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**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.

NOMENCLATURE | |

R (_{n}F ) | – A set of all n ´ n rhotrices over a field F . |

h(R) | – Heart of rhotrix R |

_{x}ni | – A rhotrix row vector of size n |

_{x}nj | – A rhotrix column vector of size n |

M (_{n}F ) | – A set of all n ´ n matrices over a field F |

**CHAPTER ONE**

**GENERAL INTRODUCTION**

**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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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 w = | n +1 | and | n Î 2Z ^{+} +1. The | row-column method | for | |

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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, 3

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** _{n}* (

*F*)

*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.

**RESEARCH AIM AND OBJECTIVES**

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

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

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

- To characterize Green‟s relations in the rhotrix semigroup.

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

**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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**DEFINITION OF TERMS**

The following definitions are useful in the subsequent chapters:

**Definition 1 (Matrix Set)**

A matrix set of all *M* _{n}_{´}* _{m}* (

*C*) is a collection of rectangular arrays, called

*n*´

*m*dimensional

matrices with entries from a set of all complex numbers. Thus,

ì | éa | a | a | … … … … | a | ||||||||

ï | ê | 11 | 12 | 13 | 1n | ||||||||

ï | _{ê}21^{a} | 22^{a} | 23^{a} | … … … … | 2^{a}n | ||||||||

ïê | : | : | : | … … … … | : | ||||||||

ï | ê | : | : | : | : | : | : | : | : | ||||

ï | ê | ||||||||||||

A | (C) = | ï_{ê} | |||||||||||

n´m | í | ||||||||||||

ï^{ê} | : | : | : | : | : | : | : | : | |||||

ï | ê | ||||||||||||

ï^{ê}a | … … | … | … … | … a | |||||||||

ï^{ê} | m1 | mn | |||||||||||

ï | ê | : | : | : | : | : | :: | : | : | ||||

ï^{ê} | : | : | : | : … … … | : | ||||||||

î | ë |

: ù | ü | ||||||

ú | ï | ||||||

: _{ú} | ï | ||||||

: ú | ï | ||||||

ú | ï | ||||||

: _{ú} | ï | ||||||

ú :a , a | ,…,a | mn | ,…ÎC | ï | (1.1) | ||

ú | 1112 | ý | |||||

ï | |||||||

: _{ú} | |||||||

ï | |||||||

: ^{ú} | |||||||

ï | |||||||

ú | ï | ||||||

: ú | |||||||

ï | |||||||

ú | |||||||

ï | |||||||

: _{û} | |||||||

þ |

**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 2´2 -dimensional matrices introduced by (Atanassov and Shannon, 1998). Matrix-tertion and Matrix-noitret are denoted by T and N and respectively defined as

ìa | b | ü | |

ï | ï | ||

T = í | :a,b, c ÎC ý | ||

ï | c | ï | |

î | þ |

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and

ì | a | ü | |

ï | ï | ||

N = í | :a, b, c ÎC ý | ||

ï | c | ï | |

î^{b} | þ |

**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*, *y* Î *S* , *x* *y *Î* S *.

S2: Associative property, that is, for all *x*, *y*, *z* Î *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 e Î S such that x e = e x = x | for all x ÎS . An element 0 Î S is called zero |

element of S if x 0 = 0 x = 0 for all | x ÎS 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 *S* ^{1} and

*S *^{0} to respectively denote the semigroup with identity or zero adjoined if necessary. We defined 1 *s* = *s* 1 = *s* , 1 1 =1 and 0 *s* = *s* 0 = 0 0 = 0 for all *s* Î *S* . Thus,

ìS | if S hasidentityelement, | (1.2) | ||

S^{1} = í | È{1} | |||

î^{S} | otherwise | |||

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and

ì | S | if S has zero element, | (1.3) | |

S ^{0} = í | ||||

_{î}S È{0} | otherwise |

**Definition 5 (ideal)**

A nonempty subset *I* of a semigroup *S* is called a *left ideal* if *SI* Í *I* , a *right ideal* if

*IS *Í* I *, and a (two-sided)* ideal *if it is both a left and a right ideal.

Alternatively, a nonempty subset I Í S | is | ||

i. | a left ideal of S, if for all a Î I and s Î S, | sa Î I ; | |

ii. | a right ideal of S, if for all | a Î I and s Î S, | sa Î I ; |

- an ideal of
*S*, if for all*a*Î*I*and*s*Î*S*,*sa*,*as*Î*I*.

**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*^{2} = *a* .

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**Definition 8 (Regular semigroup)**

An element *a* of a semigroup *S* is called *regular* if there exists *x* Î *S* such that *axa* = *a* . The semigroup *S* is called *regular semigroup* if all its elements are regular, that is

“a Î S, $x Î S ‘ 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 *A *Î* M *(*n*,* F *)* *there

exists *B* Î *M* (*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 *a* Î *S* 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**,** **, **ℋ and are

defined on *S *as follows:

*a *ℒ* b if *and only if ($*x*,* y *Î* S *^{1} )*a *=* xb *and *b* = *ya* ;* a *ℛ* b *if and only if ($*u*,* v *Î* S *^{1} )*a *=* bu *and* b *=* av *;

*a b *if and only if ($*x*,* y*,* u*,* v *Î* S *^{1} )*a *=* xby *and* b *=* uav *;* a b *if and only if ($*c *Î* S*)* aLc *and* c*Â*b *;

and

ℋ **=** ℒ Ç ℛ

For all *a*, *b*Î*S* .

These five relations on *S* are called Green‟s relations. See (Howie, 1995).

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**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

- Furthermore, a review of certain concepts in semigroup and Green‟s relations were discussed.

Chapter three considers the rhotrix set *R** _{n}* (

*F*) 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.