Project Materials

Algebraic Study of Rhotrix Semigroup

TABLE OF CONTENTS

Cover Page…………………………………………………………………………………………………………..i
Fly Leaf………………………………………………………………………………………………………………..ii
Title Page ……………………………………………………………………………………………………………iii
Declaration ………………………………………………………………………………………………………….iv
Certification ………………………………………………………………………………………………………..v
Dedication ………………………………………………………………………………………………………….vi
Acknowledgment ………………………………………………………………………………………………..vii
Abstract ……………………………………………………………………………………………………………..ix
Table of Contents …………………………………………………………………………………………………x
Nomenclature……………………………………………………………………………………………………..xiii
CHAPTER ONE
GENERAL INTRODUCTION
1.1 Background of the Research………………………………………………………………………1
1.2 Research Aim and Objectives……………………………………………………………………..4
1.3 Research Methodology……………………………………………………………………………….5
1.4 Definition of Terms…………………………………..……………………………6
1.5 Outline of the Thesis…….. …………………………………………………………………………..11
CHAPTER TWO
LITERATURE REVIEW
2.1 Rhotrix Theory…………………………………………………………………………………………13
2.1.1 Commutative rhotrix theory………………………………………………………………15
2.1.2 Non-commutative rhotrix theory ……………………………………………………….24
2.2 Semigroup and Green’s Relations………………………………………………………………38
2.2.1 Green‟s relations………………………………………………………………………………38
2.3 Concluding Remark…………………………………………………………………………………..42
CHAPTER THREE
RHOTRIX SEMIGROUP
3.1 Introduction ……………………………………………………………………………………………..44
3.2 The Rhotrix Semigroup R (F) n …………………………………………………………………..44
3.3 Some Subsemigroups of R (F) n …………………………………………………………………..46
3.4 The Regular Semigroup of R (F) n ………………………………………………………………51
4.3 Green’s Relations in R (F) n ………………………………………………………………………..52
CHAPTER FOUR
RHOTRIX LINEAR TRANSFORMATION
4.1 Introduction ……………………………………………………………………………………………..61
4.2 Rank of Rhotrix………………………………………………………………………………………..61
4.3 Rhotrix Linear Transformation…………………………………………………………………65
xii
CHAPTER FIVE
SUMMARY AND CONCLUSION
5.1 Summary…. ……………………………………………………………………………………………..72
5.2 Conclusion ………………………… …………………………………………………………………..73
References …………………………………………………………………………………………………………74

CHAPTER ONE

GENERAL INTRODUCTION

1.1 INTRODUCTION

The theory of Rhotrix is a relatively new area of mathematical discipline dealing with algebra and analysis of array of numbers in mathematical rhomboid form. The theory began from the work of (Ajibade, 2003), when he initiated the concept, algebra and analysis of rhotrices as an extension of ideas on matrix-tersions and matrix-noitrets proposed by (Atanassov and Shannon, 1998).

Ajibade gave the initial definition of rhotrix of size 3 as a mathematical array that is in some way, between two-dimensional vectors and 22 dimensional matrices. Since the introduction of the theory in 2003, many authors have shown interest in the usage of rhotrix set, as an underlying set, for construction of algebraic structures.

Following Ajibade‟s work, (Sani, 2004) proposed an alternative method for multiplication of rhotrices of size three, based on their rows and columns, as comparable to matrix multiplication, which was considered to be an attempt to answer the question of “whether a transformation can be made to convert a matrix into a rhotrix and vice versa” posed in the concluding section of the initial article on rhotrix.

This method of multiplication is now referred to as “row-column based method for rhotrix multiplication”. Unlike Ajibade’s method of multiplication that is both commutative and associative, Sani‟s method of rhotrix multiplication is non-commutative but associative.

It was shown in (Sani, 2004) that there exists an isomorphic relationship between the group of all invertible rhotrices of size n and the group of all invertible ww dimensional matrices, where

1

n
w and 2 1  n Z . The row-column method for multiplication of base rhotrices was later generalized to include rhotrices of size of n by (Sani, 2007).

Thus, two methods for multiplication of rhotrices are presently available in the literature of rhotrix theory. From now on, we shall refer to the method for multiplication of rhotrices defined by Ajibade as “commutative method for rhotrix multiplication” and the row-column method for multiplication of rhotrices defined by Sani as “non-commutative
method for rhotrix multiplication”.

Mohammed (2007a) adopted the commutative method for rhotrix multiplication to propose classification of rhotrices and their expression as algebraic structures of groups, semigroups, monoids, rings and Boolean algebras.

Based on non-commutative method for rhotrix multiplication, the aim of transforming rhotrix to a matrix and vice-versa was completely achieved in (Sani, 2008), where he proposed a method of converting rhotrix to a special form of matrix called “coupled matrix”. This coupled matrix was used to solve two different systems of linear equations simultaneously, where one is an nn system while the other one is an (n 1)(n 1) .

Following this idea, (Sani, 2009) presented the solution of two coupled matrices by extending the idea of a coupled matrix presented in his earlier work to a general case involving m n and (m1)(n 1) matrices.

It is noteworthy to mention that any research work by interested author(s) in the literature of rhotrix theory is based on either commutative method or non-commutative method for rhotrix multiplication.

So in the presentation of our algebraic study of rhotrix semigroup, we shall adopt the non-commutative technique for multiplication of rhotrices having the same size. The reason behind our choice is that an algebraically non-commutative semigroup offers an exciting platform for carrying out mathematical research in semigroup theory.

One of the well known areas of Mathematics is semigroup theory. It deals with the study of algebra of a set that is closed under an associative binary operation. Semigroup theory has been well developed by researchers, since before the twentieth century.

Many concepts in semigroup theory were analogous to group theory, but the concept of Green‟s relations and many others are developed independently. This makes semigroup theory a well deserved area of research.
The concept of Green‟s relations was first initiated by Green in 1951. These are five equivalence relations defined on a semigroup and they have played a vital role in the development of semigroup theory.

Since the introduction of these equivalence relations, they became standard tools for investigating the structure of any given semigroup. In fact, these relations are so important that, on encountering a new class of semigroups, almost the first question one asks is what are the Green‟s relations like? In certain classes of semigroups, these five equivalence relations turn out to be equal. For instance, in a commutative semigroup, the five relations reduce to one.

This research work is dealing with the algebraic study of rhotrix semigroup. A rhotrix set, R (F) n of size n over a field F was considered, together with the binary operation of non-commutative method for rhotrix multiplication, in order to construct a certain algebraic system termed as “Rhotrix Semigroup”.

Properties of this semigroup were identified and characterize its Green’s relations. Furthermore, as comparable to regular semigroup of square matrices, we show that the rhotrix semigroup is also a regular semigroup. Toward achieving the characterization of Green‟s relations in the rhotrix semigroup, it was found necessary to introduce two concepts; rank of a rhotrix and
rhotrix linear transformation.

1.2 RESEARCH AIM AND OBJECTIVES

The aim of this research is to initiate the concept of rhotrix semigroup. The following objectives were set:

a) To develop the basic fundamental algebra necessary for studying the concept of „rhotrix semigroup‟ as new paradigm of science.

b) To identify and study the properties of rhotrix semigroup as analogous to other types of semigroups in the literature.

c) To characterize Green‟s relations in the rhotrix semigroup.

d) To investigate the existence of any isomorphic relationship between certain rhotrix semigroup and certain matrix semigroup.

1.3 RESEARCH METHODOLOGY

The method adopt in this thesis is to consult all necessary and relevant papers in the literature on fundamentals of Rhotrix theory, Matrix theory and Semigroup theory in order to obtain background information for developing the theory of rhotrix semigroup.

These papers are thoroughly reviewed to cover major works done on rhotrix. In the thesis also, the non-commutative method for rhotrix multiplication was adopted. In the first stage of the work, review of development made on rhotrix theory was documented. This will serve as reference for further research works.

Next, focuses on the algebraic study of rhotrix semigroup, in which we construct and show that the set of all rhotrices of size n, together with the non-commutative rhotrix multiplication operation forms a semigroup.

The properties of this rhotrix semigroup were identified and characterized its Green‟s relations. Towards achieving that, the concept of rhotrix rank and rhotrix linear transformation was introduced and presented at the final stage of the work.