A Classical Group of Neutrosophic Triplet Groups Using {Z2p, ×}

Vasantha W.B,Ilanthenral Kandasamy,Florentin Smarandache

doi:10.3390/sym10060194

Vasantha W.B, Ilanthenral Kandasamy + Show 1 more

Open Access

https://doi.org/10.3390/sym10060194

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Journal: Symmetry	Publication Date: Jun 1, 2018
Citations: 9	License type: CC BY 4.0

Affiliation: Vellore Institute of Technology University

Abstract

In this paper we study the neutrosophic triplet groups for a ∈ Z 2 p and prove this collection of triplets a , n e u t ( a ) , a n t i ( a ) if trivial forms a semigroup under product, and semi-neutrosophic triplets are included in that collection. Otherwise, they form a group under product, and it is of order ( p − 1 ) , with ( p + 1 , p + 1 , p + 1 ) as the multiplicative identity. The new notion of pseudo primitive element is introduced in Z 2 p analogous to primitive elements in Z p , where p is a prime. Open problems based on the pseudo primitive elements are proposed. Here, we restrict our study to Z 2 p and take only the usual product modulo 2 p .

Highlights

Fuzzy set theory was introduced by Zadeh in [1] and was generalized to the Intuitionistic FuzzySet (IFS) by Atanassov [2]
It has not been feasible to relate this neutrosophic set to real-world problems and the engineering discipline. To implement such a set, Wang et al [11] introduced a Single-Valued Neutrosophic Set (SVNS), which was further developed into a Double Valued Neutrosophic Set (DVNS) [12] and a Triple
This paper studies the neutrosophic triplet groups introduced by [10] only in the case of { Z2p, ×}, where p is an odd prime, under product modulo 2p

Summary

Introduction

Fuzzy set theory was introduced by Zadeh in [1] and was generalized to the Intuitionistic Fuzzy. Real-world, uncertain, incomplete, indeterminate, and inconsistent data were presented philosophically as a neutrosophic set by Smarandache [3], who studied the notion of neutralities that exist in all problems. It has not been feasible to relate this neutrosophic set to real-world problems and the engineering discipline. The paper discusses weak neutrosophic duplets in BCI algebras Notions such as the neutrosophic triplet coset and its connection with the classical coset, neutrosophic triplet quotient groups, and neutrosophic triplet normal subgroups were defined and studied by [20]. Using the notion of neutrosophic triplet groups introduced in [10], which is different from classical groups, several interesting structural properties are developed and defined in this paper.

Basic Concepts

Semi-Neutrosophic Triplets and Their Properties

Discussions and Conclusions