Algebraic Study of Rhotrix Groups
Chapter One
RESEARCH AIMAND OBJECTIVES
The aim of this research work is to construct rhotrix groups bearing in mind the following objectives:
i. To construct finite algebraic structures that satisfy the axioms of groups using rhotrix sets as underlying sets.
ii. To systematize ideas in group theory to rhotrix group theory,using already known concepts such as cosets, order, subgroups, Lagrange’s theorem of a group and so on.
CHAPTER TWO
LITERATURE REVIEW
INTRODUCTION
The concept of rhotrix was first introduced in(Ajibade, 2003),as an extension of ideas on matrix tertions and matrix noitrets suggested by Atanassov and Shannon (1998).Ajibade(2003) discussed the initial algebra and analysis on rhotrices and also set up some relationships between rhotrices and their hearts.
The initial concept of rhotrix introduced by Ajibade was presented as objects which lie in some way between 2 × 2 -dimensional and 3 3 -dimensional matrices defined as:
Addition of rhotrices
Theaddition of rhotricesas introduced in Ajibade(2003) for rhotrices of size three is followed here. The operation of addition of two or more rhotrices is found by adding the corresponding elements of the rhotrices. Let R and S be two base rhotrices, then their sum is gotten by adding the corresponding ij a of R and ij b of S.
Properties ofRhotrix Addition
The following properties hold for any rhotrices A, B, C.
i. Existence of additive identity: Given a zerorhotrix O of the same size, we have A+𝑂 = 𝑂 + 𝐴 = 𝐴.
ii. Existence of additive inverse : There exists a unique rhotrix −𝐴 such that
A A O which is the additive inverse of A, where O is a zero rhotrix having the same size as A.
iii. Commutativity: A+𝐵 = 𝐵 + 𝐴.
iv. Associativity: 𝐴 + 𝐵 + 𝐶 = 𝐴 + 𝐵 + 𝐶 .
Usaini and Tudunkaya (2011a) constructed certain fields of fractions over rhotrices by extending the work of Mohammed (2009).The construction which was done step by step and at each step, a particular algebraic property was shown. Usaini and Tudunkaya (2012) later discovered that the field as constructedin(Mohammed, 2009)
was only possible for set of hearty rhotrices of the same size as in(Mohammed, 2007a).
Mohammed and Tella (2012) presented the categorized rhotrix sets and rhotrix spaces over real and complex fields. The idea was to systematize ways of characterizing rhotrices over the field of numbers and their expressions as rhotrix set spaces, stimulated through their work.
Mohammed and Tijjani (2011) defined metric or distance function between elements in a rhotrix set to the set of real numbers. The idea was extended to construction of metric topological spaces in their work.
Tudunkaya and Makunjuola (2012a), proposed a certain quadratic extension as an extension to the work of Mohammed(2007b). Thereafter, rhotrix polynomial and polynomial rhotrices were proposed by Tudunkaya (2013).
Usaini and Tudunkaya (2011b) presented some notes on rhotrices and the construction of finite fields. Tudunkaya and Makunjuola (2012b) worked on properties of certain finite fields constructed over rhotrices.
CHAPTER THREE
FORMATION OF RHOTRIX GROUPS
INTRODUCTION
Let G be a non- empty set equipped with a binary operation denoted by
If for any
a b G , ,
a b or more conveniently ab represents the element of G obtained by applying the said binary operation between the elements a and b of G taken in order. The algebraic structure 𝑮 is a group if the binary operation satisfies the following axioms.
a) For any a,b∈ 𝑮 a b∈ 𝑮 implies closure axiom.
b) 𝒂 ∗ 𝒃 ∗ 𝒄 = 𝒂 ∗ 𝒃 ∗ 𝒄 for all a, b, c ∈ 𝑮 implies associativity axiom.
c) If there exists e ∈ 𝑮 such that for all a∈ 𝑮 e a a e a. implies identity axiom.
d) If for all a∈ 𝑮,there exists b ∈ 𝑮 such that a b b a e implies inverse axiom.
A group is said to be abelian or commutative if in addition to all of the above axioms, it
also satisfies ab ba, for all a,b∈ 𝑮.
RHOTRIX GROUP
A rhotrix set G for which the law of composition is defined and on which a binary operation, is defined forms a rhotrix group if the following conditions are satisfied.
CHAPTER FOUR
ISOMORPHISMS OF RHOTRIX GROUPS
INTRODUCTION
Rhotrix is an innovation by way of extension from matrix tersions and matrix noitrets. Rhotrix is a new area of study relating to linear mathematical algebra. From our results so far, there are sets of rhotrix algebraic structure which satisfy the axioms of group and so they form a rhotrix group. We intend to demonstrate by way of example, the result in (Mohammed, 2007a, theorem 1) which said there is an isomorphism between group of unit heart rhotrices of the
same size and zero heart rhotrices of the same size. Furthermore, some important properties of isomorphic mappings would be discussed.
CHAPTER FIVE
SUMMARY, CONCLUSION AND RECOMMENDATIONS
SUMMARY
In this thesis, we have shown that already known algebra structures can be systematized into rhotrix groups. As the facts about groups are developed, we shall often observe analogies with facts we already know, and will serve as mathematical model on all spheres of studies irrespective of the course. Experience provides ample verification of the assumption that in learning mathematics it is pedagogically essential to develop concepts carefully, and to proceed from familiar to the unfamiliar , from specific to general and from concrete to abstract. In this work we developed rhotrix groups using residue classes of modulo two, three and five as our underlying sets, and then generalize it to modulo n. The multiplication table for rhotrix group and rhotrix maps were developed using already
known theorems. The subgroups of the rhotrix groups developed were also identified and the order of each element of the different residue classes were shown.
CONCLUSION
From this work, the rhotrix groups we developed was used to verify Langrange’s theorems. It was shown that cyclic rhotrix groups could be developed from the cyclic rhotrix multiplication table presented. It was equally shown that
there exists homomorphisms and isomorphisms between certain rhotrix groups.
RECOMMENDATIONS
I wish to recommend thatother theorems of groups be verified using rhotrices. The study of rhotrix algebra should be encouraged at all levels of both learning and research. In institutions of higher learning, it should be included in and studied alongside matrices. To researchers in science related fields, this innovation is a challenge and they should try to incorporate it in their various disciplines.
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, 4:2, 175-179.
- Aminu A. (2009) On the linear systems over rhotrices, Notes on Number theory and Discrete Mathematics,15:4, 7-12.
- AminuA. (2010a) The equation R X over rhotrices, International Journal
ofmathematical Education in science and technology41:1, 98-105. - AminuA. (2010c) An example of linear mappings: Extension to rhotrices, International Journal of Mathematical Education in Science andTechnology, 41:5, 691-698.
- 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: exercise in mathematical enrichment International Journal of Mathematical Education in Science and Technology, 29:898-903.
