MENGUE MENGUE, DAVID JOEL ( )

i

CONTRIBUTIONS TO THE THEORY OF

PARASTROPHS AND DERIVATIVES

OF LOOPS

BY

MENGUE MENGUE, DAVID JOEL

(079075102)

MASTER OF SCIENCE IN PURE MATHEMATICS, UNIVERSITY OF MAIDUGURI, 2001

A THESIS SUBMITTED TO THE SCHOOL OF POSTGRADUATE STUDIES UNIVERSITY OF LAGOS

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE AWARD OF THE

DEGREE OF DOCTOR OF PHILOSOPHY (Ph.D.) IN MATHEMATICS

DEPARTMENT OF MATHEMATICS UNIVERSITY OF LAGOS

ii

SCHOOL OF POSTGRADUATE STUDIES UNIVERSITY OF LAGOS

CERTIFICATION This is to certify that the thesis:

CONTRIBUTIONS TO THE THEORY OF PARASTROPHS AND DERIVATIVES

OF LOOPS Submitted to the

SCHOOL OF POSTGRADUATE STUDIES UNIVERSITY OF LAGOS

FOR THE AWARD OF THE DEGREE OF DOCTOR OF PHILOSOPHY (Ph.D.) IS A RECORD OF ORIGINAL RESEARCH CARRIED OUT

BY

MENGUE MENGUE, David Joel IN THE

DEPARTMENT OF MATHEMATICS

iii

DEDICATION

iv

ACKNOWLEDGEMENTS

This is to express my unreserved gratitude and appreciation to all who have contributed both directly and indirectly to the successful completion of this study. First of all, I thank the Almighty GOD for his mercies, guidance and protection during this programme. May his name be praise for ever.

I am indeed grateful to my first supervisor Dr. S. O. Ajala. I got admitted into Unilag through him. My daily meetings and discussions with him for the past four years really taught me a lot. His guidance, wise counsel, advice and suggestions were a great deal, invaluable and greatly improved this study. Sir, I learnt a lot from your intellectual cross pollination.

The deep sense of appreciation goes to my second supervisor, Prof. J. A. Adepoju; I am more than grateful to you for your unparalleled assistance in reading through the manuscript line by line despite your tight and busy schedule. This research has always received your intellectual touch before passing to the next table for further reading and assessment. Sir, your fatherly advice, encouragement, support and suggestions became what make this research real and effective. I will forever remain committed to your course.

My third supervisor, Dr. J. O. Adeniran told me that the word “Ph.D.” stands for Patience, Hard work and Drive. One must have the drive to cope with all challenges, seen and unseen, to succeed in this journey. Sir, all of these made sense to me in the cause of my sojourn at Unilag. I got a lot of academic material from him, around him and through him. Sir, your very detailed comments were helpful and responsible for this scholarly output.

I am greatly indebted to my following special lecturers: Dr. J.O. Olaleru and Dr. S.A. Okunuga for their time to input ideas into those grey areas and for their scholarly comments on almost every part and stage of this study. Also to Dr. T.G. Jaiyeola from O. A. U Ile-Ife, I acknowledge the role you played in this research. Your donation, accommodation, suggestions were very useful to me. They all help increase the scope of this research work.

And finally, special thanks go to my HOD, Prof. R. O. Okafor and all my other lecturers, my senior and junior colleagues, the entire non-academic staff of the Department, friends and family members too numerous to mention here, who had rendered help and services at various time and in different capacity toward the successful completion of this work. I say GOD will reward you all. Please, do keep the flag flying and higher, success is sure at last.

v TABLE OF CONTENTS Pages Title pages i Certification ii Dedication iii Acknowledgement iv Table of contents v Abstract vii

CHAPTER ONE: Introduction 1.1 Background of Study 1

1.2 Statement of Problem 2

1.3 Aims and Objectives of the Study 2 1.4 Scope/ Limitation of the Study 2

1.5 Significance of Study 2

1.6 Research Questions 3

1.7 Operational Definition of Terms 4

CHAPTER TWO: Literature Review 2.1 Summary of the genesis of loops theory 7

2.2 Parastrophs of quasigroups 17

2.3 Derivatives 20

2.4 Basics Facts 21

CHAPTER THREE: Methodology 26

CHAPTER FOUR: Results and Discussion 4.1 Introduction 31

4.2 Parastrophs of Extra loops 35

4.3. Derivatives of Extra loops 56

Moufang loops 64

vi

Conjugacy closed loops 84

4.4 Some area of applications 100 CHAPTER FIVE: Conclusion

5.1 Summary of findings 103 5.2 Conclusion 103 5.3 Contributions to knowledge 104

5.4 Further work 104

vii ABSTRACT

1

CHAPTER ONE

One of the features of the twentieth century Mathematics has been its recognition of the power of the abstract approach. This has given rise to a large body of new results and problems and has, in fact, led researchers to open whole new areas of Mathematics whose very existence had not even been suspected.

In the wake of these developments, there has come not only a new approach to Mathematics but a fresh outlook, and along with this are simple new proofs of difficult classical results. The isolation of a problem into its basic essentials has often revealed to us the proper setting, in the whole scheme of things, of results considered to have been special and unrelated and has shown us interrelations between areas previously thought to have been unconnected. Precisely, the theory of quasigroups and loops is one of the fairly young disciplines which take their roots from geometry, algebra and combinatorics. In geometry it arose from the analysis of web structures, in algebra, from non-associative products and in combinatories, from Latin squares. For example, when someone asks: “What is a loop?” The simplest way to explain to him is to say: “It is a group without associativity”. This is true, but it is not the whole truth. It is essential to emphasize that quasigroups and loops theory is not just a generalization of group theory but a discipline of it own, originating from and still revolving within the three above mentioned basic research areas. The subject has grown by relating these aspects to an increasing variety of new fields, and is now developing into a thriving branch of Mathematics.

2

Derivatives could be considered as algebraic structures generated from an initial main algebraic structure as we shall see.

The next diagram named after Hasse, highlights the basics and most popular loops of Bol-Moufang type. A summary of their relative position and connecting lines is given as is usual in a Hasse diagram.

{ { {

Figure: 1 a Hasse diagram

1.2

Statement of the Problem

The literature on quasigroups and loops theory confirm that, every quasigroup belongs to the set of six quasigroups of the initial algebraic structure, called conjugates, adjugates or parastrophs. The fact that the derivatives of a quasigroup are quasigroups with left and right identity elements is also well known. However, the existence, the nature and the properties of those two algebraic structures namely parastrophs and derivatives of loops of Bol-Moufang type are not yet known. It is, therefore of mathematical interest to carry out an investigation on the existence, nature and properties of those structures to fill the gap and to improve our knowledge as well as to enhance the literature on parastrophs and derivatives of Bol-Moufang type of loops.

1.3

Aims and Objectives of the Study

3

1. The construction of six functions that are generating the Parastrophs of a given Extra loops and satisfying the Extra identities.

2. The extension of the concept of Parastrophs and Derivatives of Quasigroup

3. The establishment of the nature of the Derivatives of Moufang and Conjugacy closed loops.

4. Find the necessary and sufficient conditions for the Parastrophs of and Extra loop to become Extra loops

5. The extension of Jaiyeola and Adeniran (2006) results.

1.4

Scope/ Limitation of the Study

This study, though in the general area of Algebra, is concerned with quasigroups and loops theory in general and more specifically with loops of Bol-Moufang type. Throughout this research work, Extra loop is the case study.

1.5

Significance of the Study

This research work will add new results to the literature related to the entire quasigroups and loops theory. This study shall demonstrate that from a single initial Extra loops, one can obtain several and different Extra loops called Parastrophs and Derivatives. The resulting different algebraic structures have important applications.

1.6

Research Questions

Throughout this study, the following research questions will guide the study. 1-) What do we mean by Derivatives and Parastrophs of a quasigroup?

2-) Do Parastrophs and Derivatives of an Extra loop exist?

3-) If the answer to 2-) is in the affirmative, then the researchers will investigate whether, the Parastrophs of an Extra loop are Extra loops?

4-) If the answer to 3-) is negative, then under what conditions will the Parastrophs of an Extra loop be Extra loops?

4

1.7

Operational Definitions of Terms

In this section, the researcher shall lay down operational definitions of terms that are employing in this study.

Translation maps: For a groupoid ( ), any permutation on such that for all , ( ) is called left translation map. Similarly ( ) is called right translation map for all .

Quasigroup: This is any groupoid where the translation maps are bijective. Loop: This is a quasigroup with the identity element.

Loop of exponent two: A loop ( ) is said to be of exponent two if for all we have that is _.

Extra loop: Any loop that satisfies any of the following equivalent identities for all , (i) (( ) ) ( ( )), (ii) (( ) ( )) (( ) ), (iii) (( ) ( )) ( ( )) is called an Extra loop.

Parastrophs: For any quasigroup ( ) ( ) it is possible to associate five other algebraic structures called parastrophs of ( ). If we denote the quasigroup operation (function) by the letter , then with the quasigroup operation , we can associate the following binary operations with and elements of .

( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )_{( )}

.

In other words ( ) ( ) . For example, ( )_{( ) ( ) that is,}

( )₍

( ) ( )) ( ) ( ) . We shall also find it convenient to

employ the alternative notation where , for parastrophic operations.

5 -1_{( ) .} /, and ( ) -1_{( )( ),} (-1 ) . /, and ( ) ( -1 ) ( ), (-1( _{)) .} /, and ( ) ( -1 ( )) ( ).

Isotopism: A triple ( ) of bijections from a set to a set is called an isotopism of a groupoid ( ) into a groupoid ( ) provided ( ) for all . ( ) is then called an isotope of a groupoid ( ), and groupoids ( ) and ( ) are said to be isotopic to each other.

An isotopism on is called an autotopism of a groupoid ( ).

Derivatives: If ( ) is a non associative quasigroup, then a fixed element determines a new operation ( ) on G such that:

( ) ( )

The operation ( ) depends entirely on , so we write: ( ) and call it the left derivative of with respect to . We thus have:

( ) ( ) ( ) So that we obtain the isotopism: ( ( ) ( )) or the isotopism ( ( ) ( ) ) We can then write ( ( ) _{( ) ).}

Similarly, we also have what is called the right derivative of . Fix such that:

( ) ( ) Then ( ) is called the right derivative of with respect to Thus: ( ) ( ) ( ) implies the following isotopisms:

( ( ) ( )) or ( ( ) ( ) ) and this implies that: ( ( ) _{( ) ) .}

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Derivative of a quasigroup: The derivative of a quasigroup ( ) with respect to an element , is a quasigroup generated from the initial quasigroup, leaving invariant with a new function belonging to * +.

Some Notations

RIP: Right Inverse Property LIP: Left Inverse Property IP: Inverse Property

AIP: Automorphic Inverse Property

= End (of a proof).

: Difference between : A is a subloop of G.

7

CHAPTER TWO

Looking back on the first few years of the history of quasigroups and loops theory, one can see that every decade added a new and important phase in its development. Almost each of the periods is represented in our literature by a classic book of that era, which to this day remains the main reference source in its respective area: Blaschke and Bol (1938) in Germany, Bruck (1958) in England, Pickert (1955) in Germany and Belousov (1967) in Russian.

This work shall shed light on the original motivations for the first publications, on quasigroups by Moufang and Bol. Let us recall that: Hyperbolic geometry was discovered almost simultaneously by Lobatschevski and Bolyai in 1820s, and would subsequently combine with Riemannian geometry (announced in 1866) to form the field of Non-euclidean Geometry. Similarly, around the turn of the century, an entirely new conception of Space-Time emerged out of the Lorentz transformation of 1895, which replaced earlier Galilean notions, and Einstein‟s Special Relativity of 1905. We now know that these two ideas, Non-Euclidean Geometry and Curved Space-Time, are not utterly unrelated and both helped to prepare the ground for the notion of associativity. The example of an abstract non-associative system was Cayley numbers, constructed by Arthur Cayley in 1845. Later they were generalized by Dickson to what we know as Cayley-Dickson algebras.

Another class of non-associative structures was systems with one binary operation. One of the earliest publications dealing with binary systems that explicitly mentioned non-associativity was Suschkewitsch (1929). In his paper, Suschkewitsch observes that, in the proof of the Lagrange theorem for groups, one does not make use of the associative law. So he rightly conjectures that it could be possible to have non-associative binary systems which satisfy the Lagrange property. He constructed two types of such so-called “general groups”, satisfying his postulate A or postulate B. In Suschkewitsch‟s approach, one can detect some early attempts in the direction of modern loop theory as a generalization of group-theoretical notions. His “general groups” seem to be the predecessors of modern quasigroups as isotopes of groups.

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was never realized, but in the process the significance of non-associativity began to emerge. It was around this time that Artin proved a theorem that Ruth Moufang would later use in her famous paper on quasigroups.

Artin’s theorem: In an alternative algebra, if any three elements

multiply associatively, they generate a subalgebra. Very exciting developments were also under way in differential geometry. Many young and talented mathematicians were attracted in this area by Wilhelm Blaschke innovative work. Blaschke‟s book on the subject, Web Geometry, co-authored with Bol, came out only in 1938, but was preceded by many separate publications on this topic by himself and his followers, including 66 papers of the series “Webs and Groups”. Bol alone contributed 14 papers to this series. Among the earliest were papers by Thomsen and Reidemeister, whose names we now know from corresponding web-configurations. The title of the series, “Webs and Groups”, subtitled “Topological Questions of Differential Geometry”, was also significant for having neatly combined the three main areas from which quasigroups and loops theory emerged. Bol, among others, made significant contributions to this area of research.

From the point of view of quasigroups and loops theory, all these developments culminated in the appearance of two papers that defined the two most important classes of loops as we know them now, Moufang loops and Bol loops: Moufang (1935), and Gerrit Bol (1937). Together, these papers marked the formal beginning of quasigroups and loops theory. In her paper, Moufang defined a structure, which she called a Quasigroup Q*, satisfying the following postulates: (1), (2) closure, existence of an identity element and unique inverses (3) ( ) ( ) and ( ) ( ).

(4) ( ( )) ( ( )) . She also defined a system Q**, believing it to be different from Q*. Q** satisfies an additional

identity:

(5) ( ) ( ) (( ) )

Bol soon showed that (4) implies (5), and Bruck (1944) later proved that they both are equivalent to two other identities: (6) (( ) ) ( ( ))

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Moufang proved that Q* is diassociative that is “the subquasigroup generated by any two elements is associative” and satisfies a theorem that echoes Artin‟s theorem and is now known as Moufang‟s theorem. Moufang’s theorem:

If ( ) for some then generate a subgroup of ( ).  She also gave a geometric interpretation to A.T for projective planes. The next most important paper on the subject of quasigroups appeared two years after Moufang‟s: Gerrit Bol (1937). Bol‟s approach is from a web-geometrical point of view. He constructed three new configurations U1, U2, U3 and asked whether the closure of these three figures implied

associativity. He answered that question in the negative, and showed that the three U figures together imply only the law ( ( )) (( ) ) , which is precisely one of the Moufang identities. To demonstrate this fact, Bol gives an example constructed by Zassenhaus. This example (of order 81) was, in fact, the first example of a non-associative commutative Moufang loop.

Further, Bol explained the algebraic meaning of each of the U figures and showed that U1 and U2 correspond to laws that we now call the right Bol and the left Bol identities,

respectively: (( ) ) (( ) ) and ( ( )) ( ( )). It was Zassenhaus, again, who soon constructed the first example of a right Bol loop.

Bol also proved that the following properties implied by the U figures: U1 the right inverse property: ( )

U2 the left inverse property: ( )

U3 the anti-automorphic inverse law: ( )

Bol showed that U1 and U2 together imply U3, and when all three are closed, one

obtains Moufang‟s quasigroup Q*

. Bol also demonstrated that Moufang‟s Quasigroup Q* satisfies the flexible law: ( ) ( ) .

Thus, Bol practically split the Moufang identity into two, showing that, in our language, a loop is Moufang if and only if it is both right and left Bol. The irony was that Bol was not originally aware of Moufang‟s work when he wrote his paper. In a footnote, to his paper he acknowledged that Moufang‟s paper only came to his attention after he had practically completed his article. Here we have yet another example of a simultaneous development of similar ideas from different perspectives.

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After the demise of quasigroups and loops theory in Germany, it was the United States that became the new center of research on the subject. Why the United States? There were mainly two reasons.

One important factor was that many prominent mathematicians had to leave Germany when Hitler came to power, either being Jews themselves or having Jewish spouses. The other important reason was that a strong interest in non-associative structures already existed in the United States, particularly at the University of Chicago. Leonard Dickson, whose name we know from Cayley-Dickson, algebra, was teaching at Chicago, along with his former student Abraham Adrian Albert. In this way, Chicago became a new center of quasigroups and loops theory research in the 1940s, just as Hamburg had become in the previous decade. In addition to alternative algebra research, there were already several American publications on the theory: Hausmann and Ore (1937), Murdoch (1939) and Garrison (1940). All three authors already used the term “quasigroup” in a broader sense, the way it is being use now, and not just as Moufang‟s Q*

system. The first authors still assumed the existence of an identity element, whereas Murdoch already considered the case of a one-sided identity.

It was at this point that the terminology quasigroup theory underwent a historic change. It became apparent that it was necessary to distinguish between two classes of quasigroups: those with and those without identity element. A new name was needed to designate the system with identity. This occurred around 1942, among people of Albert‟s circle in Chicago, who coined the word “loop” after the Chicago loop.

The first publications introducing the term “loop”, were the two very important that Albert (1943) wrote : Quasigroups. I and Quasigroups. II. In addition to the introduction of the new term “loop”, a highly significant aspect of the Quasigroups. I paper was the introduction of the concept of isotopy for quasigroups. Definition 2.1.1

A triple ( ) of bijections from a set into a set is called an isotopism of a groupoid ( ) into a groupoid ( ) provided ( ) for all . ( ) is then called an isotope of ( ), and groupoids ( ) and ( ) are called isotopic to each other.

11

text on loops.

One can see that during this period, from the 1940s through the 1960s, the basic algebraic frame of loop theory was erected. Loop and Quasigroups theory had gained a firm ground that would allow it to move in new directions and flourish in other places.

One of the many new directions in which quasigroups and loops theory began to move was toward the universal algebra approach.

In England, the extension of its concepts to quasigroups and loops became the life work of Trevor Evans from the late 1940s onward. By defining quasigroups as algebras with three operations, including the left and the right division, Evans (1949) was able to consider quasigroups as varieties of -algebras, * +, and apply to them many of the notions and tools of universal algebra. Still another area of research in England at the time was that of Latin squares. The subject of Latin squares is, much older than quasigroups and loops theory. Mutually orthogonal Latin Squares were already studied by Euler in the XVII century from a combinatorial point of view. However, as the theory developed, there appeared connections between the combinatorial and several quasigroup-theoretical aspects of Latin squares. For example, combinatorial structures such as bloc designs or Steiner triple systems can be associated with algebraic varieties of Steiner quasigroups and totally symmetric loops. Quasigroups may be defined combinatorially or equationally. A (combinatorial) quasigroup ( ) is a non-empty set equipped with one binary operation

12 M E N G U M M E N G U E E M U N G N N G M U E G G U E M N U U N G E M Table1 Table 2

From an Algebraic point of view, the combinatorial definition of a quasigroup has some serious disadvantages. In particular a homomorphic image of a combinatorial quasigroup need not be a quasigroup. An (equational) quasigroup ( ) is defined as a non-empty set , equipped with three binary operations ( ) ( ) ( ), satisfying the identities:

(IL): ( ) ; (IR): ( ) ; (SL): ( ) ; (SR): ( ) Another area of research in England at the time was the problem of different classes of quasigroups. For example, the work done by I.M.H. Etherington on entropic quasigroups, which satisfy the identity: ( ) ( ) ( ) ( ). Etherington (1964) showed that totally symmetric entropic quasigroups are naturally connected to the geometry of plane cubic curves. The subject of cubic hypersurfaces was later fully developed by Yuri Manin (1968) in the Soviet Union. For the Soviet Union and the former Soviet bloc countries, Belousov‟s role in the success of quasigroups and loops theory, and his 1967 book, Foundations of the Theory of Quasigroups and loops, can rightly be compared with the role that Bruck and his Binary Systems had played in the United States a decade or two earlier. Unfortunately, Belousov‟s excellent book is not as widely known as Bruck‟s in the West, since it is written in the Russian language and has never been translated.

While Bruck‟s emphasis was on loops, in Belousov‟s work, the weight shifted toward quasigroups in general. Belousov (1965, 1967), showed that the equational definition of quasigroup implies the combinatorial definition.

Just as Bolousov was the main figure in this field in Russian, the main figures in the United States in the 1940s were Reinhold Baer and Marshall Hall. Both were working on the idea of projective planes as planes over some algebraic systems. They complemented each other‟s work by using different approaches. Hall (1943) worked on the correspondence

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between projective planes and algebraic systems such as double loops and ternary rings and fields. This approach proved to be very useful for finite cases. In finite cases, collineations groups also lead to the questions of combinatorics, block designs and Steiner triple systems. He proved that the combinatorial definition implies the equational definition of a quasigroup. We hereby combined the above two results as in Smith (2007).

Theorem 2.1.1(Smith, 2007)

( ) is an equational quasigroup if and only if ( ) is a combinatorial quasigroup.

Proof.

It must be shown that knowledge of any two of such that

(2.1.2)

specifies the third uniquely. Now the existence and uniqueness of given and corresponds to the functionality of the multiplication (2.1.1). Suppose that and are given. By (SL), is one solution of equation (2.1.2), on the other hand, if is a solution then, ( ) by (IL), so the solution is unique. The existence and uniqueness of as a solution of (2.1.2) given and follows similarly.

Conversely, suppose that ( ) is a combinatorial quasigroup. For given elements and of , define as the unique solution of (SR), and as the unique solution of (SL). This defines a right division and left division that make ( ) an equational quasigroup. 

In this study the combinatorial definition of a quasigroup shall be more useful to us than equational definition. But it will be reframed as follows.

Definition 2.1.2

For a groupoid ( ), we can define two mappings ( ) and ( ) for all by: ( ) ( ) ( ) and ( ) ( ) ( ) . These are called translation maps. If the translation maps are bijective for all , then ( ) is called a quasigroup. A quasigroup with identity element is a loop. For a loop ( ), call ( ) * ( ) ( ) for all + the left nucleus of ( ). The middle and right nuclei and are obtained by shifting to the right. Each of them forms a subloop, but this subloop does not have to be normal. This is also true for the nucleus ( )

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There are different classes of quasigroups: totally symmetric, distributive, abelian etc… and through them different geometric and combinatorial systems can be defined and studied. The classes of entropic and left-distributive quasigroups were also studied by some Japanese mathematicians, including Takasaki in Harbin, as early as the late 1930s and early 1940s.

Coming back to Europe in the early 1930s, Fisher and Yates (1934) in England showed that every 6x6 Latin square belongs to a set of six (so-called) “adjugates”, which we now know as “conjugate” or “inverse” quasigroup operations, or “Parastrophs”. The concept of Parastrophs has a geometric motivation and in general was presented by A. Sade in France in the 1950s. Sade (1959) introduced the concept of “parastrophes” and it has been noted that every quasigroup ( ) belongs to a set of six quasigroups, called adjugates by Fisher and Yates (1934) , conjugates by Stein (1957 and 1956) and Belousov (1983) and parastrophes by Sade (1959).They have been studied by Artzy (1963) ,Charles Linder and Dwight Steedley (1975). Other contribution to the field include Pflugfelder (1990), Chein, Pflugfelder and Smith (1990) and Dene and Keedwell (1974). The most recent studies of the parastrophs of quasigroups (loops) are by Sokhatskii (1995a) and Sokhatskii (1995b), Duplak (2000) and, Shchukin and Gushan (2004), and Jaiyeola (2008). For a quasigroup ( ), its parastrophs could be denoted by ( ) * +, hence one can take ( ) ( ). A quasigroup which is equivalent to all its parastrophs is called a totally symmetric quasigroup which was introduced by Bruck (1944) while its loop counterpart is called a Steiner loop. Subsequently, in the early 1960s, Rafael Artzy introduced isostrophes as products of parastrophs and isotopes. Autostrophisms and autostrophy group was studied by Belousov (1971) and Pflugfelder (1990).

Remark 2.1.1

This section is pointing out that some authors use functional notation for operation on a set : instead of writing one writes ( ) . In this case the quasigroup ( ) is denoted by ( ). For operation ( ) and ( ) one use symbols and -1 , i.e., if ( ) then ( ) and -1

( ) . One can determine three other algebraic structure associated with the operation denoted by -1_{( ), (}-1 _{) and ( ( )) .}

If ( ) then -1_{( )( ) , ( ) ( ) , and ( ( )) ( ) .}

, -1 _{, ,} -1_{( ), ( ) and ( ( )) as above defined are quasigroups they are}

called conjugates or adjugates or parastrophs.

15 Definitions 2.1.3

Let ( ) be three maps define from a quasigroup ( ) into a quasigroup ( ). If for all , we have ( ) ( ) ( ) then ( ) is called an homotopism. An homotopism of bijective maps is an isotopism. The concept of isotopism is a generalization of that of isomorphism.

An isotopism on ( ) is called an autotopism. Let denote the identity map on . An autotopism ( ) is called a principal isotopism. Principal isotopy (just as isotopy) is an equivalence relation on any non-empty set of quasigroups. The importance of principal isotopy lies in the fact that up to isomorphism the principal isotopes of a groupoid ( ) account for all of the isotopes of ( ). Specifically, one has the following

Theorem 2.1.2 (Plugfelder, 1990)

If ( ) and ( ) are isotopic groupoids, then ( ) is isomorphic to some principal

isotope of ( ).

Proof.

Since ( ) and ( ) are isotopic groupoids, there exist bijections from to so that ( ) is an isotopism of ( ) onto ( ). To prove this theorem it suffices to produce a closed binary operation ( ) for , permutations and on , and a bijection of onto so that (i) ( ) is a principal isotopism of ( ) onto ( ) and (ii) is an isomorphism of ( ) onto ( ). In other words, one seeks a ( ) through which ( ) can be factored in accordance with the following diagram 2.

( ) ( ) ( ) ( ) ( )

( )

Thus, one makes ( ) ( ) and so the following selections are evident: , and for all . One can verify that this data has the desired property.  Theorem 2.1.3 (Keedwell and Shcherbacov, 2005)

Let ( ) ( ) be isotopisms from ( ) ( ) then ( ) ( ) are isotopisms from ( ) ( ) and ( ) .

16

The first statement is obvious from the definitions. For the second statement, we have: ( ) ( ) ( )  Definition 2.1.4

A collection of permutations , ( - , - and are permutations of the set G, is called an autostrophism of a quasigroup ( ) if and only if

( ) or in our alternative notation if and only if

Thus ( ) is an isotopism from a quasigroup ( ) into its parastrophs ( ). Often an autostrophism , - is called a -autostrophism or an autostrophism .

Theorem 2.1.4 (Keedwell and Shcherbacov, 2004)

The set of all autostrophisms ( ) of a quasigroup ( ) form a group with respect to the operation: , - , - , -.

Proof:

Let , - and , - where and ( ) ( ) be autostrophisms of a quasigroup ( ) then:

( ) (2.1.3) And

( ) (2.1.4) For all . Let then . From equation (2.1.3) we have ( ) (2.1.5)

Let then from equation (2.1.5) we obtained:

( ) ( ) . By virtue of equation

(2.1.4) this condition is valid for all and is the condition that , ( )- is an autostrophism of ( ) that is , - is an

17

evidently , - is an identity element. We also observe that [ _{( ) ] , -}

[ (, - ) ] , - , - and , - [ _{( ) ] [ ( )}

] , - so the autostrophism , - has a two-sided inverse: [ _{( ) ]. Finally,}

the operation ( ) is associative since: (, - , -) , - , - , - [ ( ) ] [ ] and , - (, - , -) , - [ ] [ ( ) ] [ ]. 

Duplak (2000) gave the generalization and the simplification of the methods used in (Belousov, 1983) where we have: 2.2 Parastrophs of quasigroups

Definition 2.2.1 (Duplak, 2000)

Let ( ) be a fixed quasigroup, * + and ∑( ) * +, where . Further, let ( ) ( ) ( ) ( ) ( )

_{Then the relation given by Table 3 holds. This table is read as follow :}

18

From this table 3, it follows that ( ) _{( ) for all * + and for}

. If ( ) is a quasigroup, then the mappings ( _{) are called}

translations of ( ). Every operation in ∑( ) is called a parastroph of ( ). If a quasigroup ( ) satisfies a given identity, for example

( ) (2.2.1) then in general each of its parastrophs will satisfy different conjugate identities. Thus, for example, (1) is equivalent to if we denote and (i.e. ), then

(( ) ( )) . (2.2.2) Hence, ( ) satisfies (1a) if and only if ( ) satisfies (2.2.2). If (2.2.2) is written with elements of ( ), then we obtain

( ) . (2.2.3) Thus (2.2.3) is a conjugate identity to (2.2.1). Furthermore, from (2.2.3) we have:

, i.e. . Whence ( ) and if we denote

19

Table 4 Table 5

The identities (2.2.1) and (2.2.4) may be written as , and respectively with respect to table 3 and table 4.

It can be observed that, Duplak (2000) limited his work to quasigroups. A “lowest” representation of classes of a parastrophic equivalence in quasigroups satisfying identities of the type ( ( )) , where is a parastroph of for all and are terms in ( ) and its parastrophs that not contain variable . The originality of these representations is that there are listed for by a personal computer. In other words, Duplak (2000) gave a generalization and a simplification of the methods used in Belousov (1983). Keedwell and Shcherbacov (2004) focused on quasigroups which have an inverse property. They show that each such quasigroup satisfies a generalized parastrophic identity and that, when investigating properties related to the nuclei, quasigroups which possess any type of inverse property can all be treated in the same way. By means of their approach using autostrophies, they obtained results concerning isomorphisms between or equality of these nuclei. Also, they find conditions for a groupoid which satisfy a generalized parastrophic identity to be a quasigroup. Some of these results generalize their results in Keedwell and Shcherbacov (2003, 2004). Meanwhile Jaiyela (2008) focused on group and he studied the relationship between the parastroph of the holomorph of a quasigroup and the holomorphs of the parastroph of the same quasigroup . He proved that ( ) is an associative quasigroup if and only if any one of four particular parastroph of ( )

20

obeys a Khalil condition or Evans‟ generalized associative law or Belousov‟s balanced identity or Falconer‟s generalized group identity. It was shown that isotopy-isomorphy is a necessary and sufficient condition for any two distinct quasigroups ( ) and ( ) * + to be parastrophic invariant relative to the associative law. 2.3 Derivatives

Very little or few materials were found on Derivatives. Pflugfelder (1990) briefly gave the notations and some properties. It was shown by Pflugfelder that the left and right derivatives of any quasigroups are also quasigroups. Derivatives on Central loops, left Central loops, right Central loops and their isotopes were first studied by Jaiyeola and Adeniran (2006) and it was shown that isotopes exists under the form ( ) and ( ). They also proved that the Derivatives of commutative Central loop are Central loops. It was shown that Central loops are isotopic to some finite indecomposable groups of the classes and that the center of such central loops have a rank of 1,2 or 3. Mengue Mengue and Ajala (2010) proved that if a quasigroup is associative then the left and right derivatives are isomorphic to the quasigroup.

21

of the nucleus of any finite extra loops is even and | | , so that | | The five nonassociative Moufang loops of order 16 are all extra loops by Chein (1974) and Goodaire, Sean and Maitreyi (1999). Among these five is the Cayley loop 1845, which is the oldest known example of a nonassociative loop.

The Cayley loop is usually described by starting with the octonion ring ( ), and restricting the multiplication to * +,where the are the standard basis vectors. Restricting to * + or the does not yields an extra loop (it is Moufang ,but not CC). In fact there are no nonassociative connected smooth extra loops. There are also no nonassociative compact connected extra loops, since is Boolean, and hence totally disconnected. Furthermore, ( ) is a group whenever | | . Loop extensions are used to construct an infinite associative Extra loop with trivial center and a non-associative Extra loop ( ) of order 512 such that ( ) is non-non-associative. By Kinyon and Kunen (2008), there are exactly 16 non-associative Extra loops of order for each odd prime .

2.4 Basics Facts

We collected some facts from the literature. In particular, we point out that an Extra loop yields four Boolean groups which elucidate the loop structure. One is the quotient by the nucleus:

For every quasigroup, we may define the left nucleus ( ), the middle nucleus ( ), the right nucleus ( ), and the center ( ):

Definition 2.4.1

a) For any quasigroup ( ), and : ( ) if and only if ( ) ( ) . ( ) if and only if ( ) ( ) ( ) if and only if ( ) ( ) ( ) if and only if . The nucleus of ( ) is ( ) ( ) ( ) ( ). ( ) ( ) ( ).

22

Bol loop has autotopisms of the form ( ( ) ( ) ( ) ( )). [ Then the equality ( ( ) ( ) ( ) ( )) ( ( ) _{( ) ) )( ( ) ( )) shows that the right}

Bol loop is obtainable by nuclear amalgamation]. Keedwell (2009) shows that, if ( ) has an inverse property, then this implies the existence of further autotopisms.

It is easy to verify the following equivalents, in terms of autotopy. Let ( ) ( ) denote the set of all autotopisms of ( ). The set of all autotopisms of ( ) is a subgroup of ( ( )) where ( ) denote the group of all permutations of the set , and we let denote the identity element of ( ).

Lemma 2.4.1

Let ( ) be a loop and put ( ) * for all }. Then we have ( ) ( ) ( ) ( ) . Proof.

Let belongs to ( ) . Then ( ) ( ) ( ) ( ) ( ) ( ) ( ) for all , and so . The rest is

similar. 

Lemma 2.4.2

For any quasigroup (or loop with identity element 1) ( ): 1. ( ) * ( ( ) ( )) ( )+.

( ) * ( ( ) _{( ) ) ( )+.}

( ) * ( ( ) ( )) ( )+. 2. If ( ) ( ), then ( ) and ( ). Proof.

For (2), ( ) , so taking gives . Then let and to obtain , so that ( ) ( ). 

Autotopies are useful for producing automorphisms: Theorem 2.4.1

In any loop ( ), if and either ( ) ( ) or ( ) ( ) , then and is an automorphism. Proof.

23

Theorem 2.4.2 (Kinyon and Kunen, 2004) . Let ( ) be an Extra loop with nucleus ( )

a. For each . b. is a Boolean group.

c. Every finite subloop of ( ) of odd order is contained in . d. Every element of ( ) of finite odd order is contained in . Proof.

The theorem, particularly (a), is due to Fenyves (1969). Considered as a Moufang or CC-loop, an Extra loop has a normal nucleus, so (b) follows from (a) and the fact that a Moufang or CC-loop of exponent 2 is a Boolean group. (c) follows from (b) (since maps the subloop to {1}), and (d) follows from (c).  Corollary 2.4.1 Every finite Extra loop has the Lagrange property; that is, the order of every subloop divides the order of the loop. Proof

This follows from the fact that is a group, so that both and have the Lagrange property; see Bruck (1971, ChapV.2, Lemma 2.1). This corollary holds for all CC-loops ( ), because Basarab (1991) has shown that is an abelian group; see also Kinyon, Kunen and Phillips (2004) for an exposition of Basarab‟s proof, and also Drapal

(2004) for related results.  Another Boolean group is generated by the associators:

Definition 2.4.2 For each ( ), define the associator ( ) by ( ( )) ( )

(( ) ). Let ( ) be the subloop of ( ) generated by all the associators. In an Extra loop ( ), ( ) ( ) ( ), since ( ) is a group.

Furthermore, by Kinyon, Kunen and Phillips (2004), we have:

24

a. ( ) is invariant under all permutations of the set * +. b. ( ) ( ) for all and ( ). c. ( ) ( ).

d. ( ) commutes with each of .

e. ( ) ( ( )) and ( ) is a Boolean group.  Note that lemma 2.4.3 shows that the associator ( ) determines a totally symmetric mapping from ( ) into ( ). If | | then Theorem 2.4.3 will show that ( ) ( ) ( equivalently, ( ) is a group); this fail for some ( ) of order 512. For any finite nonassociative Extra loop, | ( ) ( )| The properties we have listed for associators actually characterize Extra loops:

Lemma 2.4.4 Suppose that ( ) is a loop with the following properties:

a. ( ) is flexible, that is, ( ) for all . b. Every associator is in the nucleus.

c. The square of every associator is .

d. ( ) is invariant under all permutations of * +.

e. ( ) commutes with each of . Then ( ) is an Extra loop.

Proof. , ( )- ( ) ( ) ,( ) - ( ) ( ) ,( ) -



25

respectively, ( ) * ( ) + and , ( ) * ( ) +

Also for ( ), define ( ) ( ) ( ) ( ) and . ( ) ( ) ( ) _{( )}

It is easily seen that ( ) and that is the group generated by:

* ( ) +; likewise for ( ) and . . Lemma 2.4.5

For any Extra loop ( ):

a) All permutations in and are automorphisms of ( ) b) ( ) ( ) ( ) ( ). c) ( ) ( ) ( ) ( ) d) ( ) . e) is a Boolean group. f) ( ) ( ). Proof.

(a) Is due Goodaire and Robinson (1982), and (b), (c) are from Kinyon, Kunen and Phillips (2004); these are true for all CC-loops. (d) is also from Kinyon, Kunen and Phillips (2004), and (e) is immediate from (b), (c), (d). Also, Kinyon, Kunen and Phillips (2004) showed that ( ) ( ) _{holds in all CC-loop, so (f) follows, using (c) and lemma 2.4.3. }

Besides the left and right inner mappings, we have the middle inner mapping ( ) ( ) ( ) . In any CC-loop, the group generated by the middle inner mappings

coincides with the group generated by all inner mappings (see Drapal (2004)). . Lemma 2.4.6

In any Extra loop ( ) with ( ) and ( ): a. ( ) if and only if ( ).

b. or each , ( ) ( ) ( ). c. 𝓣: ( ) is a homomorphism.

d. Each ( ) maps , so that and is a group. e. Each ( ( )) is the identity on .

26

(a) is from Drapal (2004), and holds for all CC-loops. (b) is due to Goodaire and Robinson (1982), and (c) is from Kunen (2000). Both are true in CC-loops. ( ) ( ) is due to Fook (1976), and is true in all Moufang loops. Note that by the remark preceding the lemma, to prove that is normal, it is sufficient to show that ( ) ( ) . (e) follows from (c) and

(d), since , so ( ) is the identity on A by lemma 2.4.3.. Our last Boolean group is related to the two others. In an Extra loop ( ) with ( ), set

* +

Note that this subgroup of is the kernel of the natural homomorphism ( ) ( ) ( ), and so ( ) . Lemma 2.4.7

Let ( ) be an Extra loop. Then ( ) ( ), a direct product. Hence is a

Boolean group . Proof.

Obviously ( ) and conversely if ( ) , then . By lemma 2.4.5.(f), . If , write ( ) for . Since ( ) , and so

( ). Since ( ) and ( ), the product ( ) is direct. Since ( ) ( ) is a Boolean group (an isomorphic copy of A), and so is a Boolean group by lemma 2.4.5.(e).  Theorem 2.4.3(Kinyon and Kunen, 2004)

27

CHAPTER THREE

METHODOLOGY

The research methodology used in this study is literature based. The results obtained made use of existing relevant results in the theory of quasigroups and loops. We defined the concepts, and then used their properties to obtain our results. The methodology we use for our proofs is the direct implication. That is, a standard combination of direct and indirect inference arguments. This work used previous and relevant articles published in reputable journals to identify open problems and possible ways of solving them. Kunen (1996) results

are useful to this study. Let us introduce some of the results necessary for this work. Definition 3.1(Kunen, 1996)

If is a prime and , let ( ) be the structure , with a product operation ( ) defined by: .

Lemma 3.1

( ) is a quasigroup, and is not a loop unless .  Definition 3.2(Kunen, 1996)

In a quasigroup, define the functions and by: ( ) ( ) . Lemma 3.2 (Kunen, 1996)

If ( ) is a constant, then this constant is a right identity. If ( ) is a constant, then this constant is a left identity. 

The mirror of an equation is obtained by writing it backwards. The following are Fenyves, F (1968, 1969) Extra loop axioms:

1-) ( ( )) ( ) ( );

2-) ( ) ( ) (( ) );

3-) (( ) ) ( ( )) We shall show that there are equivalent in loops. Since each of them implies that a quasigroup is a loop, they are also equivalent in quasigroups. Observe that the first axiom is mirror of the second, while the third is its own mirror.

Theorem 3.1 (Kunen, 1996)

The relation (a):( ( )) ( ) ( ), implies that a quasigroup is a loop. Proof:

28

( ). Now, we have ( ) ( ) for all .

Next, we show that ( ) is always an idempotent. To see this, apply equation (a): ( ) ( ( ) ( ( ) ) ( ) ( ( ) ( )) ( ( )) ( ( ) ( )) and cancel to get ( ) ( ) ( ).

Finally, we show that ( ) is a constant, which must then be an identity element. To see this, fix , and we show that ( ) ( ). First, fix such that ( ). Applying equation (a), we get ( ( )) ( ) ( ). ( ) Apply ( ) with yields ( ) ( ), and hence ( ). Thus, is an idempotent, so applying ( ) with yields ( ( )) , so ( ) , which implies that ( ) ( ).  Theorem 3.2

The relation (b): (( ) ) ( ( )), implies that a quasigroup is a loop. Proof:

This relation is its own mirror, so that each time we prove a result, we also have the mirror of the result. First note that

( ) ( ( ) ) , ( ( )) ( ) . To prove ( ), use equation (b) to get .( ( )) ( )/ ( ( ) ( ( ) )) and cancel. Next note that

( ) ( ) ( ), ( ) ( ) ( ). To prove ( ), apply ( ) and equation (b) to get:

.( ( ) ( )) / ( ) ( ) . ( ) ( ( ))/ ( ( ) ) ( ) and cancel. Next, we show that ( ) ( ). To see this, fix , and let ( ) and ( ), . Applying equation (b) (with and ), along with ( ) and ( ),

we get the following : ( ( )) (( ) ( )) . (( ) )/ ( ) ( )

and we cancel to get: ( ) . Thus, ( ) ( ); squaring both sides and applying ( ) and ( ) yields ( ) ( ) . Thus, ( ) ( ) ( ) (by ( )), so ( ) .

We now have ( ) ( ) ( ) by ( ), and we proceed to prove the mirrors ( ) ( ( )) , ( ( ) ) ( ) .

29

( ( ) ) ( ) .( ( ) ( )) / ( ) ( ) . ( ) ( ( ))/. (3.2) We also have the mirror equation, ( ) ( ( )) (( ( ) ) ( )) ( ). Putting these together, we have ( ) ( ( )) ( ( ) [ ( ) ( ( ))]) ( ). Now, in a quasigroup, for all there exist such that , ( ) ( ( )) -, so we have ( ( ) ) ( ). Now, using ( ) and ( ) in (3.2), we get ( ) . Then ( ) ( ); so ( ) is a constant, which is then the identity element.  Theorem 3.3 (Kunen, 1996)

If the relation (c): ( ) ( ) (( ) ) holds in a quasigroup ( ) then

that quasigroup is a loop. 

Definition 3.3

A loop ( ) with identity element is called a cross inverse property loop (C.I.P. loop), if any elements satisfy the relation

( ) (3.3) Some properties of C.I.P. loops can be derived directly from (3.3). For example

( ) (3.4) and ( ) (3.5) Definition 3.4

A loop ( ) with identity element is called a weak inverse property (W.I.P. loop) if it satisfies the identical relation

( ) . (3.6) Theorem 3.4

Every C.I.P. loop has a W.I.P. Proof.

Let ( ) be a cross inverse property loop. Then ( ) satisfies (3.3) and (3.5). Thus, we have ( ) ( ) , which gives us (3.6). 

The converse of this theorem is not true. Now let ( ) be a W.I.P. loop. Then for all ( ) and (3.7)

( ) . (3.8) Theorem 3.5(Pflugfelder, 1990)

30

(i) and (for ) are automorphisms of ( ). (ii) If ( ) is an autotopism of ( ), then ( ) and ( ) are also autotopisms of ( ). Proof.

(i) Let us write the identical relation (3.7) as ( ) (3.9) Setting ( ) and , we can write (3.9) as ( ) ( ( ) ) . Since ( ( ) ) ( ) , we get a new identical relation ( ) (3.10) From (3.9) with and ( ) , we get (( ) ) ( ) . Combining this relation with (3.10), we obtain ( ) or

( ) (3.11) From (3.11) and from the fact that ( ) , we can conclude that is indeed an automorphism of ( ).

To prove that is an automorphism, one proceeds similarly by iterating twice the identical relation (3.8).

(ii) Let ( ) be an autotopism of ( ). Then ( ) for any . Setting and , we can write (3.7) as ( ) ( ) , or

( ) . (3.12) Now let ( ) ( ) for some . Substituting this value of into (3.12), we have (( ) ) . Since ( ) , we can write

( ) (3.13) For all . This means that ( ) is an autotophism of ( ). 

Definition 3.5

31

CHAPTER FOUR

RESULTS AND DISCUSSION

4.1 Introduction

A non-empty set with single binary operation is called magma. In algebra, a quasigroup is an algebraic structure resembling a group in the sense that “division” is always possible. Quasigroups differ from groups mainly in that they need not be associative. A quasigroup with an identity element is called a loop. Our main interest is the study of one particular loop namely an Extra loop, we shall focused on its derivatives and parastrophs. Using the definition of an Extra loop, we shall show that under some conditions each of it parastrophs is also an Extra loop. Using one of its properties, we are going to prove that the left and the right derivative of an Extra loop are also Extra loops.

It may be possible to obtain new quasigroups and loops from existing quasigroups and loops. The concept of parastroph has a well-known geometrical motivation. We are concerned with the algebraic aspect of it and so this work will not discuss the geometric aspect even though we would clarify some relevant terms and notations with geometric intuition as appropriately related to this work.

Definition 4.1.1 (PFLUGFELDER, 1990)

A k-web W consists of a set of points and a subset of the power set of whose elements are called lines. In there are k subsets called pencils ( , a natural number) such that the following axioms hold:

a-) Each lines belongs to just one of the pencils of W. b-) Each points belongs to just one lines from each pencil.

c-) Any two lines from distinct pencils have exactly one point in common, whereas lines from the same pencil are disjoint.

32

( ) in our example is the -parastroph of ( ) or ( ) is said to be -parastrophic to ( ). On the other hand, if one has a quasigroup, one can construct a 3-web which belongs to this quasigroup.

The following is obvious in view of the existence of the six permutations of pencils. Theorem 4.1.1 (Sade, 1959)

There are six quasigroups parastrophic to every quasigroup.  Definition 4.1.2

a-) Let ( ) and ( ) be two quasigroups. The triple ( ) such that

( ): ( ) ( ) are bijections is said to be an isotopisms if and only if we have x y =(x y) .

Thus, H is called an isotope of G, and H and G are said to be isotopic. According to Pflugfelder, (1990), every group is a G-loop (i.e. A loop that is isomorphic to all its loop isotopes). Hence, every loop isotope of a group is a group but this is not true for quasigroup isotopes of a group, they are not necessarily associative.

b-) Let ( ) be the automorphism group of a loop (quasigroup) ( ), and the set ( ) ( ). If we define another operation “ on H such that,

( ) ( ) ( ) ( ) ( ) , then ( ) ( ) is a loop (quasigroup) according to Bruck (1944) and it is called the holomorph of ( ). It can be observed that a loop is a group if and only if its holomorph is a group. More details can be found in Robinson (1964, 1971).

Throughout the next section, if ( ) is an extra loop then the corresponding

(algebraic) parastrophs will be denoted by: ( ) ( ) ( ) ( ) ( ) ( ) each of the operations will be defined accordingly.

33

on G such that ( ) .

Similarly, ( ) is called a right inverse property quasigroup (RIPQ) if it has the right inverse property (RIP) i.e. if there exists a bijection

on G such that ( )

A quasigroup that is both a LIPQ and a RIPQ is said to have the inverse property (IP) hence it is called an inverse property quasigroup (IPQ).

The same definitions hold for a loop and such a loop is called a left inverse property loop (LIPL), right inverse property loop (RIPL) and inverse property loop (IPL) respectively. Central loop, Moufang loops and extra loops are examples of inverse property loops (IPL). There are some classes of loops which have properties which can be considered as variations of the inverse property such as the following. e-) A loop ( ) is called an automorphic inverse property loop (AIPL) if and only if it obeys the identity

( ) or ( ) for all .

f-) A loop ( ) is called a semi-automorphic inverse property loop (SAIPL) if and only if it obeys the identity

(( ) ) (( ) ) (( ) ) .( ) / We also used the well-known fact that if is a permutation such that ( ) for all then . To see this, let be an arbitrary given element of ( ). For each given element , there exist an element such that . Thus,

( ) ( ) That is, ( ) A magma ( ) in which the translation maps are bijective is called a quasigroup. From every quasigroup ( ) one can define five others quasigroups. They are:

( ) ( ) ( ) ( ) ( ) and those quasigroups are called conjugate or Parastroph of ( ).

Definition 4.1.3

Let ( ) be a quasigroup, if there exist an element such that for all ,

34

an identity element is called a loop. Definition 4.1.4 (Fenyves, 1968)

A loop ( ) is said to be an Extra loop if and only if the following identity (( ) ) ( ( )) holds for all . Theorem 4.1.2

If ( ) is an autotopism of the inverse property loop ( ), then ( ) ( ) and ( ) are also autotopism. (Where is the map on , defined by _).

Theorem 4.1.3 (Fenyves, 1968)

For a loop ( ) the following identities are equivalent: 1.) (( ) ) ( ( )).

2.) ( ) ( ) (( ) ).

3.) ( ) ( ) ( ( )) . .

Moreover, if the loop ( ) satisfies any one of these identities, then ( ) has the inverse

property.

Proof.

Suppose that for a loop ( ), 1.) Holds. For each define by (Where 1 is the identity element of ). Then from 1.), writing _{and we get}

( _{) ( ) and hence . Thus, also ( ) holds. For}

arbitrary given elements , with the help of the solutions of the equations

_{and and using 1.), we get}

( ) _{(( ) ) ( ( )) ,} that is ( ) _{; (4.1.1)} and ( ) ( ( )) (( _{) ) , that is ( ) (4.1.2)} respectively.

35

( ( ) ( ) ( ) _{( ) ) ( ( ) ( ) ( ) ( ) ) also is an autotopism of}

( ). That is for all , ( _{) (( ) ) ( ) . Taking}

we obtain ( ) ( _{) ( ( )) and replacing by , this give}

( ) ( ) ( ( )) Thus 1.) implies 2.)

Now let assume that the loop ( ) satisfies 2.). Define _{by for all}

. Then from 2.), with ( _{) ( ) ( ( )) or}

_{( ) (4.1.3)}

so ( ) has the left inverse property. From (3) by , ( ) and this implies , that is ( ) . Moreover in 2.), we write ( ) instead of and using (4.1.3), we obtain ,( ) _{- ( ) ( ) therefore ( )}

_{and so ( ) .}

Hence ( ) has the inverse property. Thus every step in the proof of 1.) ) is reversible. This means that 2.) implies 1.).

Assume again that ( ) satisfies 2.), then ( ) is an inverse property loop. Taking inverses both sides of 2.), and replacing _{by respectively we obtain the}

identity 3.). Similarly, 3.) implies 2.). This completes the proof.  Definition 4.1.5

A loop is an Extra loop if and only if it is both Conjugacy closed loop (a CC-loop) and a Moufang loop.

4.2 Parastrophs of Extra loops

Extra loops were first introduced via the above identities by Fenyves (1968, 1969), who proved the equivalences of 1.), 2.) and 3.). Goodaire and Robinson (1982, 1990) showed that definition 4.1.5 is equivalent, and this definition is often more useful in practice, since one may combine results in the literature on CC-loop and on Moufang loops to prove theorems about extra loops. Work on Extra loops and quasigroups can be found also in Kweedwell (2009), Phillips and Vojtechovsky (1991).

Throughout this section on parastrophs definition 4.1.5 and theorem 4.1.3, shall be used. The methodology shall be to study one parastroph after another.

CASE:1

Let ( ) be a parastroph of an Extra loop ( ) define by: ( ) ( ) (-1_{( )) . The}

36 Theorem 4.2.1

If ( ) is an Extra loop then ( ) is an Extra loop To prove this theorem, this section shall establish the following:

Lemma 4.2.1

a-) If ( ) is an Extra loop then (( ) ) ( ( )) for all . b-) If ( ) is an Extra loop then ( ) ( ) (( ) ) for all . c-) If ( ) is an Extra loop then ( ) ( ) ( ( )) for all

Proof.

a-) For all ,

(( ) ) (( ) ) ( ( )) ( ( )) (4.2.1) and ( ( )) ( ( )) (( ) ) (( ) ) (4.2.2) Merely looking at (4.2.1) and (4.2.2), and using our hypothesis ( The fact that ( ) is an Extra loop) this imply (4.2.1)=(4.2.2). Thus, ( ) does satisfy the identity: (( ) ) ( ( ))  b-) Let , ( ) ( ) ( ) ( ) ( ) ( ) (4.2.3) and (( ) ) (( ) ) ( ( )) ( ( )) (4.2.4) Merely looking at (4.2.3) and (4.2.4), and using the hypothesis we observed that

(4.2.3)=(4.2.4). Hence ( ) do satisfy the identity ( ) ( ) (( ) ). c-) ,

37 ( ) ( ) (4.2.5) and ( ( )) ( ( )) (( ) ) (( ) ) (4.2.6) Merely looking at (4.2.5) and (4.2.6), and using the fact that ( ) is an Extra loop we obtained that (4.2.5)=(4.2.6). Thus, ( ) satisfy the identity ( ) ( )

( ( )) . 

Theorem 4.2.1 can now be proved.

Using the fact that for all , this gives:

a-) If ( ) is an Extra loop then (( ) ) ( ( )) b-) If ( ) is an Extra loop then ( ) ( ) (( ) ) c-) If ( ) is an Extra loop then ( ) ( ) ( ( )) and with the result of Kunen (1996) that we proved as Theorems 3.1, or 3.2 or 3.3 the proof is

complete. 

Theorem 4.2.2

If ( ) is an Extra loop of a quasigroup ( ) then ( ) is an Extra loop. To prove this theorem, we need some Lemmas. Let us recall that a loop ( ) is an Extra loop if it satisfies one of the following equivalent identities

1-) (( ) ) ( ( )) 2-) ( ) ( ) (( ) ) 3-) ( ) ( ) ( ( )) For all .

To prove the theorem the following shall be established

Lemma 4.2.2