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 DECEMBER, 2011 i 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 Author’s Name -------------------------------------------- Date ----------- First Supervisor’s Name -------------------------------------------- Date ---------- Second Supervisor’s Name -------------------------------------------- Date ----------- Third Supervisor’s Name -------------------------------------------- Date ----------- First Internal Examiner -------------------------------------------- Date ----------- Second Internal Examiner -------------------------------------------- Date ----------- External Examiner Name --------------------------------------------- Date ----------- PG School Representative --------------------------------------------- Date ----------- ii DEDICATION This Thesis is dedicated to God Almighty, for His mercy, protection and provisions, for enabling me to complete this programme successfully. Also, to my late brother Martin, my wife Alima, my children and parents for making this Ph.D. Degree Programme a huge success. I wish the Lord grant them a peaceful lifetime in order to reap the fruit of their labour and their support financially, spiritually and otherwise. iii 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. I thank you all, I pray that Almighty GOD will keep on blessing you and your respective household, long life and prosperity. I also wish you ALL THE BEST. iv 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 Central loops 77 v 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 REFERENCES 105 vi ABSTRACT This Thesis investigates the nature of the parastrophs and derivatives of loops both of Bol-Moufang (Extra, Moufang, Central loops) and non Bol-Moufang (Conjugacy Closed loops) type in general. Extra loops is the case study. By using Fenyves (1968, 1969) definition of Extra loops and the results of Goodaire and Robinson (1982, 1990), this work shows that the parastrophs and derivatives of an Extra loop exist. Taking into consideration Ken Kunen (1996) results, it has been established here that two of the six parastrophs of any Extra loop are loops of Bol-Moufang type and precisely Extra loops. The remaining four could also become Extra loops provided the initial Extra loop is an involution or exponent two and has the automorphic inverse property. The converse was also found to be true and detail proofs and statements of those multiple new theorems are provided. Through those proofs, this study confirms the claim of Fevyves (1968) over the equivalency of the three Extra identities in loops. A new isotopic invariant for loops is obtained, namely the property of being Extra. However, the property of being an inverse property loop is not an isotopic invariant for loops in general. The only class of loops which has inverse property as isotopic invariant is the class of Moufang loops. In a group (an associative quasigroup, a commutative Extra loop, an associative loop) and in an Extra loop, the left and the right derivatives with respect to any element of the initial algebraic structures preserve the algebraic structure. vii CHAPTER ONE INTRODUCTION 1.1 Background to the study 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. Generally speaking, a loop is a quasigroup with identity element. The varieties of loops are multiple, such as the variety of Bol-Moufang type. Phillips and Vojtechovsi (2005) thought that an identity is said to be of Bol-Moufang type if: (i) the only operation appearing in is the binary operation, (ii) the number of distinct variables appearing in (and thus in ) is 3, (iii) the number of variables appearing in (and thus in ) is 4, (iv) the order in which the variables appear in coincides with the order in which they appear in . Here is an example (( ) ) ( ( )). Loop varieties defined by one identity of Bol-Moufang type include Extra loops, Bol loops, Moufang loops, Central loops, etc…just to name a few. Eight of them were known, but as research is on-going, the number has increased to about sixteen (Jaiyeola, 2009). On the other hand, Parastrophs and 1 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.