A BASIC STUDY OF RHOTRICES
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, integeral domains and fields are carefully studied with some basic definitions.
Table of contents
DECLARATION ii
CERTIFICATION iii
DEDICATION iv
ACKNOWLEDGMENT v
ABSTRACT vi
Table of contents vii
CHAPTER ONE
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
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
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 n 14
3.1.2 Set of All Rational Rhotrices of size n 14
3.1.3 Set of All Real Rhotrices of Size n 15
3.1.4 Set of all Integer Rhotrices of size n 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
4.0 Introduction 23
4.1 Results 23
CHAPTER FIVE
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 R of dimension three was defined as
Where h(R) = c, the element at the perpendicular intersection of the two diagonals of R 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:
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.
1.4 Project Outline
The first Chapter gives the basic introduction to rhotrices, with the definition of rhotrix and definition of terms in rhotrices. In Chapter two, we discuss the types of rhotrices and operation on rhotrices, addition, and multiplication, determinant and inverse. Algebraic structures having rhotrix set as underlying set are presented in chapter three. In Chapter Four, a number of results and discussions are presented. Finally, chapter five contains conclusion, Recommendations and references.
1.5 Definitions of Terms
i. Binary Operation on a Set
Let G be a non-empty set and let G×G={(a,b):a,b┤ ├ ∈G}. If f: G×G → G then f is said to be a binary operation on the set G.
We use symbols +,∙, etc. to denote binary operation on a set. Thus ‘+’ will be a binary operation on G if a+b ∈ G,a∙b ∈ G and a+b is unique.
A binary operation on a set G is sometimes also called a binary composition in a set G.
ii. Definition of a Group
Let * be a binary operation and let G be a non empty set equipped with the binary operation denoted by *,i.e a*b represents the element of G obtained by applying the said binary operation between the elements a and b of G taken in that order.
Thus the set G is said to be a group if the following axioms are satisfied:
For all a,b∈G,a*b∈G i.e closure axiom.
For all a,b,c∈G,(a*b)*c=a*(b*c). i.e, associative property.
There exists a ∈ G such that for any a∈G,a*e=e*a=a. i.e, Identity axiom.
For any a∈G,∃ a^(-1)∈G,such that a*a^(-1)=a^(-1)*a=e, i.e, inverses exist.
Furthermore, if a*e=e*a then the group is said to be commutative.
iii. definition of rhotrix
A rhotrix of size-n, is a rhomboidal array of real numbers that lies in some way between n×n dimensional matrices and (2n-1)×(2-1)dimensional matrices. In essence, a rhotrix is defined as an object that lies between 2×2 and 3×3 dimensional matrices. It consists of m and n columns and we say that the rhotrix is m×n. It should be noted that the name rhotrix is as a result of the rhomboidal nature of the arrangement of its entries. i.e,
a,b,c,d ∈R (Ajibade, 2003).
iv. Definition of Ring
A ring is a set R together with two operations (+) and (∙) satisfying the following axioms:
R is an abelian group under addition. That is R is closed under addition. There is an additive identity (called 0), every element a∈R has an additive inverse –a ∈R, and addition is associative and commutative.
R is closed under multiplication, and multiplication is associative.
For every a,b∈R a∙b∈R.
For every a,b,c ∈R,a∙(b∙c)=(a∙b)∙c
Multiplication is distributive over addition:
For every a,b,c∈R, a∙(b+c) = a∙b+a∙c
A ring is usually denoted by (R,+,∙) and often it is written as R when the operations are understood.
v. Definition of Field
A field is a set F together with two operations, addition denoted by +and multiplication denoted by . such that the following axioms are satisfied:
Closure of F under addition and multiplication
Associativity of addition and multiplication
Commutativity of addition and multiplication
Additive and multiplicative inverses.
Distributive of multiplication over addition.
vi. Definition of Semigroup
A non-empty set S is called a semigroup when the operation defined on it is closed and associative.
vii. Definition of Monoid
A monoid is a semigroup that contains an identity element.
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