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
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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
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CHAPTER FIVE
SUMMARY AND CONCLUSION
5.1 Summary…. ……………………………………………………………………………………………..72
5.2 Conclusion ………………………… …………………………………………………………………..73
References …………………………………………………………………………………………………………74
CHAPTER ONE
ENERAL 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 22 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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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 ww
dimensional matrices, where
2
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 nn system while the other one is an (n 1)(n 1) .
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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 (m1)(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,
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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.
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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.
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1.4 DEFINITION OF TERMS
The following definitions are useful in the subsequent chapters:
Definition 1 (Matrix Set)
A matrix set of all M (C) nm is a collection of rectangular arrays, called nm dimensional
matrices with entries from a set of all complex numbers. Thus,
a a a C
a a
a a a a
a a a a
A C mn
m mn
n
n
n m : , ,…, ,…
: : : : … … … : :
: : : : : :: : : :
… … … … … … :
: : : : : : : : :
: : : : : : : : :
: : : … … … … : :
… … … … :
… … … … :
( ) 11 12
1
21 22 23 2
11 12 13 1
(1.1)
Definition 2 (Matrix-tertion and Matrix-noitret)
Matrix-tertion and Matrix-noitret can be defined as mathematical arrays that are in some
way between 2-dimensional vectors and 22 -dimensional matrices introduced by
(Atanassov and Shannon, 1998). Matrix-tertion and Matrix-noitret are denoted by T and
N and respectively defined as
a b cC
c
a b
T : , ,
7
and
a b cC
b c
a
N : , ,
Definition 3 (Semigroup)
A nonempty set S together with a binary operation defined on S is called a semigroup
if (S, ) satisfied the following properties:
S1: Closure property, that is, for all x, yS , x yS .
S2: Associative property, that is, for all x, y, z S , (x y) z x (y z) .
Definition 4 (Monoid and Zero elements)
The semigroup (S, ) is called a Monoid if it has an identity element. That is, if there
exists eS such that x e e x x for all xS . An element 0S is called zero
element of S if x 0 0 x 0 for all xS and (S, ) is called a semigroup with zero.
If (S, ) has no identity or zero element, then it is easy to adjoin an extra identity or zero
to S , in order to form a monoid or semigroup with zero respectively. We write 1 S and
0 S to respectively denote the semigroup with identity or zero adjoined if necessary. We
defined 1 s s 1 s , 111 and 0 s s 0 0 0 0 for all sS . Thus,
S otherwise
S if S hasidentityelement
S
1
, 1 (1.2)
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and
S otherwise
S if S has zero element
S
0
, 0 (1.3)
Definition 5 (ideal)
A nonempty subset I of a semigroup S is called a left ideal if SI I , a right ideal if
ISI, and a (two-sided) ideal if it is both a left and a right ideal.
Alternatively, a nonempty subset I S is
i. a left ideal of S, if for all aI and sS, sa I ;
ii. a right ideal of S, if for all aI and sS, sa I ;
iii. an ideal of S, if for all aI and sS, sa ,asI.
Definition 6 (Subsemigroups)
A subset H of a semigroup (S, ) is called a subsemigroup of (S, ) , if (H, ) is also a
semigroup under the same binary operation.
Definition 7 (Idempotent)
An element a of a semigroup S is called an idempotent element if a a 2 .
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Definition 8 (Regular semigroup)
An element a of a semigroup S is called regular if there exists xS such that axa a .
The semigroup S is called regular semigroup if all its elements are regular, that is
aS, xS axa a . (1.4)
The semigroup M(n,F) of all n n matrices over a field F with respect to matrix
multiplication is an example of regular semigroup, that is for every AM(n,F) there
exists BM(n,F) such that ABA A.
A regular semigroup must contain idempotent elements. It follows from (1.4) that both
ax and xa are idempotents.
Definition 9 (Inverse semigroup)
An element b of the semigroup S is an inverse of aS if aba a and bab b. A
semigroup is called an inverse semigroup if every element of the semigroup has a unique
inverse.
Notice that, an element with an inverse is necessarily regular. Less obviously, every
regular element has an inverse for if there exists x such that axa a , then define
b xax and observe that
aba a(xax)a (axa)xa axa a
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and
bab (xax)a(xax) x(axa)(xax) xaxax x(axa)x xax b
An element a may well have more than one inverse. So, the idea of inverse under
discussion here is substantially more general than a group inverse.
Definition: 10 (Green’s relations)
Let S be an arbitrary semigroup. The equivalence relations ℒ, ℛ, 𝒥, ℋ and 𝒟 are
defined on S as follows:
a ℒ b if and only if (x, y S )a xb 1 and b ya ;
a ℛ b if and only if (u,vS )a bu 1 and b av ;
a 𝒥 b if and only if (x, y,u,vS )a xby 1 and b uav ;
a 𝒟 b if and only if (cS) aLc and cb ;
and
ℋ = ℒ ℛ
For all a,bS .
These five relations on S are called Green‟s relations. See (Howie, 1995).
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1.5 OUTLINE OF THE THESIS
The outline of the thesis is as follows:
Chapter one presents the general introduction of the thesis, the aim and objectives of the
study, the methodology for carrying out the research, definition of terms and then finally,
the outline for the thesis.
Chapter two focuses on a review of developments in the literature of rhotrix theory,
starting from the year 2003, when the concept of rhotrix was introduced up to the end of
2013. Furthermore, a review of certain concepts in semigroup and Green‟s relations were
discussed.
Chapter three considers the rhotrix set R (F) n of size n over a field F and together with
Sani‟s row-column based method for rhotrix multiplication, in order to initiate the
concept of non-commutative rhotrix semigroup. We identify the properties of this
semigroup and characterize its Green‟s relations. Furthermore, as comparable to regular
semigroup of square matrices, we showed that the rhotrix semigroup is also a regular
semigroup.
Chapter four introduces two concepts; rank of a rhotrix and rhotrix linear transformation,
as two necessary tools required for achieving our aim of characterization of Green‟s
relations in rhotrix semigroup. Furthermore, some properties of this rank and a necessary
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and sufficient condition under which a linear transformation can be represented by a rhotrix were also presented in the chapter. Chapter five gives the summary for the whole thesis, its conclusion and recommendations for future research direction.
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