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Linear Aspects of Equational Triality in Quasigroups Iowa State University Capstones, Theses and Graduate Theses and Dissertations Dissertations 2020 Linear aspects of equational triality in quasigroups Alex William Nowak Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/etd Recommended Citation Nowak, Alex William, "Linear aspects of equational triality in quasigroups" (2020). Graduate Theses and Dissertations. 17959. https://lib.dr.iastate.edu/etd/17959 This Thesis is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Graduate Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. Linear aspects of equational triality in quasigroups by Alex William Nowak A dissertation submitted to the graduate faculty in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Major: Mathematics Program of Study Committee: Jonathan D. H. Smith, Major Professor Tathagata Basak Jonas T. Hartwig Jack H. Lutz Timothy H. McNicholl The student author, whose presentation of the scholarship herein was approved by the program of study committee, is solely responsible for the content of this dissertation. The Graduate College will ensure this dissertation is globally accessible and will not permit alterations after a degree is conferred. Iowa State University Ames, Iowa 2020 Copyright c Alex William Nowak, 2020. All rights reserved. ii DEDICATION To my parents, who always gave me the courage to sit down and think. iii TABLE OF CONTENTS Page LIST OF TABLES . .v ACKNOWLEDGMENTS . vi ABSTRACT . vii CHAPTER 1. INTRODUCTION . .1 1.0.1 Some historical remarks motivating quantum quasigroups . .1 1.1 Overview . .3 1.2 Basic quasigroup theory . .4 1.2.1 Multiplication groups . .6 1.2.2 Quasigroup triality . .6 1.3 Notions from category theory and universal algebra . 10 1.3.1 Categories of quasigroups . 10 1.3.2 Coproducts, free products, and algebraic presentations . 11 1.3.3 Group modules, abelian group objects in slice categories . 13 1.4 Free extensions of quasigroups . 16 1.4.1 A confluent rewriting system for the normal form . 17 CHAPTER 2. QUASIGROUP MODULE THEORY . 20 2.1 The universal multiplication group, universal stabilizer and the Fundamental Theorem 20 2.2 Combinatorial differentiation and the Relativized Fundamental Theorem . 23 2.2.1 Modules over semisymmetric quasigroups . 24 2.2.2 The Relativized Fundamental Theorem . 25 CHAPTER 3. MODULES OVER H-SYMMETRIC QUASIGROUPS AND MODULES OVER H-SYMMETRIC, IDEMPOTENT QUASIGROUPS . 27 3.1 Universal multiplication groups . 29 3.1.1 Useful conventions and lemmas . 29 3.1.2 Classification of universal multiplication groups . 31 3.2 Universal stabilizers . 34 3.2.1 Left symmetric quasigroups . 38 3.2.2 Totally symmetric quasigroups . 39 3.3 Classifying the rings of representation . 40 3.3.1 H-symmetric quasigroups . 40 3.3.2 Idempotent, H-symmetric quasigroups . 46 iv CHAPTER 4. AFFINE MENDELSOHN TRIPLE SYSTEMS AND THE EISENSTEIN IN- TEGERS . 55 4.1 Abstract . 55 4.2 Introduction . 55 4.2.1 The isomorphism problem for affine MTS . 55 4.2.2 Affine MTS, entropicity, and distributivity, . 58 4.3 Background on the Eisenstein integers . 59 4.3.1 Modules over the Eisenstein integers . 59 4.4 A direct product decomposition for affine MTS . 61 4.5 Affine non-ramified MTS . 64 4.5.1 Direct product decomposition for non-ramified MTS . 64 4.5.2 Enumeration . 67 4.6 Affine ramified MTS . 68 4.6.1 Even powers of 1 + ζ ................................ 69 4.6.2 Odd powers of 1 + ζ: mixed congruence . 69 4.7 Combinatorial properties of affine MTS . 74 4.7.1 Purity . 74 4.7.2 The converse of affine MTS . 76 CHAPTER 5. TRIALITY IN THE THEORY OF QUANTUM QUASIGROUPS . 81 5.1 Quantum quasigroups and their conjugates . 81 5.1.1 Quantum conjugates . 82 5.2 Triality of quantum conjugates for H-bialgebras . 84 5.3 Conjugates of the quantum couple . 88 CHAPTER 6. CONCLUSIONS AND FUTURE WORK . 93 6.1 Rings of representation . 93 6.1.1 Future work . 93 6.2 Affine Mendelsohn triple systems . 94 6.2.1 Future work . 94 6.3 Conjugates of quantum quasigroups . 94 6.3.1 Future work . 94 BIBLIOGRAPHY . 96 v LIST OF TABLES Page Table 2.1 Partial derivatives of quasigroup words . 25 Table 3.1 Rings associated with feg, regarded as an H-symmetric quasigroup . 54 Table 3.2 Rings associated with feg, regarded as an idempotent, H-symmetric quasi- group . 54 vi ACKNOWLEDGMENTS Thank you to my advisor, Jonathan Smith, for creating the machinery that made much of this work possible. I encountered Jonathan's unconventional, uncompromising, penetrating vision of the algebraic landscape in my first semester at ISU. Since then, he has brought out the best in me, and allowed me to share in that vision. Many thanks are due to the rest of my committee. Tim's Logic Seminar has brought forth new questions and collaborators whose impact will stretch well beyond this work. Jack has always offered thoughtful questions and insights during my talks in this seminar. His course in finite-state randomness taught me new ways to ask \what is knowable?" Jonas is always incredibly generous with his time and intellect. Several discussions in his office shaped the way I thought about the presentations of rings in Chapter3. In addition to being a generally good influence on my presentation style, Tathagata keyed me in on Hensel lifting as a potential way out of the ramified case for affine MTS. I would like to acknowledge Bokhee Im for her contributions to the project that led to a triality theory for quantum quasigroups, and for treating my whole house to a very mathematical dinner at Proof. Chapter4 was conceived of and drafted at 1660 East McKinley Avenue, aided by the love of my adoptive Iowa family: Jenetta, Chelsea, Patrick and Nathan. Thank you for showing such concern for \the six cases" that evaded me throughout the Spring of 2019. I doubt that I would have survived graduate school without my partner Esperanza. She uprooted life in the Greatest City on Earth to join me in Ames, and for that, I will always be grateful. I love you Espi. Lastly, I'd like to thank the Au-Gilbert family for hosting me in quarantine during the final stages of this project. vii ABSTRACT We approach quasigroup triality from two vantages: Q-modules and quantum quasigroups. The former requires considerable exposition and a review of the Fundamental Theorems of Quasigroup Modules due to Smith. There are six varieties associated with triality symmetry. We give presen- tations for rings whose module theories coincide with the Beck modules for quasigroups in each of these six varieties, plus the idempotent extensions of each of these varieties. In three of the idempotent cases, we are able to give abstract structure theorems which present these rings as free products of PIDs and free group rings. We define a Mendelsohn triple system (MTS) of order coprime with 3, and having multiplication affine over an abelian group, to be affine non-ramified. We classify, up to isomorphism, all affine non-ramified MTS and enumerate isomorphism classes (extending the work of Donovan, Griggs, McCourt, Oprˇsal, and Stanovsk´y). As a consequence, all entropic MTS and distributive MTS of order coprime with 3 are classified. The classification is accomplished via the representation theory 2 of the Eisenstein integers, Z[ζ] = Z[X]=(X − X + 1). Partial results on the isomorphism problem for affine MTS with order divisible by 3 are given, and a complete classification is conjectured. We also prove that for any affine MTS, the qualities of being non-ramified, pure, and self-orthogonal are equivalent. We conclude with an exploration of conjugates and triality in quantum quasigroups, which offer a nonassociative generalization of Hopf algebras. We show that if Q is a finite quasigroup and G ≤ Aut(Q), then the Hopf smash product KG#K(Q) exhibits a quantum quasigroup structure which, despite being noncoassociative and noncocommutative, has a set of conjugates on which S3 acts. 1 CHAPTER 1. INTRODUCTION Quasigroups provide an algebraic account of Latin squares and offer a nonassociative generaliza- tion of group theory. In addition to being objects of pure combinatorial interest, Latin squares have found applications in in molecular biology [9], coding theory [55], and experimental design [54], to name a few areas. Our work is concerned with a triality (symmetry of threes) that is inherent to the theory of quasigroups. To be more precise, consider a set under binary multiplication (Q; ·). For · to be a quasigroup operation means that its multiplication table 2 T = f(x1; x2; x1 · x2) j (x1; x2) 2 Q g (1.0.1) has the Latin square property: for all (x1; x2; x3); (y1; y2; y3) 2 T , jf1 ≤ i ≤ 3 j xi = yigj 6= 2. Quasigroup triality refers to the fact that the Latin square property is invariant under S3-action, g which is to say that T is Latin if and only if T = f(x1g; x2g; x3g) j (x1; x2; x3) 2 T g is Latin for all g g 2 S3. The multiset fT j g 2 S3g is the set of conjugates of T . For a subgroup H ≤ S3, we say gh g that a quasigroup is H-symmetric when T = T for all g 2 S3, h 2 H. In this Latin square formulation, quasigroup triality is rather tidy, but if we want to obtain models of these H-symmetry classes it is, in some sense, necessary [19] to move to a universal algebraic and then further to a linear algebraic formulation of triality and H-symmetry.
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