Algebraic Study of Rhotrix Semigroup

Algebraic Study of Rhotrix Semigroup

Chapter One

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

CHAPTER TWO

LITERATURE REVIEW

RHOTRIX THEORY

Rhotrix theory was initiated by (Ajibade, 2003) and a rhotrix was defined as a rhomboidal form of representing array of numbers. The concept is an extension of ideas of Matrix-tersions and Matrix-noitrets proposed by (Atanassov and Shannon, 1998). (Ajibade, 2003) presented the initial concept, analysis and algebra on rhotrices, where he defined an operation of multiplication of rhotrices of size three. This operation of multiplication is known as heart-oriented multiplication and is commutative. (Sani, 2004) proposed an alternative method for multiplication of rhotrices of size three and later generalized the idea to rhotrices of size n. This alternative operation of multiplication is known as row-column based method for rhotrix multiplication and is non-commutative but associative.

Therefore, in the literature of rhotrix theory, two methods for multiplication of rhotrices having the same size are currently available and each method provides enabling environment to explore the usefulness of rhotrices as tools for carrying out mathematical research.

Based on this, we shall have our review of developments of rhotrix theory in systemic form, starting with the review of commutative rhotrix theory followed by the review of non-commutative rhotrix theory.

CHAPTER THREE

THE RHOTRIX SEMIGROUP

INTRODUCTION

This chapter focuses on the algebraic study of rhotrix semigroup. Using rhotrix set as an underlying set, and together with the binary operation of row-column method for rhotrix multiplication proposed by Sani (2007), we initiate certain algebraic system, which we termed as „Rhotrix Semigroup‟ and study its properties. In particular, we classify the rhotrix semigroup as a regular semigroup and characterise all its five Green‟s equivalence relations.

THE RHOTRIX SEMIGROUP

Let R_n (F ) be a set of all rhotrices of size n with entries from an arbitrary field F . Let the

CHAPTER FOUR

RHOTRIX LINEAR TRANSFOMATION

INTRODUCTION

In the process of achieving the main aim of the present thesis, we found it necessary to use the concept of rank of rhotrix. To the best of our knowledge, the concept of rank ofB rhotrix is not available in the literature of rhotrix theory. Therefore, we introduce in this chapter, the concept of rank of rhotrix and rhotrix linear transformation. Moreover, some properties of this rank and a necessary and sufficient condition under which a linear transformation can be represented by a rhotrix will be discussed in this chapter.

CHAPTER FIVE

SUMMARY, CONCLUSION AND RECOMMENDATION

SUMMARY

In this thesis, the algebraic study of rhotrix semigroup was initiated, using rhotrix set consisting of all rhotrices of size n, with entries from an arbitrary field (F ) , as the

underlying set, and together with the choice of non-commutative method for rhotrix multiplication, as the binary operation. The study is termed as „algebraic study of rhotrix semigroup’. The study started with a complete literature survey of the developments made in the field of rhotrix theory for a decade, starting from the year 2003, when the concept of rhotrix was introduced up to the end of 2013. Next, we presented the construction of non-commutative general rhotrix semigroup and studied its properties.

This was followed by investigation of its certain subgroups and characterizing its Green‟s relations. Parts of this are to appear in Mohammed and Balarabe (Submitted), Mohammed et al; (Accepted).

Towards achieving the main aim of this thesis, we have also introduced new concepts in the field of rhotrix theory, such as rhotrix rank and rhotrix linear transformation. A number of theorems developed have assisted in the quest to characterize Green‟s relations in the rhotrix semigroup. Part of this has appeared in Mohammed et al; (2012)

CONCLUSION

In conclusion, an algebraic study of rhotrix non-commutative semigroup R_n (F ) was initiated and presented. It was also shown that the rhotrix semigroup R_n (F ) is regular and embedded in the square matrix regular semigroup M _n (F ) . As the major contribution, the study was able to characterised all the Green‟s relations in the rhotrix semigroup which form the basis for the development of rhotrix semigroup.

RECOMMENDATIONS

For the future research direction, it seems reasonable to consider the following topics:

A search for generating sets for the finite rhotrix semigroup.
The combinatorial aspect of rhotrix semigroup can be studied.
Any area of study using matrices as tool can be extended to rhotrices analogously.
Idempotent and the product of idempotent should be studied

REFERENCES

Absalom E. E., Sani B. and Junaidu S. B. (2011), “The concept of heart-oriented rhotrix multiplication”, Global Journal of Science Frontier, 11:2, 35-46.
Ajibade A. O. (2003), “The concept of rhotrix in mathematical enrichment”, International Journal of Mathematical Education in Science and Technology, 34:2, 175-179.
Aminu A. (2009), “On the linear system over rhotrices”, Notes on Number Theory and Discrete Mathematics, 15:4, 7-12.
Aminu A. (2010a), “The Equation R_n ( X ) = B over Rhotrices”, International Journal of BMathematical Education in Science and Technology, 41:1, 98-105.
Aminu A. (2010b), “Rhotrix Vector Spaces,” International Journal of Mathematical Education in Science and Technology, 41:4, 531-578.
Aminu A. (2010c), “An example of linear mappings: Extension to rhotrices”, International Journal of Mathematical Education in Science and Technology, 41:5, 691-698.
Aminu A. (2012a), “A determinant method for solving rhotrix system of equation”, Journal of Nigerian Association of Mathematical Physics, 21: 281-288.
Aminu A. (2012b), “Cayley-Hamilton theorem in rhotrices”, Journal of Nigerian Association of Mathematical Physics, 20: 289-296.
Atanassov K.T, Shannon A.G. (1998), “Matrix-tertions and matrix-noitrets: exercises in mathematical enrichment” International Journal of Mathematical Education in Science and Technology, 29: 898-903.
Chinedu M. P. (2012), “Row-wise representation of arbitrary rhotrix” Notes on Number Theory and Discrete Mathematics, 18:2, 1-27.
Howie, J. M. (1995), „‟Fundamentals of Semigroup Theory‟‟. Oxford University Press, London
Kaurangini L. and Sani B. (2007), “Hilbert Matrix and its relationship with a special rhotrix”, Journal of Mathematical Association of Nigeria, 34:2a, 101-106.

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