TABLE OF CONTENTS
Abstract
Table of Contents
Nomenclature
CHAPTER ONE
GENERAL INTRODUCTION
1.1 Background of the Research
1.2 Research Aim and Objectives
1.3 Research Methodology
1.4 Definition of Terms
1.5 Outline of the Thesis
CHAPTER TWO
LITERATURE REVIEW
2.1 Rhotrix Theory
2.1.1 Commutative rhotrix theory
2.1.2 Non-commutative rhotrix theory
2.2 Semigroup and Green’s Relations
2.2.1 Green‟s relations
2.3 Concluding Remark
CHAPTER THREE
RHOTRIX SEMIGROUP
3.1 Introduction
3.2 The Rhotrix Semigroup Rn (F )
3.3 Some Subsemigroups of Rn (F )
3.4 The Regular Semigroup of Rn (F )
4.3 Green’s Relations in Rn (F )
CHAPTER FOUR
RHOTRIX LINEAR TRANSFORMATION
4.1 Introduction
4.2 Rank of Rhotrix
4.3 Rhotrix Linear Transformation
CHAPTER FIVE
SUMMARY AND CONCLUSION
5.1 Summary
5.2 Conclusion
References
ABSTRACT
In this thesis, we present an algebraic study of „rhotrix semigroup‟. We identify the properties of this semigroup and introduce some new concepts such as rhotrix rank and rhotrix linear transformation in order to 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.
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.....
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