On Refined Neutrosophic Matrices and Their Application in Refined Neutrosophic Algebraic Equations

Mohammad Abobala,Parimala Mani

doi:10.1155/2021/5531093

Abstract

The objective of this paper is to introduce the concept of refined neutrosophic matrices as matrices such as multiplication, addition, and ring property. Also, it determines the necessary and sufficient condition for the invertibility of these matrices with respect to multiplication. On the contrary, nilpotency and idempotency properties will be discussed.

Highlights

Neutrosophy is a new branch of generalized logic found by Smarandache to deal with indeterminacy in all fields of human knowledge
E concept of refined neutrosophic structure was supposed firstly in [6] by splitting indeterminacy I into two levels of subindeterminacies I1 and I2. is idea was used in the study of refined neutrosophic rings [7,8,9], modules [10, 11], and groups [6]
E structure of refined neutrosophic numbers is taken as a + bI1 + cI2 instead of (a, bI1, cI2). is representation is based on the theory of n-refined neutrosophic rings proposed in [12], where refined neutrosophic numbers can be represented by this form without any loss of generality or algebraic properties

Summary

Introduction

Neutrosophy is a new branch of generalized logic found by Smarandache to deal with indeterminacy in all fields of human knowledge. The concept of n-refined neutrosophic structures was defined and used in [12,13,14]. Neutrosophic matrices were a useful tool to deal with indeterminacy in many studies; we find their basic definition and properties such as ring structure, multiplication, and other applications in [15, 16]. Rough this work, we define, for the first time, the concept of refined neutrosophic matrices as a direct application of the refined neutrosophic set. We build an example to show how refined matrices can be used in refined neutrosophic equations defined in [17]. Is representation is based on the theory of n-refined neutrosophic rings proposed in [12], where refined neutrosophic numbers can be represented by this form without any loss of generality or algebraic properties E structure of refined neutrosophic numbers is taken as a + bI1 + cI2 instead of (a, bI1, cI2). is representation is based on the theory of n-refined neutrosophic rings proposed in [12], where refined neutrosophic numbers can be represented by this form without any loss of generality or algebraic properties

Preliminaries

Main Concepts

Conclusion