomorphisms for neutrosophic triplet groups. A neutro-homomorphism is the generalization of the classical homomorphism with two extra conditions. As a further generalization, we gave rise to a new field or research called Neutrosophic Triplet Structures (such as neutrosophic triplet ring, neutrosophic triplet field, neutrosophic triplet vector space, etc.). In the end, we gave main distinctions and comparison of neutrosophic triplet group with the Molaei’s generalized group as well as the possible application areas of the neutrosophic triplet groups. In this paper we improve our [13] results on neutrosophic triplet groups.
url http://fs.unm.edu/NSS/NeutrosophicTripletGroup-v2.pdf
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