A BASIC STUDY OF RHOTRICES AND ITS PROPERTIES

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Study Level: BTech, BSc, BEng, BA, HND, ND or NCE

Amount: ₦4,000.00

Account Details # ABSTRACT

This project presents a basic study of rhrotrices and its properties.An overview of the generalizations and axiomatic developments of various classifications of rhotrices having their entries as numbers on the real line are discussed. Furthermore, the extension of various classifications of rhotrices to construction of a number of commutative algebraic structures, such as semigroups, monoids, groups, rings, integral domains and fields are carefully studied with some basic definitions.

DECLARATION.. ii

CERTIFICATION.. iii

DEDICATION.. iv

ACKNOWLEDGMENT. v

ABSTRACT. vi

CHAPTER ONE. 1

1.0      Introduction. 1

1.1 Aim and Objectives of the Project 2

1.2      Methodology. 2

1.3 Significance of the Study. 2

1.4 Project Outline. 2

1.5 Definitions of Terms 3

CHAPTER TWO.. 6

2.0 Introduction. 6

2.1 Rhotrix Group. 6

2.2 Definition of Terms in Rhotrix. 6

2.2.1 Types of Heart Rhotrix. 6

2.4.2 Properties Of Rhotrix Addition. 9

2.4.4 Multiplication of Rhotrices 10

2.4.6 Identity Element of A Rhotrix. 10

2.4.7 Inverse Of A Rhotrix. 11

2.4.8 An Alternative Method for Multiplication Of Rhotrices 11

2.5 Algebraic Structure. 12

2.6 Mapping. 12

2.6.1 Types of Mapping. 13

CHAPTER THREE. 14

3.0 Introduction. 14

3.1 Generalization of Rhotrix Sets over Numbers in Real Line. 14

3.1.1 Set of All Natural Rhotrices of Size . 14

3.1.2 Set of All Rational Rhotrices of size . 14

3.1.3 Set of All Real Rhotrices of Size . 15

3.1.4 Set of all Integer Rhotrices of size . 15

3.2 Axiomatization of Rhotrix Spaces 15

3.2.1 The Axioms for the Natural Rhotrix Space. 15

3.2.2 The Axioms for the integer Rhotrix Space. 16

3.2.3 The Axioms for the Rational Rhotrix Space. 17

3.2.4 The Axioms for the Real Rhotrix Space. 18

3.3 A Number of Results 19

CHAPTER FOUR.. 23

4.0 Introduction. 23

4.1 Results 23

CHAPTER FIVE. 29

5.0 Summary. 29

5.1 Conclusion. 29

5.2 Recommendations 29

REFERENCES. 30

CHAPTER ONE

GENERAL INTRODUCTION

1.0       Introduction
Mathematics is a discipline that branched into Pure Mathematics and Applied Mathematics. Algebra is one of the aspects of Pure Mathematics and it has a lot of areas/discipline of which both Abstract and linear Algebra are among. These two aforesaid fields of Algebra have found applications in Engineering, Sciences, Arts and Social Sciences, which are essential in many aspects of real life studies.

Matrix, which deals with rectangular arrangement of numbers, is a branch of both abstract and linear Algebra.  In this work, a presentation of a relatively new method of representing arrangements of numbers in rhomboid mathematical form, known as rhotrices is made. The concept of rhotrix was first introduced by Ajibade(2003) as an extension of ideas on matrix-tertions and matrix-noitrets.  Ajibade presented the initial concept of the algebra and analysis of rhotrix and established some interesting relationships between rhotrices and their hearts. A rhotrix  of dimension three was defined as Where, the element at the perpendicular intersection of the two diagonals of  and is called the heart of R. Thus, is a specific rhotrix.

## 1.1 Aim and Objectives of the Project

The aim of this work is to carry out a basic study of rhotrices and their properties. The objectives are as below. To:

1. Investigate various classifications of rhotrices with entries from real numbers.

Extend various classifications of rhotrices over real numbers to constructions of a number of commutative algebraic structures.

1.2       Methodology

In this work, we consider the concept of rhotrices and their properties as appeared in literature. Constructions of various types of rhotrices and their expression as commutative abstract structures of semigroups, monoids, groups, rings, integeral domains and fields will be studied.

1.3 Significance of the Study.

The study is of significance because rhotrix algebraic structures can serve as tools for concretizations of abstract notions of groups, rings and fields both in teaching and research. This will facilitate better understanding of many difficult concepts and ideas arising in Group Theory, Ring Theory and Field Theory. Besides, an application of rhotrices for data encryption in Cryptography is vindicated.

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