Neutrosophic Quadruple Vector Spaces and Their Properties

Vasantha Kandasamy W.B,Florentin Smarandache,Ilanthenral Kandasamy

doi:10.3390/math7080758

Vasantha Kandasamy W.B, Florentin Smarandache + Show 1 more

Open Access

https://doi.org/10.3390/math7080758

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Journal: Mathematics	Publication Date: Aug 19, 2019
Citations: 4	License type: CC BY 4.0

Affiliation: Vellore Institute of Technology University

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.

Highlights

We just give a brief literature survey of this new field of NeutrosophicQuadruples [1]
We proceed on to define for the first time the new notion of Neutrosophic Quadruple vector spaces (NQ -vector spaces) their NQ vector subspaces, NQ bases and direct sum of NQ vector subspaces. All these NQ vector spaces are defined over R, the field of reals or C, the field of complex numbers and finite field of characteristic p, Z p, p a prime. All these three NQ vector spaces are different in their properties and we prove all three NQ vector spaces defined over R or C or ZP are of dimension 4
In this paper for the first time we define the notion of NQ vector spaces and NQ linear algebras

Summary

Introduction

We just give a brief literature survey of this new field of Neutrosophic. Several research papers on the algebraic structure of Neutrosophic Quadruples, such as groups, monoids, ideals, BCI-algebras, BCI-positive implicative ideals, hyperstructures, BCK/BCI algebras [27,28,29,30,31,32] have been recently studied and analyzed. In this paper authors have defined the new notion of Neutrosophic Quadruple vector spaces (NQ vector spaces) and Neutrosophic Quadruple linear algebras (NQ linear algebras) and have studied a few related properties. NQ vector spaces are introduced, further NQ subspaces are introduced and the notion of direct sum and NQ bases are analysed It is shown all NQ vector spaces are of dimension 4 be it defined over R or C or Z p , p a prime.

Basic Concepts

Neutrosophic Quadruple Vector Spaces and Their Properties

Neutrosophic Quadruple Linear Algebras over R or C or Z p

Conclusions and Open Conjectures