Linear Aspects of Equational Triality in Quasigroups

Home , Bialgebra, Euclidean domain, Idempotence, Principal ideal domain, Quasigroup

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

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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

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

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 {e}, regarded as an H-symmetric quasigroup . . . . . 54 Table 3.2 Rings associated with {e}, regarded as an idempotent, H-symmetric quasigroup ...... 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 presentations 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 Stanovský). 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 generalization 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 = {(x1, x2, x1 · x2) | (x1, x2) ∈ Q } (1.0.1)

has the Latin square property: for all (x1, x2, x3), (y1, y2, y3) ∈ T , |{1 ≤ i ≤ 3 | xi = yi}| 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 = {(x1g, x2g, x3g) | (x1, x2, x3) ∈ T } is Latin for all

g g ∈ S3. The multiset {T | g ∈ S3} 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 ∈ S3, h ∈ 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. This is the essential motivation for the ﬁrst three chapters of this dissertation.

In Chapter4, we see the payoﬀ of the module-theoretic perspective in quasigroup theory, as we obtain classiﬁcation results in a variety that apply to all objects of order coprime with 3, whereas pre-module work had been limited to prime-square order [15].

1.0.1 Some historical remarks motivating quantum quasigroups

Chapter5 examines a generalization of the triality theory to symmetric monoidal categories.

Some of this is joint work with Im and Smith [25]. The transportation of groups into categories of 2 vector spaces via Hopf algebras has produced an explosion of results, unifying ostensibly unrelated areas such as statistical mechanics and knot theory (cf. [16]). The study of quasigroups and loops

(quasigroups with identity) in categories of vector spaces, can be traced back to Malcev [34]. This work is the foundation of a “nonassociative Lie theory,” which may be understood as the study of locally Euclidean loops through 1.) algebras deﬁned on the tangent space at the identity of a topological loop (cf. Lie algebras), and 2.) nonassociative and noncoassociative generalizations of cocommutative Hopf algebras [39]. Perhaps the best-known object in this ﬁeld is the smooth loop structure on S7 furnished by norm-1 octonions.

Remark 1.0.1. (a) The following discussion could be carried out in any symmetric monoidal

category (sets under Cartesian product, for example), but we will stick to (K-Mod, ⊗,K),

the category of modules over a commutative unital ring K, under the tensor product.

(b) For K-modules A and B, we let τA,B : A ⊗ B → B ⊗ A; x ⊗ y 7→ y ⊗ x, and if A = B, we

abbreviate τA,A to τA. In fact, we will typically have enough context to simply write τ.

We recall the basic terminology for Hopf algebras. A Hopf algebra (A, ∇, η, ∆, ε, S) has an associative multiplication ∇ : A ⊗ A → A; x ⊗ y 7→ xy equipped with a unit η : K → A. Dually, there is a comultiplication ∆ : A → A ⊗ A; x 7→ P xL ⊗ xR endowed with a counit ε : A → K satisfying

X X xLL ⊗ xLR ⊗ xR = xL ⊗ xRL ⊗ xRR, coassociativity X X xLεxR = xLxRε = x. counitality

Moreover, ∆ has to be a homomorphism from the tensor square algebra

(A ⊗ A, (1A ⊗ τ ⊗ 1A)(∇ ⊗ ∇), η ⊗ η) to (A, ∇, η), and ε has to be an algebra homomorphism from (A, ∇, η) into the ring K. The ﬁfth structure map is the antipode, S : A → A, of the Hopf algebra.

This is a linearized formulation of a group inverse in that, for all x ∈ A,

X X xLSxR = xLxRS = xεη. 3

The basic philosophy of the nonassociative Lie theory is that if you examine topological loops, the universal enveloping algebras should, as in the case of groups, have algebra and coalgebra structures linked by an antipode, but one has to allow for nonassociativity in the algebra structure, and noncoassociativity in the dual of the enveloping algebra.

P´erez-Izquierdo’s H-bialgebras1 along with Klim and Majid’s Hopf (co)quasigroups are the standard nonassociative generalizations of Hopf algebras [42, 30]. These constructions lack self- duality in the following sense: if A is the vector space of an H-bialgebra or Hopf (co)quasigroup, then the dual space A∗, endowed with transposes of the original structure maps on A, does not necessarily constitute an H-bialgebra or Hopf (co)quasigroup. This at ﬁrst appears to be a considerable shortcoming, considering the fact that the rich structure of Hopf algebras and quantum groups often comes from axiomatic self-duality. However, in 2015, Smith provided a self-dual framework for H- bialgebras and Hopf (co)quasigroups: a quantum quasigroup is a (nonassociative, noncoassociative, nonunital, and noncounital) bialgebra (A, ∇, ∆) such that the left composite

∆⊗1A 1A⊗∇ A ⊗ A / A ⊗ A ⊗ A / A ⊗ A, (1.0.2) and its dual right composite

1A⊗∆ ∇⊗1A A ⊗ A / A ⊗ A ⊗ A / A ⊗ A (1.0.3) are invertible [52]. Furthermore, (A, ∇, η, ∆, ε) is a quantum loop if it is a (nonassociative and noncoassociative) biunital bialgebra with invertible composite maps.

1.1 Overview

Because the module theory of quasigroups involves constructions from category theory, combinatorial group theory, linear algebra, and universal algebra, a great deal of background is required.

After we introduce quasigroup theory and its inherent three-fold symmetry, Sections 1.3 and 1.4 provide the requisite terminology to discuss the module theory. Chapter2 is this discussion. We

1The nominal overlap with H-symmetry is coincdental. P´erez-Izquierdo’sterminology is a reference to the hyperalgebras of Sabinin and Miheev (cf. introduction of [42]). 4 review Smith’s Fundamental Theorems, which relate the abstract concept of Beck modules over quasigroups [1] to the module theory of certain rings. The construction of these rings for the six

H-symmetry varieties and their idempotent extensions is the goal of Chapter3.

Chapter4 is the manuscript of the paper “Affine Mendelsohn triple systems and the Eisenstein integers,” currently under review at The Journal of Combinatorial Designs. It is a study of the isomorphism problem for finite, idempotent, semisymmetric (h(123)i-symmetric) quasigroups. In the language of Chapters2 and3, this may be regarded as the classification problem for the abelian groups of MTS, the category of semisymmetric, idempotent quasigroups (technically, abelian group objects of MTS/{e}, but these are naturally isomorphic categories). Section 4.5 solves the isomorphism problem for Mendelsohn quasigroups of order coprime with 3, while Section 4.6 conjectures a complete solution. The chapter concludes (Section 4.7) with an examination of the consequences of our structure theorems on the design-theoretic properties of affine MTS.

Chapter5 expands on some of the work in [25]. In this paper, we deﬁned a notion of conjugates from quantum quasigroups. As an extension of the results from our paper [25], we provide two examples —P´erez-Izquierdo’s H-bialgebras and Smith’s quantum couple— of quantum quasigroup constructions which exhibit triality in their set of conjugates.

1.2 Basic quasigroup theory

Deﬁnition 1.2.1. Let Q be a set, and consider three binary operations on Q: · denoting multiplication, / right division, and \ left division. We say that (Q, ·, /, \) is a(n) (equational) quasigroup if and only if the following identities are satisﬁed:

(IL) y\(y · x) = x, (IR) x = (x · y)/y,

(SL) y · (y\x) = x, (SR) x = (x/y) · y.

Remark 1.2.2. (a) We convey multiplication of quasigroup elements by concatenation, and in

some instances, use both · and concatenation. Concatenation takes precedence over ·. For

example, xy · z ≡ (xy)z ≡ (x · y) · z. 5

(b) While quasigroup divisions allow us to deﬁne a universal algebraic variety of quasigroups,

and are a necessary component of the triality theory, knowledge of the multiplication speciﬁes

the entire structure. Indeed, the combinatorial quasigroup deﬁnition states that (Q, ·) is a

quasigroup if and only if the multiplication table associated with ·,

T (Q) = {(x1, x2, x1 · x2) | x1, x2 ∈ Q)},

has the Latin square property. That is, for all (x1, x2, x3), (y1, y2, y3) ∈ T (Q), we have

|{1 ≤ i ≤ 3 | xi = yi}|= 6 2.

Example 1.2.3. Perhaps the most straightforward example of a nonassociative quasigroup is oﬀered by subtraction on the integers (or any abelian group for that matter). Precisely, (Z, −, +, −) is a quasigroup, as

(IL) y − (y − x) = x, (IR) x = (x − y) + y,

(SL) y − (y − x) = x, (SR) x = (x + y) − y, for all x, y ∈ Z. The coincidence of multiplication and left division is an example of left symmetry in the triality theory of Section 1.2.2.

A subset P of a quasigroup (Q, ·, /, \) is said to be a subquasigroup of Q if it is closed under the three binary operations. A function f : Q → Q0 between quasigroups is a quasigroup homomorphism if it respects the three quasigroup operations. In fact, when checking if a function f : Q → Q0 between quasigroups is a homomorphism, it suffices to verify that it respects multiplication, or that (xy)f = xf · yf, where juxtapostion denotes the multiplication of Q and · that of Q0. It is important to note that this simplification of the definition only applies when it is known that Q and Q0 are quasigroups. There are examples of combinatorial quasigroup homomorphisms whose images are not themselves combinatorial quasigroups. Naturally, an isomorphism of quasigroups is a bijective homomorphism. Now we may define a category of quasigroups Q. One may also view

Q as a variety (v.i. Subsection 1.3.1), with axioms (IL), (SL), (IR), (SR) serving as an equational basis. 6

1.2.1 Multiplication groups

Alternatively, we can deﬁne a quasigroup structure on a set Q to be a binary multiplication whose left and right translation maps

L(y): Q → Q; x 7→ yx;

R(y): Q → Q; x 7→ xy are invertible for every y ∈ Q. We are now able to justify our labelling of the axioms (IL)-(SR).

The (IL) and (IR)-identities ensure the maps L(y) and R(y) are injective, while (SL) and (SR) tell us that L(y) and R(y) are surjective. In fact, the axioms show xL(y)−1 = y\x, and xR(y)−1 = x/y for all x, y ∈ Q.

For a given subset S ⊆ Q, let L(S) = {L(s) | s ∈ S}, and R(S) = {R(s) | s ∈ S}. The subgroup of the symmetric group SQ generated by L(Q) ∪ R(Q) is called the combinatorial multiplication group of Q, and is denoted by Mlt(Q). For any subquasigroup P of Q, we deﬁne the relative multiplication group of P in Q to be the subgroup of Mlt(Q) generated by L(P ) ∪ R(P ).

1.2.2 Quasigroup triality

Given a group (G, ·, e), one may deﬁne its opposite

(G, ◦, e, ), where ◦ : G × G → G;(x, y) 7→ y · x. Now, this is the only so-called conjugate guaranteed by group theory. In the case of abelian groups, these conjugates coincide. However, the theory of quasigroups extends this duality to a theory of triality.

Consider a combinatorial quasigroup (Q, ·). Just as in the group case, there is an opposite quasigroup (Q, ◦), where

x ◦ y = y · x for each x, y ∈ Q. But recall that the combinatorial structure (Q, ·) gives rise to an equational one

(Q, ·, /, \). The algebras (Q, /) and (Q, \) are themselves combinatorial quasigroups, as suggested 7 by Example 1.2.3, and each of these pairs has associated with it its opposite quasigroup counter- part. We denote the opposites of right and left division by // and \\, respectively. Thus, given a combinatorial quasigroup (Q, ·) there are six potentially distinct conjugate quasigroups, giving us a theory of triality. As fully ﬂedged equational quasigroups, the six conjugates appear schematically in (1.4.1) below:

(Q, ·, /, \) o / (Q, ◦, \\, //) (1.2.1) oo 3; ck OOO oooo OOOO oooo OOOO s{ ooo OO #+ (Q, \, //, ·) (Q, \\, ◦,/) gOOO oo7 OOO ooo OOO ooo OO' wooo (Q, //, \, ◦) ks +3 (Q, /, ·, \\) .

The arrows are taken from the Cayley diagram of S3

(1) o / (1 2) (1.2.2) v 7? _g HHH vvvv HHHH vvvv HHHH vvvv HHHH w vvv H ' (2 3) (1 3 2) Hc H v; HH vv HH vv HH vv H# {vv (1 2 3) ks +3 (1 3) , where single arrows denote the action of right multiplication by (12) and double arrows correspond to right multiplication by (23). The arrangement of cycles is not arbitrary. In the expression

x1 · x2 = x3, (1.2.3)

we may permute the indices according to S3-action. In fact, suppose (Q, ∗m, ∗r, ∗l) is conjugate

(Q, ·, /, \), and appears at the node corresponding to g ∈ S3 in (1.2.2). Then

x1g ∗m x2g = x3g.

For example, if σ = (12), then (1.2.3) holds if and only if x3 = x2 ◦ x1 = x1σ ◦ x2σ = x3σ. Within an abstract scheme of quasigroup conjugates, we will refer to (Q, ·, /, \) as the base quasigroup. The quasigroup (Q, \, //, ·) is the left conjugate and (Q, /, ·, \\) is the right conjugate. 8

Rounding out the list, we have the opposite quasigroup (Q, ◦, \\, //), the opposite left conjugate

(Q, \\, ◦,/), and the opposite right conjugate (Q, //, \, ◦).

1.2.2.1 H-symmetry classes

Just as in group theory, whenever a quasigroup coincides with its opposite, we say the quasigroup is commutative. Recall Example 1.2.3, in which multiplication and left division coincided; we say such quasigroups are left symmetric. When multiplication in the base quasigroup coincides with right division, it is a right symmetric quasigroup.

It is now important we emphasize the fact that, in quasigroups, multiplication determines division, and vice versa. When a quasigroup is commutative, we have · = ◦, but this also means that / = \\, and that \ = //; this is all to say, the base quasigroup coincides with the opposite, the left conjugate coincides with the opposite right conjugate, and the right conjugate coincides with the opposite left conjugate. Notice that all conjugates connected by a single arrow in (1.4.1) were identiﬁed. Put another way, two conjugates overlapped if and only if the corresponding nodes in

(1.2.2) belong to the same right coset in the quotient h(12)i\S3. In general, the action of S3 on (1.2.3) demands that identiﬁcation of conjugates respect the orbit-stabilizer theorem.

Deﬁnition 1.2.4. Let (Q, ·, /, \) be a quasigroup, and deﬁne a ternary relation

3 T = {(x1, x2, x1 · x2) | x1, x2 ∈ Q} ⊆ Q .

g g For each g ∈ S3, let T = {(x1g, x2g, x3g) | (x1, x2, x3) ∈ T }, and ﬁx C = {T | g ∈ S3}, and use

π : S3 → SC to denote the permutation representation. Given H ≤ S3, we say that (Q, ·, /, \) is H-symmetric if and only if H ≤ Ker π.

If multiplication in the base quasigroup coincides with the opposite of right division, then we say the quasigroup is semisymmetric. Now, there can be at most two distinct conjugates, for

· = //,/ = \, and \ = ◦ force / = \ = ◦, and · = // = \\. These two conjugates correspond to the right cosets of h(123)i\S3, verifying the synonymy of h(123)i-symmetry and semisymmetry. 9

If the base quasigroup coincides with all of its conjugates, then it is totally symmetric. Com- mutativity and semisymmetry are enough to guarantee totally symmetry, as are left symmetry and

0 0 0 right symmetry. In general, if Q is H-symmetric and H -symmetric, and HH = H H = S3, then Q is totally symmetric.

1.2.2.2 Examples from the various symmetry classes

Example 1.2.5 (Trivial symmetry). Any nonabelian group oﬀers us an example of a quasigroup with six distinct conjugates. Indeed, ﬁx a nonabelian group (G, ·). Multiplication in the left conjugate is given by x\y = x−1 · y, and the right conjugate multiplication corresponds to x/y = x · y−1. Since G is nonabelian, there is at least one element x ∈ G for which x 6= x−1, and we cannot have any operations overlapping.

The rest of our examples will be linear, in the sense that they arise via automorphisms on abelian groups

Example 1.2.6 (Commutative symmetry). The arithmetic mean may be regarded as a nonassociative, commutative quasigroup operation. That is, (R, ·) is a commutative quasigroup under x+y 1 1 x · y = 2 = x 2 + y 2 . In fact, in any abelian group M for which there is some automorphism R such that R + R = 1M , the operation x · y = xR + yR is a commutative quasigroup operation.

Example 1.2.7 (Left symmetry and right symmetry). We generalize Example 1.2.3 along the lines of the previous Example. Given any abelian group M and R ∈ Aut(M), the operation x·y = xR−y is left symmetric. The opposite x ◦ y = yR − x is right symmetric. In fact, a quasigroup is left symmetric if and only if its opposite is right symmetric.

Example 1.2.8 (Semisymmetry). Let R be an automorphism of an abelian group M such that

3 −1 R = −1M . Then the operation x · y = xR + yR is semisymmetric. The smallest properly

2 semisymmetric (i.e., not totally symmetric) example is given by M = (Z/2) , and   0 1   R =   . 1 1 10

Example 1.2.9 (Total symmetry). In any abelian group, the operation x · y = −x − y = −(x + y) is totally symmetric.

1.3 Notions from category theory and universal algebra

1.3.1 Categories of quasigroups

We assume familiarity with the axioms of category theory [31, Sec. I.1]. Knowledge of limit and colimit constructions (coequalizers and pullbacks) is helpful [31, Ch. III], but other than the coproducts of Subsection 1.3.2, these things are only mentioned in passing.

Birkhoﬀ’s HSP Theorem [4] —which states that a class of algebras is a variety (deﬁned by identities) if and only if it is closed under taking homomorphic images (H), substructures (S), and arbitrary direct products (P)— guarantees that the various H-symmetry classes form categories.

Thus, we will use the terms variety and category interchangeably when referring to classes of quasigroups. The variety Q is deﬁned by the axioms (IL)-(SR). Adding the identity

xy = yx (1.3.1) forms the variety C of commutative quasigroups, and the category of their homomorphisms.

The symmetry classes corresponding to the coincidence of multiplication with various divisions can actually be expressed by identities involving only multiplication; such expressions are preferable in the module theory. Using the axiom, (SL), it is easily seen that yx = y\x if and only if

y · yx = x (1.3.2) for all x, y ∈ Q. We take (1.3.2) as the deﬁning identity for the variety LS of left symmetric quasigroups. Dually,

xy · y = x (1.3.3) 11 generates the variety RS of right symmetric quasigroups. Earlier, we deﬁned semisymmetric quasigroups as those in which yx = x/y for all x, y ∈ Q. However, axiom (IR) makes it clear that

yx · y = x (1.3.4) is a perfectly good axiom for semisymmetry. The category of semisymmetric quasigroups is denoted

P. As we alluded to in the previous subsection, any two distinct nontrivial symmetries generate total symmetry. It will be best for us to use (1.3.1) and (1.3.4) as the deﬁning identities for the variety

TS of totally symmetric quasigroups. Whenever we wish to refer to an arbitrary H-symmetry class, we will use H.

This dissertation will also examine the eﬀects of idempotence in the six symmetry classes. A quasigroup Q (or any set under binary operation) is idempotent whenever

x2 = x (1.3.5) for all x ∈ Q. With two exceptions, for any subgroup H ≤ S3, the category of idempotent, H- symmetric quasigroups will be denoted HI. Idempotent, semisymmetric quasigroups form to the category MTS, while totally symmetric, idempotent quasigroups constitute the category STS.

The title MTS is an acronym for Mendelsohn triple system, a class of combinatorial designs which are in one-to-one correspondence with idempotent, semisymmetric quasigroups (these quasigroups are the subject of Chapter4). Steiner triple systems are designs that furnish totally symmetric, idempotent quasigroups.

1.3.2 Coproducts, free products, and algebraic presentations

` Let V be a category, and {Xi}i∈I a family of objects in V. An object i∈I Xi is said to be a ` coproduct of the Xi’s if there is a collection of morphisms ιi : Xi → i∈I Xi such that for any object

Y accompanied by an I-indexed family of morphisms fi : Xi → Y , there is a unique morphism ` g : i Xi → Y such that ιig = fi for all i ∈ I. Coproducts exist in all of the categories we study 12

(namely, in categories of algebras coming from varieties [45, Th. IV.2.2.3]), and they are unique up to isomorphism.

Example 1.3.1. In the category Set of functions between sets, the coproduct is given by disjoint union. We will denote the disjoint union of two sets X1 and X2 by X1 + X2.

Remark 1.3.2. We will use hG | Ri to denote the presentation for the group generated by G, subject to relations R. The free group on G, hG | ∅i, will be abbreviated to hGi. More generally, we use hG | RiV to denote a presentation for an algebra in a category V that contains free objects and coequalizers.

Example 1.3.3. In Gp, the category of group homomorphisms, coproducts are speciﬁed by the free product construction. It will be useful for us to think of free products in terms of group presentations. If hG | Ri and hG0 | R0i are presentations for two groups G and G0, respectively, then the free product G ∗ G0 is presented by hG + G0 | R + R0i.

2 Let us look at a particular free product. The order-2 cyclic group C2 has presentation hx | x =

2 2 1i. The free product C2 ∗C2 = hx, y | x = y = 1i consists of all words in the alphabet {x, y} which do not contain any consecutive repetitions of the symbols x and y. For example, x3yx = x(x2)yx reduces to xyx. Repeated free products of C2 will be very important when we study the module theory of left, right, and totally symmetric quasigroups.

Example 1.3.4. The category of unital rings, Ring, also has a free product construction that concretizes the notion of coproduct. For the rigorous, tensor algebra description of ring free products, consult Sections 1.2 and 1.4 of [2]. Just as in the case of groups, if a ring R has presentation

0 0 0 0 hG | RiRing, and R is presented by hG | R iRing, then the free product R ∗ R has presentation

0 0 hG + G | R + R iRing.

±1 The ring Z[X ] of integral Laurent polynomials is speciﬁed by

hX,Y | XY − 1 = YX − 1 = 0iRing. 13

It may also be regarded as the integral group ring of the free group on one generator, which we denote by Zhxi. Accordingly, the ring with presentation

hXi,Yi (i = 1, . . . , n) | XiYi − 1 = YiXi − 1 = 0 (i = 1, . . . , n)iRing

±1 describes the free product of n copies of Z[X ], which is isomorphic to the Laurent polynomial ∼ `n ring in n noncommuting variables, isomorphic to the free group ring Zhx1, . . . , xni = i=1 Zhxi.

1.3.3 Group modules, abelian group objects in slice categories

Our goal now is to motivate the quasigroup module theory by introducing some rather abstract notions that are, for many, simply in the background of group module theory. We bring them to the fore in order to make the fact that quasigroup module theory generalizes group module theory more apparent to the reader.

1.3.3.1 Group modules

For a group Q, we say that the abelian group M, along with a group homomorphism

ρ : Q → Aut(M); q 7→ (ρq : M → M; m 7→ mq), is a right Q-module. Associativity of automorphism composition leaves us with no opportunity to represent general quasigroups via maps such as ρ. However, recall that G-modules are in one-to-one correspondence with semidirect product constructions E = Q ρn M –where

(q1, m1)(q2, m2) = (q1q2, m1q2 + m2).

These split extensions will generalize to quasigroups. Indeed, letting p : E → Q denote the projection homomorphism mapping (q, m) 7→ q, the extension

p M / E / Q is split via the section η : Q → E; g 7→ (g, 0). Note that the above diagram is set in the category of groups Gp, and the projection p : E → Q is an object of what’s known as the slice category Gp/Q of groups over Q. As we shall soon see, an emphasis on categorical structure permits the extension of the theory of modules to quasigroups. 14

1.3.3.2 Abelian group objects in slice categories

Our discussion of abelian group objects is rather informal. We seek only to motivate the deﬁnition of quasigroup module and will avoid any discussion of the interplay between terminal objects and symmetries in Cartesian categories (cf. [31, Sec. III.6]).

An abelian group A may be viewed as an object of Set with structure maps

0 : A0 → A; ∗ 7→ 0;

+ : A2 → A;(a, b) 7→ a + b;

− : A → A; a → −a which allow for the algebraic axioms to be expressed diagrammatically. Suppose we want to describe

0 the fact that 0 is an identity with respect to addition. Let lA : A × A → A;(∗, a) 7→ a and

0 rA : A × A → A;(a, ∗) 7→ a. Then 0 + a = a = a + 0 for all a ∈ A translates to the following commutative diagram

0×1A A0 × A / A × A r O + rr l rr 1 ×0 A rrr A rrr AAyro × A0. rA In a category V with ﬁnite products, we deﬁne an abelian group object of V to be an object A with accompanying morphisms 0 : A0 → A, + : A2 → A, − : A → A, called zero, addition, and negation, respectively. These structure maps adhere to the abelian group axioms.

A concrete category is one in which all of the objects are sets, and morphisms are “structure- preserving” functions. The following discussion need not be restricted to concrete categories, but it will make exposition simpler. Let Q be an object of a concrete category V. The slice category

V over Q, denoted V/Q, takes as its objects the class of all arrows p : E → Q in V with codomain

Q. The arrows f :(p1 : E1 → Q) → (p2 : E2 → Q) of V/Q, once again, come from the morphism 15

class of V; those are, arrows f : E1 → E2 for which the diagram

f E1 / E2 (1.3.6)

p1 p2 Q / Q 1Q commutes in V. The product of two objects p1, p2 : E1,E2 → Q in V/Q is the pullback p1 ×Q p2 :

E1 ×Q E2 → Q, where E1 ×Q E2 = {(x, y) | p1(x) = p2(y)}, and p1 ×Q p2 is the restriction of p1 × p2 to E1 ×Q E2. The empty product (i.e., the terminal object) of V/Q is the identity 1Q : Q → Q. An abelian group object in V/Q is denoted (p : E → Q, +, −, 0), where + : E ×Q E → E, − : E → E, and 0 : Q → E.

To give an example, we return to Gp. We will show that p : E → Q, the projection of the split extension of Q by the module M onto Q, is an abelian group object in Gp/Q. The domain of p ×Q p is

E ×Q E = {((q, m1), (q, m2)) | q ∈ Q, m1, m2 ∈ M} , and we deﬁne addition accordingly:

+Q : E ×Q E → E; ((q, m1), (q, m2)) 7→ (q, m1 + m2).

Commutativity and associativity of +Q follow from those properties holding in M. The section

η : Q → E; q 7→ (q, 0) is the zero of +Q. To see neutrality of η with respect to +Q, note Q ×Q E =

{(q, (q, m)) : q ∈ Q, m ∈ M)} and E ×Q Q = {((q, m), q) | q ∈ Q, m ∈ M}; deﬁne $l : Q ×Q E →

E;(q, (q, m)) 7→ (q, m) and $r : E ×Q Q → E; ((q, m), q) 7→ (q, m) so that

η×Q1E Q ×Q E / E ×Q E O +Q rr $l rr 1E ×Qη rrr rrr EEoyr × Q $r Q commutes. The natural choice for negation is −Q : E → E;(q, m) 7→ (q, −m).

Conversely, let η, +Q, and −Q denote the structure maps of p : E → Q, an abelian group object in Gp/Q. Deﬁne M = Ker p, which is an abelian group by commutativity of the addition +Q. 16

The right conjugation action of Q on M comes from

η −Q η ρ : Q → Aut(M); q 7→ (ρq : M → M; m 7→ (q ) mq ).

In summary, Q-modules are equivalent to abelian group objects in the slice category Gp/Q. It is in the spirit of this equivalence that we deﬁne modules of general quasigroups.

1.4 Free extensions of quasigroups

The purpose of this section is to give an overview of Smith’s version of the normal form theorem

[18] for free, H-symmetric quasigroup extensions [51].

Let X be a set, and suppose U ⊆ X3 is a ternary relation. Then (X,U) is a partial Latin square if for all (x1, x2, x3), (y1, y2, y3) ∈ U, |{1 ≤ i ≤ 3 | xi = yi}|= 6 2. If Q is a quasigroup, then the ternary multiplication table of Q, T (Q) = {(x1, x2, x1 · x2) | x1, x2 ∈ Q}, is a partial Latin square. For H ≤ S3, a partial Latin square (X,U) is H-symmetric if (x1, x2, x3) ∈ U if and only if (x1h, x2h, x3h) ∈ U for all h ∈ H. An H-symmetric quasigroup Q is an H-symmetric extension of an H-symmetric partial Latin square (X,U), if X ⊆ Q (via ιQ : X,→ Q), and U ⊆ T (Q). Such an extension is free if for any H-symmetric quasigroup P extending (X,U) via the embedding

ιP : X,→ P , there is a unique H-morphism f : Q → P so that ιQf = ιP . Smith’s version of the normal form theorem states that for any H-symmetric partial Latin square (X,U), a free H- symmetric extension exists, is unique up to isomorphism, and —most importantly— the theorem explicitly describes a normal form for the elements of the extension.

Smith’s normal form relies on a postﬁx notation for expressing the six conjugate operations of

2 2 a quasigroup. Let σ = (12) and τ = (23), so that S3 = hσ, τ | σ = τ = 1, στσ = τστi. Fix a base quasigroup (Q, ·, /, \). Denote x · y = xyµ for all x, y ∈ Q. Multiplication in the conjugate

g corresponding g ∈ S3 (recall the diagrams (1.4.1), (1.2.2)) is expressed as xyµ . This is summarized in the diagram below: 17

x · y = xyµ o / x ◦ y = xyµσ (1.4.1) m 2: dl QQ mmm QQQQ mmmm QQQQ mmmm QQQQ rz mmm QQ $, x\y = xyµτ x\\y = xyµστ hQQ m6 QQQ mmm QQ mmm QQQ mmm QQ( vmmm x//y = xyµτσ ks +3 x/y = xyµτστ .

In the postﬁx notation, a quasigroup Q is H-symmetric if for all g ∈ S3, and all h ∈ H, the identity xyµg = xyµgh holds (in fact, it is suﬃcient that xyµ = xyµh for all h ∈ H [51, Lmm. 6]).

Example 1.4.1. It is necessary to see how the postﬁx notation extends to expressions with multiple operations. One reads from the inside out, beginning with the innermost µg. For instance, the word z\(xy · y) translates to z xyµy µµτ .

1.4.1 A conﬂuent rewriting system for the normal form

Let µS3 = {µ, µτ , µτστ , µσ, µστ , µτσ}. Given a partial Latin square (X,U), we will write out the elements of the free H-symmetric extension using the free monoid (X +µS3 )∗ (words in the alphabet

X + µS3 ). There are two rewriting rules which do not alter the length of a word in (X + µS3 )∗:

• Replace the subword uvµg with vuµσg. This is called a σ-replacement.

• Replace the subword uvµg with uvµgh. This is called an H-replacement.

Two words are σ-equivalent if they are related by a (possibly empty) ﬁnite sequence of σ-replacements; they are H-equivalent if they are related by a (possibly empty) ﬁnite sequence of H- replacements. The smallest equivalence relation generated by these two is called (σ, H)-equivalence.

Example 1.4.2. Let Q be a hστi-symmetric (semisymmetric) quasigroup, and treat x, y ∈ Q as subwords of (Q + µS3 )∗. Then just as x/y = y//x in Q, the words xyµτστ and yxµτσ are

(σ, hστi)-equivalent under the sequence of replacements

2 xyµτστ → yxµσ(τστ) = yxµ(στ) = yxµ → yxµτσ. 18

Deﬁnition 1.4.3. Consider the ordered set of basic quasigroup opertations {µ < µτστ < µτ } =

{· < / < \}. A word in (X + µS3 )∗ containing only basic operations is said to be primary if no word in its H-equivalence class contains a lesser operation.

There are two reduction rules which shorten the length of words in (X + µS3 )∗:

• Replace the subword u uvµgµτg with v. This is called a hypercancellative reduction.

g • If (x1, x2, x3) ∈ U, replace any occurrence of the subword x1gx2gµ with x3g. This is called a table reduction.

The hypercancellative law for g = 1 corresponds to the quasigroup axiom (IL). For g = σ, this is

(IR); when g = τ, this expresses (SL); and g = τσ is the postﬁx formulation of (SR). The choice of g = τστ corresponds to the fact that

(y/x)\y = x in quasigroups, and g = στ describes the consequence

y/(x\y) = x of the quasigroup axioms. The reduction rules form an equivalence class on (X + µS3 )∗; two words are equivalent if and only if one can be obtained from the other by a (possibly empty) ﬁnite sequence of reductions.

Deﬁnition 1.4.4. Let V denote the smallest equivalence relation on (X + µS3 )∗ formed by (σ, H)- equivalence and the reduction equivalence. A word of (X + µS3 )∗ is in normal form if it is the minimal-length primary representative of its V -equivalence class.

Theorem 1.4.5 (Smith’s Version of Evans’ Normal Form Theorem). Let (X,U) be an H-symmetric partial Latin square. The free H-symmetric extension of (X,U) is deﬁned on the set of normal form representatives for the equivalence classes of V on (X + µS3 )∗.

A partial Latin square (X,U) is idempotent if (x, x, x) ∈ U for all x ∈ X. Add the reduction rule 19

• Replace the subword uuµg with u to the previous two. Let V 0 denote the equivalence relation on (X + µS3 )∗ generated by (σ, H)- equivalence, hypercancellative, table, and idempotence reductions. Normal forms with respect to

V 0 are also deﬁned to be minimal-length primary representatives. We have the following Corollary to the normal form theorem:

Corollary 1.4.6. Let (X,U) be an idempotent, H-symmetric partial Latin square. The free, idempotent H-symmetric extension of (X,U) is deﬁned on the set of normal form representatives for the equivalence classes of V 0 on (X + µS3 )∗. 20

CHAPTER 2. QUASIGROUP MODULE THEORY

Fix a nonempty quasigroup Q. The category of quasigroup homomorphisms contains pullbacks

[45, Th. IV.2.2.3], so we can take direct products in the slice category Q/Q. Let (p1 : E1 →

Q, +1, −1, 01) and (p2 : E2 → Q, +2, −2, 02) be abelian group objects in Q/Q.A Q/Q-morphism f : E1 → E2 is a homomorphism of abelian groups if +1f = (f ×Q f)+2, −1f = f−2, and

01f = 02. We use Q-Mod(Q) to denote the category of abelian group objects in Q/Q, and refer to these objects as Q-modules (in the sense of Beck [1]). Smith’s Fundamental Theorem of

Quasigroup Modules describes an equivalence between Q-Mod(Q) and the category of modules over a particular group ring associated with Q. If Q belongs to a subvariety V of Q, then there is even a procedure for producing a ring whose modules are equivalent to the abelian group objects of V/Q (denoted Q-Mod(V)). The equivalence between Q-Mod(V) and the modules over this ring is the essence of Smith’s Relativized Fundamental Theorem of Quasigroup Modules. The ﬁrst account of these results appears in [49], while Chapter 10 of [50] gives a concise overview of the theory. We will adhere to the latter source in our exposition, with some input from Madariaga and

P´erez-Izquierdo’ssummary of the theory from [32]. This Chapter will include a review of Smith’s application of the Fundamental Theorems to Q, and the author’s to P.

2.1 The universal multiplication group, universal stabilizer and the Fundamental Theorem

Let Q be a nonempty quasigroup, and use Q[X] to denote the free h(1)i-extension of

(Q + {X},T (Q)). Then Q is a subquasigroup of Q[X], and we use U(Q; Q) to denote hL(Q) ∪ R(Q)(q ∈ Q)i ≤ Mlt(Q[X]), the relative multiplication group of Q in Q[X]. This group is called the universal multiplication group of Q in Q. It acts transitively on Q by xL(q) = qx and xR(q) = xq. 21

We forgo a formal proof of this next Theorem (many will be given regarding universal multiplication groups in other categories in Chapter3), but a brief justiﬁcation is warranted.

Theorem 2.1.1. Let Q be a quasigroup. Then the universal multiplication group U(Q; Q) is free on the set L(Q) + R(Q).

By deﬁnition, L(Q)∪R(Q) is a generating set for U(Q; Q), but these sets are disjoint since XL(q) = qXµ 6= qXµσ = Xqµ = XR(q) (we don’t have hσi-symmetry) for all q ∈ Q. Showing that the set of relations between these generators is empty comes down to showing that, on basic words, the hypercancellative reductions come from quasigroup axioms, and that these are in one-to-one correspondence with the reductions R(q)R(q)−1,R(q)−1R(q),L(q)L(q)−1,L(q)−1L(q) → 1 in the free group. Again, we emphasize that arguments such as these will be fully ﬂeshed out in Chapter

Let (p : E → Q, +, −, 0) be a Q-module, and q ∈ Q be arbitrary. The ﬁber over q is the set

Eq = {x ∈ E | (x)p = q}. Define 0q to be the image of q under 0 : Q → E. Since 0 is a morphism of the slice category, 0p = 1Q, and, thus, 0q ∈ Eq. Now, the commutativity of (1.3.6) with f = +, − ensures these maps preserve fibers, so the restriction of + and − to Eq, along with 0q, endow Eq with the structure of an abelian group. Because the quasigroup homomorphism 0 is a section of p (making it injective), its image 0Q = {0q | q ∈ Q} is subquasigroup of E which is isomorphic to Q. Now, fix a preferred element e ∈ Q, and let x0 ∈ E such that x0p = e (which we can do because p is a retraction of 0). Then for any x ∈ E, we have (x/0(x0\x)p)p = xp/(0(x0\x)p)p = xp/(x0\x)p = (x/(x0\x))p = x0p = e, meaning x/0(x0\x)p ∈ Ee. Thus, the arbitrary element x = (x/0(x0\x)p)0(x0\x)p is a product of an element of Ee and 0Q, so Ee × Q → E;(xe, a) 7→ xe0a is a bijection. The universal multiplication group U(Q; Q) acts on E by permutations: xL(q) = 0qx and xR(q) = x0q. Note that

(xe + ye)0a = xe0a + ye0a (2.1.1) for any a ∈ Q because + is a quasigroup homomorphism. Consider the element of the point-

−1 w stabilizer w = R(e\q)L(q/e) ∈ U(Q; Q)e ≤ U(Q; Q). For any xe ∈ Ee, we have ((xe) )p =

w (0q/e\(xe0e\q))p = 0q/ep\(xe0e\q)p = (q/e)\(e · (e\q)) = (q/e)\q = e; conclude (xe) ∈ Ee. In fact, 22

U(Q;Q)e this holds for any w ∈ U(Q; Q)e. Conclude that Ee ⊆ Ee, in conjunction with (2.1.1), prove

U(Q; Q)e acts on Ee by abelian group automorphsims. From a Q-module, we have furnished a

U(Q; Q)e-module.

Conversely, suppose U(Q; Q)e acts on an abelian group M by automorphisms. The set E = M × Q becomes a quasigroup under the multiplication

R(q2) L(q1) (m1, q1)(m2, q2) = (m1 + m2 , q1q2), (2.1.2)

where the actions of R(q2) and L(q1) come from the induced module M ⊗ZU(Q;Q)e ZU(Q; Q), and it is clear that the projection p : E → Q;(m, q) 7→ q is a quasigroup homomorphism. The maps +Q : E ×Q E → E; ((m1, q), (m2, q)) 7→ (m1 + m2, q), −Q : E → E;(m, q) 7→ (−m, q), and

0Q : Q → E; q 7→ (0, q) are also quasigroup homomorphisms, and thus furnish an abelian group object of Q/Q. We have proved the Fundamental Theorem:

Theorem 2.1.2 (Fundamental Theorem of Quasigroup Modules). Let Q be a nonempty quasigroup, and ﬁx e ∈ Q. Use G to denote the universal multiplication group of Q in Q, and Ge the stabilizer of e. The category of Q-modules is equivalent to the category of modules over ZGe.

A crucial object was introduced in the above discussion. The permutation action of U(Q; Q) on the split extension E comes from the fact that Q is naturally a U(Q; Q)-set under xR(q) = xq, and xL(q) = qx. This action is transitive; for example, x = y(y\x) = yR(y\x) whenever x, y ∈ Q.

Refer to this action as the multiplicative action of U(Q; Q) on Q, and due to its transitive nature we may refer to the universal stabilizer U(Q; Q)e of Q in Q, where e is any point in Q. Consider

−1 Te(q) := R(e\q)L(q/e) ; (2.1.3)

−1 Re(q, r) := R(e\q)R(r)R(e\qr) ; (2.1.4)

−1 Le(q, r) := L(q/e)L(r)L(rq/e) . (2.1.5)

These are members of U(Q; Q)e. We interpret them in terms of the Cayley graph of Q. The action of (2.1.3) on e amounts to a circuit of two nodes starting at our base vertex e and moving forwards 23 along the edge of right multiplication by e\q to q and returning to e backwards along the edge of left multiplication by q/e. The actions of (2.1.4) and (2.1.5) are three point trips, with the former moving from e to q, then along to a possibly distinct point qr, and returning to e backwards along the edge directing e to qr via right multiplication by e\(qr). Then (2.1.5) is a similar trip along arrows given by left multiplications. The circuits are pictured below.

rq L(rq/e) 7 g L(r)

L(q/e) e 3+ q R(e\q)

R(e\qr) R(r) ' qr w

These elements of the stabilizer are prototypical in the sense that U(Q; Q)e is free on

{Te(q),Re(q, r),Le(q, r) | q, r ∈ Q}

[50, Th. 2.3]. The following theorem provides a more explicit description of the basis.

Remark 2.1.3. In describing bases for the stabilizer of a particular point e, we will frequently

# make reference to the set Q r {e}. For brevity, we will let Q = Q r {e}.

Theorem 2.1.4. Let Q be a nonempty quasigroup, and ﬁx e ∈ Q. The universal stabilizer of Q in

Q is free on the set

# {T (q) | q ∈ Q} + {Re(e, e)} + {Re(q, r),Le(q, r) | (q, r) ∈ Q × Q}. (2.1.6)

2.2 Combinatorial diﬀerentiation and the Relativized Fundamental Theorem

The rest of the Chapter is an exposition of Smith’s extension of Theorem 2.1.2 to arbitrary categories of quasigroups. First, we need to generalize some of the terminology from the previous

Section. Fix a variety/category of quasigroups V, and let Q be a member of this class. The free

V-extension of Q is deﬁned to be the coproduct in V of Q with the free V-object on the singleton

{X}; this object is denoted Q[X]V, and in the case that V = H, this deﬁnition agrees with the 24

notion of free H-symmetric extension of Section 1.4. The coproduct map ιQ : Q → Q[X]V is injective [50, Lmm. 2.1], so Q may be regarded as a subquasigroup of Q[X]V.

Deﬁnition 2.2.1. Let V be a variety of quasigroups and Q a member of V.

(a) The relative multiplication group of Q in Q[X]V is the universal multiplication group of Q in V.

(b) If Q is nonempty, then U(Q; V)e, the stabilizer of some point e ∈ Q under the multiplicative action of U(Q; V) on Q, is the universal stabilizer of Q in V.

2.2.1 Modules over semisymmetric quasigroups

Let Q be a nonempty, semisymmetric quasigroup. In hopes of obtaining an abelian group object in P/Q, we will look at modules over U(Q; P)e. The semisymmetric universal multiplication group and stabilizer have been computed already by the author (cf. Theorems 3.6 and 3.7 of [41]).

Theorem 2.2.2. Let Q be a semisymmetric quasigroup.

(a) The universal multiplication group U(Q; P) is free on the set R(Q) of right multiplications.

(b) If Q is nonempty, and e ∈ Q, then the universal stabilizer U(Q; P)e is free on the set

# {Re(e, e),Te(q),Re(q, r) | (q, r) ∈ Q × Q, qr 6= e}. (2.2.1)

We ﬁx the abbreviation G = U(Q; P). Suppose that M is a Ge-module, and we apply the split extension construction of (2.1.2) to E = M × Q, but now we recognizing that in P, L(q) = R(q)−1:

−1 R(q2) R(q1) (m1, q1)(m2, q2) = (m1 + m2 , q1q2). (2.2.2)

Recall we want to situate E in P, but —working out the entries of [(m1, q1)(m2, q2)](m1, q1)— we

−1 see that (2.2.2) is a semisymmetric operation if and only if R(q2)R(q1) + R(q1q2) annihilates M G , the induced module. Choose {1,R(xe) | x ∈ Q#} as a transversal to G in G, and Frobenius Ge e −1 −1 −1 reciprocity dictates that R(q1e)R(q2)R(q1)R(q2e) + R(q1e)R(q1q2) R(q2e) must annihilate 25

Table 2.1 Partial derivatives of quasigroup words

∂u ∂u u ∂x ∂y x 1 0 x2 R(x) + L(x) 0 y · yx R(yx) + R(x)L(y) L(y)2 yx · y R(x)R(y) + L(yx) L(y)R(y) the restriction of M G to G for all q , q ∈ Q. This is a rough sketch for the proof of [41, Th. 4.5], Ge e 1 2 which we reproduce below.

Theorem 2.2.3. Suppose Q is a nonempty, semisymmetric quasigroup, G = U(Q; P), so that Ge is the universal stabilizer in P. Let J be the two-sided ideal in ZGe generated by

{R(ye) R(x)R(y) + R(yx)−1 R(xe)−1 | (x, y) ∈ Q2}. (2.2.3)

Then Q-Mod(P) is equivalent to the category of modules over the quotient ring ZGe/J.

2.2.2 The Relativized Fundamental Theorem

The proof of Theorem 2.2.3 involves a method of “combinatorial diﬀerentiation” that generalizes

−1 the procedure by which we found the ZU(Q; P)-elements R(q2)R(q1) + R(q1q2) . The theory is outlined in [50, Sec. 10.4]. We will simply show the “product rule” for diﬀerentiating expressions involving quasigroup multiplication, as all the identities with which this thesis is concerned may be expressed multiplicatively.

Suppose u and v are quasigroup words (words in the free quasigroup on some generating set) containing the variable x. We deﬁne the partial derivative of uv with respect to x recursively:

∂w = δ ∂x w,x for any length-one word w, while

∂(uv) ∂u ∂v = R(v) + L(u). ∂x ∂x ∂x

The partial derivatives for several words coming from H-symmetry are given in Table 2.1. 26

In the next Theorem, we will employ the following notation: if u is a quasigroup word in the variables x1, . . . , xn, and q1, . . . , qn is a sequence of elements in a quasigroup Q, then u(q1, . . . , qn)

∂u is the result of evaluating u in Q at (x1, . . . , xn) = (q1, . . . , qn). In a similar fashion, (q1, . . . qn) ∂xh denotes the result of evaluating the partial derivative of u with respect to xh at (x1, . . . , xn) =

(q1, . . . , qn) in ZU(Q; V).

Theorem 2.2.4 (The Relativized Fundamental Theorem of Quasigroup Modules). Let V denote a variety of quasigroups deﬁned by the identities B = {ui = vi}i∈I . If Q is a nonempty quasigroup in

V, G = U(Q; V), and Ge denotes the universal stabilizer of Q in V, then the category of Q-modules is equivalent to the category of modules over ZGe/J, where J is the two-sided ideal generated by ∂u ∂v −1 ρ(e, qh) (q1, ··· qn) − (q1, . . . , qn) ρ(e, v(q1, . . . qn)) | u = v ∈ B, qh ∈ Q , (2.2.4) ∂xh ∂xh

−1 for {ρ(e, q) = R(e\e) R(e\q) | q ∈ Q}, a transversal to Ge in G. 27

CHAPTER 3. MODULES OVER H-SYMMETRIC QUASIGROUPS AND MODULES OVER H-SYMMETRIC, IDEMPOTENT QUASIGROUPS

2 Let Q be a totally symmetric, idempotent quasigroup. Set B = {{q, p, qp} | (q, p) ∈ Q r Qb}, where Qb = {(x, x) | x ∈ Q}. The pair (Q, B) constitute a Steiner triple system (STS), meaning that for every two-element subset {q, p} ⊆ Q, there is exactly one triple {a, b, c} ∈ B so that

{p, q} ⊆ {a, b, c}. Conversely, any Steiner triple system (Q, B) corresponds to a totally symmetric, idempotent quasigroup, where, for distinct q, p ∈ Q, deﬁne qp to be the third element of the unique block containing {q, p}. We refer to totally symmetric, idempotent quasigroups as Steiner quasigroups [35].

Example 3.0.1. Let V be a vector space over F3. The operation x · y = −(x + y) endows V with the structure of a Steiner quasigroup.

The module theory of Steiner quasigroups is relatively well-behaved. The following theorem is a collection of results due to Smith [50, Thms. 11.3-11.5]

Theorem 3.0.2. Let Q be a nonempty Steiner quasigroup with corresponding STS (Q, B). Let Be

# denote the set of blocks containing e, and B = B r Be.

(a) The universal multiplication group has presentation

a C2, (3.0.1) Q

where each generating involution comes from R(q).

(b) The universal stabilizer is isomorphic to

a a a C2 ∗ hxi ∗ hx, y, zi, (3.0.2) Q Be B# 28

where the order-2 generators come from Re(q, q), the free generators over Be are of the form

# Re(a, e), and for a block {q, p, qp} ∈ B , Re(q, p),Re(qp, q),Re(p, qp) represent free generators.

the category of modules over F3Ge/J, where J is the two-sided ideal generated by

# {1 + Re(q, e) | q ∈ Q} + {Re(q, p)Re(qp, q)Re(p, qp) + 1 | {q, p, qp} ∈ B } (3.0.3)

Our goal in this chapter is to obtain a version of Theorem 3.0.2 for the six H-symmetric varieties and their idempotent extensions. Since a quasigroup is right symmetric if and only if its opposite is left symmetric, we will skip the former class. The particular varieties we treat are

QI, C, CI, P, MTS, LS, LSI, and TS.

We shall break up our work in a way that reﬂects the partitioning of Theorem 3.0.2. Section

3.1 is devoted to computing universal multiplication groups. The following section, 3.2, gives presentations for universal stabilizers. In Section 3.3, we compute the the ideal generators to describe the quotient ring of the Relativized Fundamental Theorem. The combinatorial diﬀerentiation procedure makes it relatively easy to obtain a description of ideal generators that looks like (2.2.3).

What is less straightforward is describing the ideal generators in terms of basis elements from the universal stabilizer, as (3.0.3) does. It requires some tedious calculations and case analysis. The author begs the reader their forgiveness.

In certain cases, we will be able to give abstract structure theorems for the rings of representation; these cases are CI, MTS, STS. Other varieties in all likelihood have very similar structure theorems, but the cases and subcases and subsubcases we ran into in seemed not worth pursuing; diﬀerent methods to alleviate such issues are discussed in Chapter6. For example, we will show that the ring described above in 3.0.2(c) is isomorphic to the free product

a a a F3 ∗ Z ∗ Zhx, yi. (3.0.4) Q Be B# 29

3.1 Universal multiplication groups

3.1.1 Useful conventions and lemmas

Lemma 3.1.1. Let Q be a quasigroup belonging to a variety V. The universal multiplication group

U(Q; V) is a quotient of the free group hL(Q) + R(Q)i on the disjoint union of two copies of Q.

Moreover, if V is a subvariety of C, P, or TS, then U(Q; V) is a quotient of the free group hR(Q)i on Q.

Proof. By deﬁnition, {L(q),R(q) | q ∈ Q} is a generating set for U(Q; V). For a commutative

−1 quasigroup Q, L(w) = R(w) for every w ∈ Q[X]C. If Q is semisymmetric, L(w) = R(w) . A totally symmetric quasigroup is both commutative and semisymmetric, so L(w) = R(w) =

−1 −1 L(w) = R(w) for all w ∈ Q[X]TS. Therefore, in these cases, the generating set for U(Q; V) may be reﬁned to R(Q).

Let

F : G → U(Q; H) (3.1.1) denote the canonical projection aﬀorded by Lemma 3.1.1, where G is either hL(Q) + R(Q)i or

ε1 ε hR(Q)i, depending on the variety. Suppose g = E(q1) ··· E(qk) k is a a generic element of G. We use the fact that for any u ∈ Q[X]H and q ∈ Q

uL(q) = quµ

uL(q)−1 = quµτ

uR(q) = quµσ

uR(q)−1 = quµτσ to obtain

g1 gk qk ··· q1uµ ··· µ (3.1.2) as a standard representative for ugF . 30

Remark 3.1.2. It is important the reader keep in mind that we are working with elements of ∗ hL(Q) + R(Q)i and Q + {X} + µS3 that are not necessarily in the normal forms of U(Q; H) and

Q[X]H. As long as there is a conﬂuent rewriting system specifying normal forms in Q[X]H, this ambiguity is resolvable.

Lemma 3.1.3. Let Q be a quasigroup belonging to an H-symmetric variety H. Suppose that g1, . . . , gk ∈ {1, σ, τ, τσ} and q1, . . . , qk ∈ Q. Set

g1 gk w = qk ··· q1Xµ ··· µ ∈ Q[X]H. (3.1.3)

No table reductions can be applied to w.

g Proof. A table reduction would require that w contains a subword of the form q1q2µ , where

g g q1, q2 ∈ Q. Since X/∈ Q, q1Xµ 1 does not respond to this challenge. Since q1Xµ 1 cannot be

g g g expressed as an element of Q, q2q1Xµ 1 µ 2 does not permit table reductions with respect to µ 2 . Continue by induction on k.

Lemma 3.1.4. Suppose that Q is an H-symmetric, idempotent quasigroup. For any g ∈ U(Q; H), ∗ a (3.1.2)-type representation of Xg, which we shall call w ∈ Q + {X} + µS3 , is irreducible under the rewriting rules deﬁning Q[X]H if and only if it is irreducible under the rewriting rules deﬁning

Q[X]HI.

Proof. ( =⇒ )

Suppose

g1 gk w = qk ··· q1Xµ ··· µ , (3.1.4) where q1, . . . , qk ∈ Q, and g1, . . . , gk ∈ {1, σ, τ, τσ}. Our hypothesis ensures us that none of the hypercancellative or table reductions apply to subwords of w. If we show that no idempotent reductions

Ig uuµg / u can be applied to subwords of w, then we are done. By way of contradiction, suppose u ∈

S ∗ g Q + {X} + µ 3 , and g ∈ S3 so that uuµ is a subword of (3.1.4). Then there is some 1 ≤ l ≤ k 31

for which g = gl, and

g g1 gl uuµ = ql ··· q1Xµ ··· µ . (3.1.5)

g1 g g Since X 6= q1, we cannot have l = 1. On the other hand, if (3.1.5) is of the form (qlql−1 ··· q1Xµ ··· µ l−1 ) µ l ,

g1 g we still arrive at a contradiction. Indeed, for multiple reasons, qlql−1 ··· q1Xµ ··· µ l−1 cannot be ∗ equal to uu for some u ∈ Q + {X} + µS3 ; for one, X appears only once as a subword of w.

( ⇐= ) That the rewriting rules deﬁning Q[X]H are a subset of those deﬁning Q[X]HI is all one needs to observe.

3.1.2 Classiﬁcation of universal multiplication groups

In this section, we will classify the universal multiplication groups associated with each of the

12 varieties. Recall that Q and STS have been classiﬁed by Smith [50, Ths. 2.2, 11.3], and that the author described U(Q; P) via [41, Th. 3.3]. Note also, that complete descriptions of universal multiplication groups in LS and LSI apply to RS and RSI, simply switching the roles of left and right translation maps.

Our proofs follow a template. Recall (3.1.1), the canonical surjection aﬀorded by Lemma 3.1.1.

We will show that when H is I, C, CI, or MTS, F is an isomoprhism. Towards this end, ﬁx an

ε1 ε gF gF element g = E(q1) ··· E(qk) k ∈ KerF , so that X = X. We have a representation of X from ∗ the monoid Q + {X} + µS3 as in (3.1.2):

gF g1 gk X = X ≡H qk ··· q1Xµ ··· µ . (3.1.6)

g1 g Now, if qk ··· q1Xµ ··· µ k is fully reduced, then it must, in fact, equal X. This is due to the absence of symbols from µS3 in X. There is no way to obtain X from a word containing operation symbols without applying reductions. Hence, the process by which we obtained (3.1.6) involved an empty set of symbols, so g = 1. Even in the cases of total and one-sided symmetry —where we are

g1 g not showing F to be an isomorphism— this will be the general setup: show that qk ··· q1Xµ ··· µ k is fully reduced, proving injectivity of a certain homomorphism that is surjective by construction.

Theorem 3.1.5. Let Q be an idempotent quasigroup. The universal multiplication group U(Q; I) is free on the disjoint union L(Q) + R(Q). 32

ε1 ε Proof. Here, the domain of F is hL(Q) + R(Q)i. Let g = E(q1) ··· E(qk) k be a reduced free group word, and by way of proving injectivity, suppose gF acts trivially on Q[X]I. Theorem 2.2

∼ g1 gk of [50], which states that U(Q; Q) = hL(Q) + R(Q)i, demands qk ··· q1Xµ ··· µ is fully reduced

g1 g with respect to Q[X]Q. By Lemma 3.1.4, qk ··· q1Xµ ··· µ k is fully reduced under the rewriting rules of Q[X]I. Therefore, F is an isomorphism.

Theorem 3.1.6. Let Q be a commutative quasigroup. The universal multiplication group U(Q; C) is free on R(Q).

ε1 ε Proof. The domain of F is now hR(Q)i. Fix g = R(q1) ··· R(qk) k ∈ hR(Q)i, and suppose gF acts trivially on Q[X]C. Then

gF g1 gk X = X ≡C qk ··· q1Xµ ··· µ . (3.1.7)

As an element of the free group on R(Q), there are no subwords of g of the form R(q)±1R(q)∓1, where q ∈ Q. In terms of (3.1.7), this means that there is no 1 ≤ l ≤ k − 1 such that gl = σ and gl+1 = τσ, or vice versa. Such occurrences are the only way that hypercancellation could be

g1 g applied to qk ··· q1Xµ ··· µ k . By Lemma 3.1.3, no table reductions are possible on (3.1.7). It

g1 g follows that X = qk ··· q1Xµ ··· µ k , and g = 1.

Corollary 3.1.7. Let Q be a commutative, idempotent quasigroup. The universal multiplication group U(Q; CI) is free on R(Q).

Proof. The argument here is identical to that found in the proof of Theorem 3.1.5, with C playing the role of Q, and Theorem 3.1.6 being referenced instead of the work of Smith in [50].

Theorem 3.1.8. Let Q be a Mendelsohn quasigroup. The universal multiplication group U(Q; MTS) is free on R(Q).

Proof. Once again, the argument matches the proof of Theorem 3.1.5, but instead of citing [50,

Th. 2.2], we turn to Theorem 2.2.2, which states that U(Q; P) =∼ hR(Q)i. 33

Theorem 3.1.9. Let Q be a left-symmetric quasigroup. The universal multiplication group U(Q; LS) has presentation   −1 ∼ a hR(q) ∈ R(Q),L(q) ∈ L(Q) | L(q) = L(q) i =  C2 ∗ hR(Q)i. (3.1.8) Q Proof. Denote the group presented in (3.1.8) by G. Because (uv)L(u) = u(uv) = v = u\(uv) =

−1 −1 (uv)L(u) for all u, v ∈ Q[X]LS, L(u) = L(u) . So each word in U(Q; LS) can be expressed as a product of left and right translations, where each L(q) is an involution. Thus, we have a surjective map F 0 : G → U(Q; LS). There are no q ∈ Q for which L(q)L(q) and R(q)±1R(q)∓1

0 are subwords of an arbitrary reduced group word g ∈ G. So if gF acts trivially on Q[X]LS, then

0 gF g1 gr the LS-equivalent of X , qk ··· q1Xµ ··· µ —where, now, g1, . . . , gk ∈ {1, σ, τσ}— is fully reduced, for the absence of L(q)L(q) and R(q)±1R(q)∓1 rules out hypercancellations. Conclude F 0 is an isomorphism.

Corollary 3.1.10. Let Q be a left-symmetric, idempotent quasigroup. The universal multiplication group U(Q; LSI) has presentation   −1 ∼ a hR(q) ∈ R(Q),L(q) ∈ L(Q) | L(q) = L(q) i =  C2 ∗ hR(Q)i. Q Proof. Follows from the previous theorem as Corollary 3.1.7 follows from Theorem 3.1.6.

Theorem 3.1.11. Let Q be a totally symmetric quasigroup. The universal multiplication group

U(Q; TS) has presentation

−1 ∼ a hR(q) ∈ R(Q) | R(q) = R(q) i = C2. (3.1.9) Q ` −1 Proof. Proceed as in the proof of Theorem 3.1.9, with G = Q C2. Since L(u) = L(u) = R(u) = −1 0 R(u) for any u ∈ Q[X]TS, we get a surjective group homomorphism F : G → U(Q; TS). For any g ∈ G, we have

gF 0 σ σ X ≡TS qk ··· q1Xµ ··· µ , q1, . . . , qk ∈ Q, and ql 6= ql+1 for all 1 ≤ l ≤ k − 1. Clearly, such a word is not subject to hypercancellations, so if XgF 0 = X, then g = 1, and F 0 is also a monomorphism. 34

From the results of this subsection, the following theorem is apparent.

Theorem 3.1.12. Let H ≤ S3, and Q be an idempotent, H-symmetric quasigroup. Then

U(Q; H) =∼ U(Q; HI).

3.2 Universal stabilizers

Remark 3.2.1. (a) We will establish a variation of (2.1.3), as it appears when computing the generating sets for certain universal stabilizers. Let U(Q; V)e be the universal stabilizer for a nonempty quasigroup Q. Because e = q(q\e) = (e(e\q))(q\e),U(Q; V)e contains the permutation

Te(q) := R(e\q)R(q\e). (3.2.1)

(b) If G is a subgroup containing the set S, let hSiG denote the subgroup of G generated by S.

Lemma 3.2.2. Let H ≤ S3. Suppose Q is a nonempty, idempotent, H-symmetric quasigroup and ∼ that e ∈ Q. Then U(Q; H)e = U(Q; HI)e.

Proof. We proceed at risk of being tedious. For q ∈ Q, let E(q)ε denote a generic element of

ε U(Q; H), and Ee(q) one of U(Q; HI). The former acts on Q[X]H, and the latter on Q[X]HI. The various proofs of Subsection 3.1.2 reveal E(q)ε 7→ Ee(q)ε extends to an isomorphism of U(Q; H) onto U(Q; HI). The restriction of this map to U(Q; H)e maps onto U(Q; HI)e. Indeed, both

ε1 ε E(q1) ··· E(qk) k ∈ U(Q; H) and its image in U(Q; HI) result in the same quasigroup word

g1 gk qk ··· q1eµ ··· µ (3.2.2)

(each gi ∈ {1, σ, τ, τσ}) when acting on e, and since Q[X]H and Q[X]HI extend the same Latin square, (3.2.2) is H-equivalent to e if and only if it is HI-equivalent to e.

In order to classify the universal stabilizers for the 12 varieties, it now suﬃces to describe them for the 6 triality varieties. Stabilizers for Q and P are classiﬁed. We devote the rest of this section to classifying stabilizers in the varieties C, RS, and TS (the stabilizers in LS are described by those in RS via a switch of left/right multiplications). 35

We use Serre’s work in [47] as our main source for groups acting on (Cayley) graphs (of quasigroups) and trees, and, thus, feel it necessary to review some of this book’s terminology and notation.

Let G be a free group with basis G, and H ≤ G a subgroup. By the Nielsen-Schreier Theorem

[47, Th. 5], H is free. We review a procedure for determining a free basis G0 for H (cf. [47, Prop. 16]).

Begin by establishing a Schreier transversal to H in G, which is a complete set of distinct coset representatives closed under initial subwords; call it T . Closure under initial subwords means for

ε1 εk Qj εi each s1 ··· sk ∈ T , all partial products i=0 si , 0 ≤ j ≤ k, belong to T . Proceed to deﬁne

W = {(t, s) ∈ T × G | ts∈ / T }.

Then

0 G = {ht,s | (t, s) ∈ T × G} (3.2.3)

−1 —where ht,s = tsu , and u can be any member of T for which Hts = Hu— is a free basis for H. Furthermore, if H has ﬁnite index [G : H], then we have the Schreier Index Formula [47, Cor. 5]:

rankH = [G : H](rankG − 1) + 1. (3.2.4)

Theorem 3.2.3. Let Q be a nonempty, commutative quasigroup. Fix e ∈ Q. The universal stabilizer U(Q; C)e is free with basis

n # o Re(e, e),Re(q, r), Te(q) | (q, r) ∈ Q × Q, qr 6= e . (3.2.5)

2 If |Q| = n < ∞, then U(Q; C)e has rank n − n + 1.

Proof. Set

T = {1} ∪ {R(e\q) | q ∈ Q#}.

−1 Note R(e\q1)R(e\q2) ∈ U(Q; C)e if and only if e = (e(e\q1))/(e\q2) = q1/(e\q2) if and only if ∼ ∼ q1 = q2 (/ is a quasigroup multiplication). Then since T =Set Q =Set H\G, T is a transversal, and it is easily seen to be Schreier. It follows

W = {(1,R(e\e))} ∪ {(R(e\q),R(r)) | (q, r) ∈ Q# × Q}. 36

For (t, s) = (1,R(e\e)), we can let u = 1. This ht,s corresponds to Re(e, e) in (3.2.5). If (t, s) =

(R(e\q),R(r)), and qr 6= e, u = R(e\qr) works, accounting for each Re(q, r) in (3.2.5). In the case that qr = e, u = 1 gives an appropriate ht,s = Te(q), for r = q\e.

Finally, if |Q| = n, then orbit-stabilizer theorem tells us U(Q; C)e has index n, and so

2 rank(U(Q; C)e) = n(n − 1) + 1 = n − n + 1.

Remark 3.2.4. Let Q be a nonempty, semisymmetric quasigroup. Recall from Theorem 2.2.2 that

U(Q; P) = hR(Q)i, just as in the commutative case. Furthermore, U(Q; P)e is also free on

# {Re(e, e),Re(q, r), Te(q) = Te(q) | (q, r) ∈ Q × Q, qr 6= e}

[41, Th. 3.7].

In LS and TS, universal multiplication groups are no longer free, so the Nielsen-Schreier method cannot yield presentations of the corresponding stabilizers. The Reidemeister-Schreier algorithm takes as its input a group G with presentation hG | Ri and a ﬁnite-index subgroup H, producing a presentation hH | Si for H. Crowell’s specialized Reidemeister-Schreier algorithm applies to the case when G acts transitively on a set Q, and H = Ge for some e ∈ Q [12, Th. 6.2]. We’ll review the notation and terminology of Crowell’s method.

The derived group of a right permutation representation of a group G on a nonempty set Q has presentation

−1 Q ∧ G := Q × G | (q, g1g2) (q, g1)(qg1, g2) . (3.2.6)

Elements of the generating set are denoted q ∧ g. For each q ∈ Q and g ∈ G, we have [12, Prop. 1.5]

q ∧ g = 1 ⇐⇒ g = 1, (3.2.7) and

(q ∧ g)−1 = qg ∧ g−1. (3.2.8)

For the sake of brevity and clarity, we’ll adapt the remainder of our exposition of Crowell’s paper to the particular situation of a quasigroup Q under the transitive action of G = U(Q; H). In 37 analogy with the Schreier transversal for our free group cases, ﬁx

T = {e ∧ R(e\q) | q ∈ Q#}. (3.2.9)

Then, by Theorem 5.4 of [12] the derived group can be speciﬁed as the free product

∼ Q ∧ U(Q; V) = U(Q; V)e ∗ hT i. (3.2.10)

Moreover, if we have a presentation U(Q; V) = hE(qj) | ui = vii for the universal multiplication group, the universal stabilizer has presentation

D # E U(Q; V)e = q ∧ E(qj)(q ∈ Q) | q ∧ ui = q ∧ vi, e ∧ R(e\r) = 1 (r ∈ Q ) . (3.2.11) R

In (3.2.11), the R-subscript is our way of saying that there are “extra relations,” and these are the derived group relations of (3.2.6).

Example 3.2.5. (a) Let Q be a nonempty, commutative quasigroup. By Theorem 3.2.3, U(Q; C) =

hR(Q)i, and U(Q; C)e is free on (3.2.5). We will verify this fact using Crowell’s procedure. Indeed,

D # E U(Q; C)e = q ∧ R(r)(q, r ∈ Q) | e ∧ R(e\r) = 1 (r ∈ Q ) . R

For q 6= e and r 6= q\e, q ∧ R(r) represents Re(q, r), while Te(q) appears as q ∧ R(q\e). The

ﬁnal generator, Re(e, e), comes from e ∧ R(e\e).

(b) Now, if Q is a nonempty, left symmetric quasigroup, then U(Q; LS) = hR(Q)+L(Q) | L(q) =

L(q)−1i. In the case of left symmetry, it will be convenient for us to use the transversal

T 0 = {e ∧ L(e/q) | q ∈ Q#} (3.2.12)

instead of T . Then

D E q ∧ L(p), r ∧ R(s)(q, p, r, s ∈ Q) | q ∧ L(p) = q ∧ L(p)−1, e ∧ L(e/q) = 1 (r ∈ Q#) R (3.2.13)

serves as a presentation for U(Q; LS)e. 38

3.2.1 Left symmetric quasigroups

We will establish some notation. Let Q be a left symmetric quasigroup containing the point e.

# 3 Deﬁne I(q) = Q r {e/q, q/q}. For each q ∈ Q , deﬁne a subset of Q : Bq = {(q, p, pq) | p ∈ I(q)}.

Fix S = ∪Q# Bq. We establish an equivalence relation on S as follows:

(a, b, ba) ≡ (c, d, dc) ⇐⇒ b = d and {a, ba} = {c, dc}. (3.2.14)

Let F denote the set of ≡-equivalence classes.

Lemma 3.2.6. Let Q be a left symmetric quasigroup. Then each equivalence class in E has precisely

n−1 two constituents. In particular, if |Q| = n < ∞, then |F| = 2 .

Proof. Suppose (a, b, ba) ≡ (c, b, bc) ≡ (d, b, bd). If a = bc, and a = bd, then c = d, so (c, b, bc) =

(d, b, bd). This proves the ﬁrst statement of the Lemma. Now, assume Q is ﬁnite. The right

# cancellative property demands |Bq| = |I(q)| = n − 2 for all q ∈ Q . The Bq’s are pairwise disjoint, so |S| = (n − 1)(n − 2). Conclude

(n − 1)(n − 2) |F| = 2 (n − 1)! = 2(n − 3)! n − 1 = . 2

The next Lemma follows immediately from the left symmetric identity r · rq = q.

# Lemma 3.2.7. Let Q be a left symmetric quasigroup, and ﬁx q, r ∈ Q . Suppose p ∈ I(q), and

−1 that s ∈ I(r). Then (q ∧ L(p)) = r ∧ L(s), or q ∧ L(p) = r ∧ L(s) (in U(Q; LS)e) if and only if (q, p, pq) ≡ (r, s, sr).

Theorem 3.2.8. Let Q be a nonempty, left symmetric quasigroup. Fix e ∈ Q. The universal stabilizer U(Q; LS)e is isomorphic to 39

a a a C2 ∗ hxi ∗ hxi. (3.2.15) Q F Q2

The involutions are represented by Le(q, q/q); the free group generators are Le(q, p) for each

# # (q, p, pq) ∈ F, Re(e, e), Re(q, p) for each (q, p) ∈ Q × Q, and Te(q) for each q ∈ Q . If Q has ﬁnite order n, then U(Q; LS)e is isomorphic to the free product of n copies of C2 with the free

3n2−3n+2 group on 2 generators.

Proof. We will reﬁne the presentation (3.2.13). Clearly, we can throw out all generators coming from

T 0. Notice, too, that (q ∧ L(p))−1 = pq ∧ L(p) accounts for all crossed product relations coming from generators of the form q ∧ L(p). In particular, for all q ∈ Q, we have order-2 generators from (q ∧ L(q/q))−1 = (q/q)q ∧ L(q/q) = q ∧ L(q/q). Finally, (q ∧ L(e/q))−1 = (e/q)q ∧ L(e/q) = e∧L(e/q) = 1, so we can also eliminate these generators from our presentation. We have established

a D # −1 −1E C2 ∗ q ∧ L(p)(q ∈ Q , p ∈ I(q)) | q ∧ L(p) = q ∧ L(p) = (pq ∧ L(p)) ∗ hr ∧ R(s)iR Q as a valid presentation for U(Q; LS)e. Our choice to exclude q/q from a given I(q) means that each relation q ∧ L(p) = (pq ∧ L(p))−1, corresponds to two distinct generators, so that

D # −1 −1E q ∧ L(p)(q ∈ Q , p ∈ I(q)) | q ∧ L(p) = q ∧ L(p) = (pq ∧ L(p)) is a free group on a set of choices selecting one generator with respect to each relation. By Lemma

3.2.7, the set F represents precisely such a choice, whence the reﬁnement

∼ a a U(Q; LS)e = C2 ∗ hxi ∗ hr ∧ R(s)iR. (3.2.16) Q F

We are left to prove hr ∧ R(s)iR is free. This follows from [12, Prop. 1.9], upon interpreting hr ∧ R(s)iR as the derived group of the subgroup hR(Q)i ≤ U(Q; LS) acting on Q.

2 n−1 2 3n2−3n+2 If Q is ﬁnite and has order n, then Lemma 3.2.6 ensures |F + Q | = 2 + n = 2 .

3.2.2 Totally symmetric quasigroups

Let Q be a nonempty, totally symmetric quasigroup. Recall Q is left-symmetric, and

−1 U(Q; TS) = hL(Q) | L(q) = L(q) i ≤ U(Q; LS). More speciﬁcally, the presentation of U(Q; TS)e 40 is simply (3.2.13), minus the generators coming from Q ∧ hR(Q)i. The proof of Theorem 3.2.8 thus provides all the requisite arguments for our next Theorem. Do recall that in a totally symmetric quasigroup, all operations coincide, so that q/q = q2.

Theorem 3.2.9. Let Q be a nonempty, totally symmetric quasigroup. Fix e ∈ Q. The universal stabilizer U(Q; TS)e is isomorphic to

a a C2 ∗ hxi. (3.2.17) Q F 2 The involutions are represented by Le(q, q ); the free group generators are Le(q, p) for each

(q, p, pq) ∈ F. If Q has ﬁnite order n, then U(Q; TS)e is isomorphic to the free product of n copies n−1 of C2 with the free group on 2 generators.

Remark 3.2.10. If Q is totally symmetric, idempotent (Steiner), then the presentation of Theorem

3.2.9 reﬂects the block structure of the associated Steiner triple system (Q, B) [50, Sec. 11.3]. Indeed,

2 2 ﬁx q 6= e. By idempotence, I(q) = Q r {qe, q}, and e = e 6= qe 6= q = q, so that, what’s more,

I(qe) = Q r {qe · e, qe} = I(q). Then (q, e, qe) ≡ (qe, e, q) correspond to the block {q, e, qe} ∈ B.

That is, there is one free group generator for each block containing e. Concretely, it is Re(q, e).

If p 6= e ∈ I(q), then qp 6= e, q ∈ I(qp), and qp ∈ I(p). Hence, {q, p, qp} ∈ B is a block not containing e, and we have three distinct free group generators associated with this block: Re(q, p),

Re(qp, q), and Re(p, qp), corresponding to (q, p, qp), (qp, q, p), and (p, qp, q), respectively.

3.3 Classifying the rings of representation

3.3.1 H-symmetric quasigroups

Recall Smith’s Fundamental Theorem of Quasigroup Modules 2.1.2 states that given a quasigroup Q, abelian group objects in Q/Q are equivalent to modules over the ring ZU(Q; Q)e, which is the free group ring over (2.1.3)-(2.1.5). In order to obtain the ring of representation for Q in an

H-symmetric variety H, we have to compute the two-sided ideal J generated by ∂u ∂v −1 R(e\xh) (q1, . . . , qn) − (q1, . . . , qn) R(e\v(q1, . . . , qn)) (3.3.1) ∂xh ∂xh 41

for each qh ∈ Q and u = v deﬁning H, and then describe the quotient ZU(Q; H)e/J. The astute reader may have noticed a discrepancy between (3.3.1) and (2.2.4). Smith’s choice of transversal

ρ(e, q) turns out to be overkill for the H-symmetric setting, so {1,R(e\x) | x ∈ Q#} leads to the alternative formulation for the ideal in this Chapter.

Remark 3.3.1. Let V be a variety of quasigroups and Q a nonempty constituent of said variety.

We will abbreviate our expression of the quotient ring ZU(Q; V)e/J, where J is generated by

(3.3.1), to ZVQ.

Let H ≤ S3. Recall that if z ∈ {x, y} appears at most once in u = v, then generators coming from xh = z are extraneous [50, Lmm. 10.3]. Hence, J = 0 for H = Q, C. What’s more, for the semisymmetric, left symmetric, and totally symmetric cases, we need only diﬀerentiate with

∂v respect to y, and with v = x, ∂y = 0. We have established the following process for computing J in the semisymmetric, left symmetric, and totally symmetric cases:

∂u −1 1. Determine R(e\y) ∂y (q1, q2) R(e\x) .

2. Express the generators of (1) in terms of the group generators Re(q, r),Le(q, r),

Te(q) of the previous Section.

Whenever x = e, or y = e, we can ignore R(e\e)±1 because these are units in the group ring

ZU(Q; H)e, so they make no contribution to the ideal.

3.3.1.1 Semisymmetric quasigroups

Recall U(Q; P)e is free on the set

# {Re(e, e),Re(q, p),Te(q) | (q, p) ∈ Q × Q, qp 6= e}, (3.3.2) and that Theorem 2.2.3 dictates J is generated by elements of the form

R(ye) R(x)R(y) + R(yx)−1 R(xe)−1. (3.3.3) 42

Deﬁne S = {(y, x, yx) | x, y ∈ Q#, yx 6= e}, and let E denote the set of orbits of S under the action

g of the order-3 cyclic group G = h(1 2 3)iS3 given by (x1, x2, x3) = (x1g, x2g, x3g) for all g ∈ G. We will now work out the various generators for the ideal J. Begin by assuming that e2 = e.

If, in (3.3.3), we have x = y = e, then –after we remove factors of R(e2)±1 = R(e)±1, we have the

2 −1 2 −1 3 3 generator R(e) + R(e) , which is the same as R(e)(R(e) + R(e) ) = R(e) + 1 = Re(e, e) + 1. Consider, now, the case where x 6= e, and y = e. Then, once again, remove R(e2) = R(e), and

(3.3.3) becomes

R(x)R(e) + R(ex)−1 R(xe)−1 = R(x)R(e)R(xe)−1 + (R(xe)R(ex))−1

−1 −1 = R(ex · e)R(e)R((ex · e)e) + Te(x)

−1 = Re(ex, e) + Te(x) .

2 Note that Re(ex, e) is part of (3.3.2), as e = e implies ex 6= e, and ex · e = x 6= e. We will now exhibit a one-to-one correspondence between generators coming from the choices

−1 x = e, and y 6= e, and the generators of the form Re(ex, e) + Te(x) obtained above. Indeed, (3.3.3) is expressed as

R(ye) R(e)R(y) + R(ye)−1 = R(ye)R(e)R(y) + 1

= (R(ye)R(e)R(ye · e)−1)(R(ye · e)R(e · ye)) + 1

= Re(y, e)Te(ye) + 1,

−1 which we could right multiply by Te(ye) to obtain the equivalent generator

−1 Re(y, e) + Te(ye) . (3.3.4)

Let x ∈ Q#. There is a unique element of Q for which x = ye, but since e2 = e, there is a unique y ∈ Q# for which x = ye. Similarly, for every y ∈ Q# there is a unique x ∈ Q# for which y = ex.

Thus, for all of (x, y) ∈ {e} × Q# and (x, y) ∈ Q# × {e}, we have a non-redundant generator of the form (3.3.4), for each y ∈ Q#. 43

It is readily veriﬁed that choosing any (x, y) ∈ Q#2 such that yx = e also yields a generator of the form (3.3.4). Thus, we are left to consider cases for which (x, y) ∈ Q#2, and yx 6= e. In fact,

R(ye) R(x)R(y) + R(yx)−1 R(xe)−1

=R(ye)R(x)R(y)R(xe)−1 + R(ye)R(yx)−1R(xe)−1

= R(ye)R(x)R(yx · e)−1 R(yx · e)R(y)R((yx · y)e)−1 + R((x · yx)e)R(yx)−1R(xe)−1

−1 =Re(y, x)Re(yx, y) + Re(x, yx) , and these are all free group generators, as x = yx · y, y = x · yx, and yx are all in Q#.

If we assume e2 6= e, then while the cases above must be altered, the nature of the ideal changes little. The only diﬀerence in generators comes from the case (x, y) = (e, e), and now,

2 2 −1 2 2 2 −1 2 −1 R(e) + R(e ) = R(e · e)R(e · e ) + R(e ) = Te(e ) + Re(e, e) . The remaining cases to consider are (e, e2); (e2, e) (which end up producing generators equivalent

2 −1 2 2 #2 to Te(e ) + Re(e, e) ); (e, y), y ∈ Q r {e, e };(x, e), x ∈ Q r {e, e };(x, y) ∈ Q , yx = e (these all produce (3.3.4)-type generators); and (x, y) ∈ Q#2, yx 6= e, and these generators match those of the last case for when Q is a pique. We have proved the following:

Theorem 3.3.2. Let Q be a nonempty semisymmetric quasigroup containing the element e. Deﬁne

3 X1 = {Re(e, e) + 1},

2 X2 = {Re(q, e)Te(qe) + 1 | q ∈ Q r {e }},

0 # X2 = {Re(q, e)Te(qe) + 1 | q ∈ Q },

X3 = {Re(q, p)Re(qp, q)Re(p, qp) + 1 | (q, p, qp) ∈ E} of ZU(Q; P)e.

2 (a) If e = e, then ZPQ is the quotient of the free group ring on (3.3.2) by the ideal generated by 0 X1 ∪ X2 ∪ X3.

2 (b) If e 6= e, then ZPQ is the quotient of the free group ring on (3.3.2) by the ideal generated by

X2 ∪ X3. 44

3.3.1.2 Totally symmetric quasigroups

Since we can use the semisymmetric and commutative identities as an equational basis for TS, a description of ZTSQ follows naturally from the previous section. In keeping with the notation of the previous section, we will use Re(q, p) for stabilizer elements, which is all the same, since L(q) = R(q) in TS. What’s more, only semisymmetry contributes to the ideal, so we need only examine generators of the form (3.3.3), with the added structure of commutativity forcing all

2 maps to be involutions. Furthermore, Te(q) = R(qe)R(eq) = R(qe) = 1Q[X] for all q ∈ Q, and R(e, e2) = R(e2)R(e2)R((e · e2)e) = R(e2)3 = R(e2). The next result, then, is a commutative specialization of Theorem 3.3.2. The reader should also recall the set

Theorem 3.3.3. Let Q be a nonempty totally symmetric quasigroup containing the element e.

2 X1 = {Re(q, e) + 1 | q = e ∨ q = e ∨ e ∈ I(q)},

X2 = {Re(q, p)Re(qp, q)Re(p, qp) + 1 | (q, p, pq) ∈ F}.

Then ZTSQ is the quotient of the group ring over (3.2.17) by the ideal generated by X1 ∪ X2.

Proof. Because right multiplication by any element is an involution, when x = y = e in (3.3.3),

2 2 we get R(e ) + 1 = Re(e, e) + 1, whether e = e , or not. The other conditions on members of X1 ensure that Re(q, e) is actually a member of the basis of Theorem 3.2.9. The same goes for X2.

3.3.1.3 Left symmetric quasigroups

For left symmetric quasigroups, \ = ·, so that (3.3.1) is equal to

R(ey)(R(yx) + R(x)L(y)) R(ex)−1 (3.3.5) in U(Q; LS)e. Once again, we will split our arguments into the case where e2 = e, and that where e2 6= e. We begin with the former. When x = y = e in (3.3.5), we have

R(e)(R(e) + R(e)L(e))R(e)−1 = R(e)2(1 + L(e))R(e)−1, (3.3.6) 45

2 but since R(e) ∈ U(Q; LS)e, 1 + L(e) = 1 + L(e ) = 1 + Le(e, e/e) is in our generating set. With y 6= e and x = e, we obtain the generator

2 −1 −1 R(ey)(R(ye) + R(e)L(y)) R(e ) = Re(y, ye) + Re(y, e)Te(ye)Re(e, e) , (3.3.7) and all of these are in the free group basis, as ye 6= e2 = e. Next, if y = e, but x 6= e,(3.3.5) simpliﬁes to

−1 1 + Te(ex)Le(ex, e)Te(x) . (3.3.8)

−1 2 But ex 6= e, so we make the substitution 1 + Te(y)Le(y, e)Te(ey) . We know e = e 6= y/y, so

Le(y, e) cannot be an involution, but in order for it to not be a free group generator, e = e/e = e/y, a contradiction. If y 6= e, x 6= e, and yx = e, then (3.3.5) is equivalent to R(ey)R(e)R(ex)−1 +

R(ey)R(x)L(y)R(ex)−1 = R(ey)R(e)R(e · ye)−1 + R(ey)R(x)R(e2)−1 R(e2) L(y)R(ex)−1 =

−1 Re(y, e) + Re(y, x)Re(e, e)Te(x) , and when yx 6= e (3.3.5) is given by

−1 Re(y, yx) + Re(y, x)Te(yx)Le(yx, y)Te(x) , (3.3.9) and Le(yx, y) is a basis element because y = e/(yx) implies x = e. Now, assume e2 6= e. Then (3.3.6) is instead

2 Re(e, e) + Te(e ), (3.3.10)

while (3.3.7) will not consist of basis elements if y = e/e, but in this instance, we get 1 + Le(e, e/e)

(after recognizing eR(e(e/e))R(e) = (e(e(e/e))e = (e/e)e = e, so R(e(e/e))R(e) ∈ U(Q; LS)e.

2 −1 With (3.3.8), we could have ex = e, but this becomes 1 + Re(, e)Te(e ) , a generator we already covered. Le(ex, e) will not be a basis element if e = e/(ex), implying e · ex = e, but e · ex = x, contradicting our assumption x 6= e. In the above cases where (x, y) ∈ (Q#)2, we did not have to worry about whether e2 = e or not. 46

Theorem 3.3.4. Let Q be a nonempty, left symmetric quasigroup containing the element e. Deﬁne

X1 = {1 + Le(e, e/e)},

0 2 X1 = {Re(e, e) + Te(e )}

−1 # X2 = {Re(q, qe) + Re(q, e)Te(qe)Re(e, e) | q ∈ Q r {e/e}},

−1 # X3 = {1 + Te(q)Le(q, e)Te(eq) | q ∈ Q , eq 6= e ∧ e = q/q ∨ e ∈ I(q) },

−1 X4 = {Re(q, e) + Re(q, p)Re(e, e)Te(p) | qp = e},

−1 # 2 X5 = {Re(q, qp) + Re(q, p)Te(qp)Le(qp, q)Te(p) | (q, p) ∈ (Q ) , qp 6= e}.

2 (a) If e = e, then ZLSQ is isomorphic to the quotient of the group ring over (3.2.15) by the S5 ideal generated by i=1 Xi.

2 (b) If e 6= 2 then ZLSQ is isomorphic to the quotient of the group ring over (3.2.15) by the ideal 0 S5 generated by X1 ∪ i=2 Xi.

3.3.2 Idempotent, H-symmetric quasigroups

Adding idempotence to a given H-symmetric variety supplements its rings of representation with a remarkable amount of structure. Indeed, we’ll demonstrate that for any nontrivial subgroup

H ≤ S3, and idempotent, H-symmetric quasigroup Q, the ring ZHIQ is a free product of principal ideal domains and free group algebras, and when the quasigroup in question is a combinatorial design, the indexing of free products comes from the block structure.

If Q is idempotent, then the ideal contains generators

R(e\x)(R(x) + L(x) − 1) R(e\x)−1 = R(e\x)R(x)R(e\x)−1 + R(e\x)L(x)R(e\x)−1 − 1

2 −1 = Re(x, x) + Te(x)Le(x, x)Te(x ) − 1

−1 = Re(x, x) + Te(x)Le(x, x)Te(x) − 1.

Thus, if Q is an idempotent quasigroup, then ZIQ is the quotient of the free group ring on the set (2.1.3)-(2.1.5), modulo the relations 47

−1 Re(x, x) = 1 − Te(x)Le(x, x)Te(x) , (3.3.11) for every x ∈ Q.

In the structure theorems below, we will appeal to the following lemma:

Lemma 3.3.5. Consider the free group algebras Shgi, Shx, yi and Shu, v, wi. We have the following

S-algebra isomorphisms:

(a) Shx, yi/(xy + 1) =∼ Shgi

(b) Shu, v, wi/(uvw + 1) =∼ Shx, yi.

Proof. (a)

Let A = Shx, yi/(xy + 1). We demonstrate that A and the homomorphism ι : hgi → A; g 7→ x, where x indicates passage through the projection Shx, yi A, possess the universal property of ∼ × Shgi and its corresponding inclusion. Namely, we exhibit a bijection AlgS(A, −) = Gp(hgi, (−) ), where (−)× assigns the group of units to each S-algebra.

Suppose ϕ : hgi → R×; g 7→ r. Deﬁne θ : Shx, yi → R to be the unital algebra map given by x 7→ r and y 7→ −r−1. Clearly, (xy + 1)θ = 0, so there is a unique θ : A → R for which

(gι)θ = xθ = r = gϕ. This, of course, requires ιθ = ϕ.

To verify uniqueness of θ, assume ψ : A → R is a unital algebra homomorphism for which

ιψ = ϕ. It follows xψ = (gι)ψ = gϕ = r = (gι)θ = xθ. Since {x±1, y±1} is an S-algebra generating set for Shx, yi, and y = −x−1, {x±1} generates A, meaning we have proven ψ = θ.

(b)

The argument for this case is identical. If

× ϕ : hx, yi → R ; x 7→ r1, y 7→ r2, 48 then the map θ : Shu, v, wi/(uvw + 1) → R assigning

u 7→ r1

v 7→ r2

−1 w 7→ −(r1r2) exhibits the desired correspondence.

3.3.2.1 Idempotent, commutative quasigroups

Let Q be nonempty, idempotent, and commutative so that U(Q; C)e is free on

# {Re(e, e),Re(q, r), Te(q) | (q, r) ∈ Q × Q, qr 6= e}.

That R(q) = L(q) for all q ∈ Q means that (3.3.11) becomes

2Re(x, x) − 1 = 0. (3.3.12)

Theorem 3.3.6 (Structure Theorem for ZCIQ). Let Q be a nonempty, commutative, idempotent # quasigroup. Deﬁne the set S = {(q, r) ∈ Q × Q | q 6= r, qr 6= e}. Then ZCIQ is isomorphic to the free product

a 1 a a ∗ hxi ∗ hxi, (3.3.13) Z 2 Z Z Q Q# S 1 k where Z 2 is the localization of Z at the monoid {2 | k ≥ 0} ⊂ Z.

# 2 Proof. Note that for any q ∈ Q , q = q 6= e, so Re(q, q) is in the universal stabilizer for all q ∈ Q.

# Hence, U(Q; CI)e is isomorphic to the free group on the disjoint union Q + S + Q , where the S

# corresponds to the Re(q, r)’s for which q 6= r, and Q corresponds to the Te(q)’s. The relations generating the ideal are in one-to-one correspondence with the free group generators coming from

∼ −1 ∼ 1 Q, and ZhRe(q, q)i/(2Re(q, q) − 1) = Z[X,X ]/(2X − 1) = Z 2 . 49

3.3.2.2 Idempotent, semisymmetric quasigroups

We require a technical lemma regarding the relationship between the blocks of an MTS Q, and the universal stabilizer U(Q; MTS)e.

Lemma 3.3.7. Let Q be a nonempty, idempotent, semisymmetric quasigroup, with corresponding MTS, (Q, B). Select a point e ∈ Q, and let B# denote the set of blocks not containing e.

# # For each q ∈ Q , deﬁne Xq = {Re(q, e),Te(qe)}, and whenever (q, p, qp) ∈ B , let Y(q,p,qp) =

{Re(q, p),Re(qp, q),Re(p, qp)} Then the free group basis (3.3.2) for U(Q; MTS)e is equal to the disjoint union

X X {Re(q, q) | q ∈ Q} + Xq + Y(q,p,qp). (3.3.14) q∈Q# (q,p,qp)∈B#

Proof. First, we show disjointedness of the various sets. It is clear that Xq ∩ Xp 6= ∅ if and only if p = q. Next, consider Re(q, p) ∈ Y(q,p,qp) ∩ Y(a,b,ab). If Re(q, p) = Re(a, b), then clearly,

(q, p, qp) = (a, b, ab), as a = q, and b = p. If Re(q, p) = Re(ab, a), then q = ab, and p = a, so qp = ab · a = b. Thus, (a, b, ab) = (p, qp, q), but the triple (p, qp, q) falls into the same block- equivalence class as (q, p, qp). If Re(q, p) = Re(b, ab), then q = b, and p = ab, so qp = b · ab = a, meaning (a, b, ab) = (qp, q, p), which, once again, lies in the same block-equivalence class as

(q, p, qp). Similar arguments may be applied to the assumptions Re(qp, q) ∈ Y(q,p,qp) ∩ Y(a,b,ab), and

Re(p, qp) ∈ Y(q,p,qp) ∩ Y(a,b,ab). Conclude Y(q,p,qp) ∩ Y(a,b,ab) 6= ∅ if and only if (q, p, qp) = (a, b, ab).

# It is clear that the construction of B forces the Xq’s to be disjoint from the Y(q,p,qp)’s, and that these are all disjoint from the singleton {Re(e, e)}.

Finally, we prove that (3.3.14) is equal to (3.3.2). Clearly, Re(e, e) and all Te(q)’s appear in

# 2 (3.3.14). Let (q, p) ∈ Q × Q so that qp 6= e. If p = e, then e = e forces q 6= e, so Re(q, p) ∈ Xq. If

#2 # (q, p) ∈ Q , q 6= p, and qp 6= e, then one of (q, p, qp), (qp, q, p), or (p, qp, q) is in B , and Re(q, p) belongs to Y(q,p,qp) = Y(qp,q,p) = Y(p,qp,q).

Proposition 3.3.8. Let Q be a nonempty Mendelsohn quasigroup containing the element e, and

# set Q = Q r {e}. With (Q, B) denoting the MTS associated with the quasigroup structure, use 50

B# to denote the set of blocks in B not containing the point e. Consider

2 X1 = {Re(x, x) − Re(x, x) + 1 | x ∈ Q}

# X2 = {Re(x, e)Te(xe) + 1 | x ∈ Q }

# X3 = {Re(x, y)Re(xy, x)Re(y, xy) + 1 | (x y xy) ∈ B }, subsets of ZU(Q; MTS)e. Then ZMTSQ is the quotient of the free group on (3.3.2) by the ideal generated by X1 ∪ X2 ∪ X3.

Proof. For x ∈ Q, we have

−1 −1 −1 R(xe)(R(x) + R(x) − 1)R(xe) = Re(x, x) + Re(x, x) − 1, accounting for all generators (3.3.11). Furthermore,

−1 2 Re(x, x)(Re(x, x) + Re(x, x) − 1) = Re(x, x) − Re(x, x) + 1. (3.3.15)

Thus, X1 corresponds to all generators arising from idempotence. Next, we reﬁne the set (3.3.1) by considering cases for (x, y) ∈ Q2. If x = y, then

R(ye)(R(x)R(y) + R(yx)−1)R(xe)−1 = R(xe)(R(x)2 + R(x2)−1)R(xe)−1

= R(xe)(R(x)2 + R(x)−1)R(xe)−1

2 −1 = Re(x, x) + Re(x, x)

−1 −1 = (Re(x, x) + 1)(Re(x, x) + Re(x, x) − 1),

and so this case is subsumed by the generators from X1. When (x, y) = (e, y) and y 6= e, we have

R(ye)(R(e)R(y) + R(ye)−1)R(e2)−1 = (R(ye)R(e)R(y) + 1)R(e)−1,

but R(e) ∈ U(Q; MTS)e, so this case reduces to elements of the form

R(ye)R(e)R(y) + 1 = R(ye)R(e)R(ye · e)−1R(ye · e)R(e · ye) + 1 51

= Re(y, e)Te(ye) + 1, (3.3.16) which are in one-to-one correspondence with X2. Considering (x, y) = (x, e) for x 6= e leads to generators

−1 Re(ex, e) + Te(x) . (3.3.17)

For each x ∈ Q#, there is a unique y ∈ Q# so that ye = x, and we can write (3.3.17) as

−1 −1 Re(e(ye), e) + Te(ye) = Re(y, e) + Te(ye) , (3.3.18) but multiplying (3.3.18) on the right by Te(ye) reveals (3.3.16) to be a repeated instance of (3.3.16)- type elements.

Finally, take (x, y) ∈ Q# × Q# with x 6= y. Notice

R(ye)R(x)R(y)R(xe)−1 = R(ye)R(x)R(yx · e)−1 R(yx · e)R(y)R(xe)−1

= Re(y, x)Re(yx, y), so that

−1 −1 −1 R(ye)(R(x)R(y) + R(yx) )R(xe) = Re(y, x)Re(yx, y) + Re(x, yx) . (3.3.19)

If yx = e, then Re(yx, y) is not a U(Q; MTS)e-basis element, but because yx = e =⇒ y = xe =⇒ x = ey,

−1 −1 Re(y, x)Re(yx, y) + Re(x, yx) = R(ye)R(ey) + Re(ey, e)

−1 = Te(y) + Re(ey, e) , which are generators covered by X2. However, when yx 6= e, every element on the right-hand side of (3.3.19) is a U(Q; MTS)e-basis element, and such generators are equivalent to those given by

X3.

Theorem 3.3.9 (Structure Theorem for ZMTSQ). Let Q be a nonempty, semisymmetric, idempotent quasigroup, with associated MTS (Q, B). Deﬁne B# to be the set of all blocks not containing e. Then ZMTSQ is isomorphic to the free product 52

a a a Z[ζ] ∗ Zhxi ∗ Zhx, yi, (3.3.20) Q Q# B# 2 where Z[ζ] = Z[X]/(X − X + 1) is the ring of Eisenstein integers.

Proof. This follows from Lemmas 3.3.5, 3.3.7 and the previous Proposition.

3.3.2.3 Idempotent, left symmetric quasigroups

A completely abstract structure theorem for ZLSIQ evades us. We will provide a idempotent version of Theorem 3.3.4. The left-symmetric form of R(e\x)(R(x) + L(x) − 1) R(e\x)−1 is

−1 Re(x, x) + Te(x)Le(x, x)Te(x) − 1. (3.3.21)

Looking to X5 from Theorem 3.3.4, when x = y, we have

−1 1 + Te(x)Le(x, x)Te(x) , (3.3.22)

and these generators are equivalent to 1 + Le(x, x). Then (3.3.21) becomes Re(x, x) − 2. Since

Re(e, e) is identiﬁed with 2 in the quotient, X2 can now be expressed as

1 R (y, ye) + R (y, e)T (ye) ≡ 2R (y, ye) + R (y, e)T (ye). e e e 2 e e e

Sets X3 and X5 carry through.

Theorem 3.3.10. Let Q be a nonempty idempotent, left symmetric quasigroup. Deﬁne

# X1 = {2Re(q, qe) + Re(q, e)Te(qe) | q ∈ Q }

−1 # X2 = {1 + Te(eq)Le(eq, e)Te(q) | q ∈ Q }

−1 # 2 # X3 = {Re(q, qp) + Re(q, p)Te(qp)Le(qp, q)Te(p) | (q, p) ∈ (Q ) r Qd}.

∼ Then ZLSIQ = K1 ∗ K2/I, where

a a 1 K = ∗ , (3.3.23) 1 Z Z 2 Q Q# 53

K2 is the free group ring on the basis

# {Te(q),Re(q, qp) | (q, p) ∈ Q × Q, q 6= p} + {Le(q, p) | (q, p, pq) ∈ E}, (3.3.24)

and I is the two-sided ideal generated by X1 ∪ X2 ∪ X3.

Proof. To begin,

±1 ±1 ±1 ∓1 ±1 ∓1 ∼ ±1 ±1 ∓1 hR ,L | L L − 1,R R − 1,L + 1,R − 2iRing = hL | L L − 1,L + 1iRing

±1 ±1 ∓1 ∗ hR | R R − 1,R − 2iRing 1 =∼ ∗ . Z Z 2

1 # The index set for Z 2 is Q because Re(e, e) is involved in other relations. The set (3.3.24) is simply all the generators of U(Q; LSI)e that are not Re(q, q),Le(q, q), and the Xi’s come from Theorem 3.3.4.

3.3.2.4 Steiner quasigroups

We conclude the chapter by proving that ZSTSQ is isomorphic to (3.0.4).

Theorem 3.3.11 (Structure Theorem for ZSTSQ). Let Q be a nonempty Steiner quasigroup with corresponding STS (Q, B). The ring ZSTSQ is isomorphic to

a a a F3 ∗ Z ∗ Zhx, yi. (3.3.25) Q Q# B#

Proof. A Steiner triple system is a Mendelsohn triple system, so the indexing is going to match

# (3.3.20) (note that there is a one-to-one correspondence between Q and Be). The generators coming from idempotence are 2Re(x, x) − 1, and those coming from semisymmetry when x = y are

Re(x, x) + 1. Recall that Te(x) = 1 in the totally symmetric case. Hence,

∼ a a a ZSTSQ = Zhxi/(2x − 1, x + 1) ∗ Zhxi/(x + 1) ∗ Zhx, y, zi/(xyz + 1). Q Q# B# 54

Table 3.1 Rings associated with {e}, regarded as an H-symmetric quasigroup

H ZH{e} Q ZhR,Li ±1 C Z[R ] ±1 3 P Z[R ]/(R + 1) ±1 LS Z ∗ Z[R ] TS Z

Table 3.2 Rings associated with {e}, regarded as an idempotent, H-symmetric quasigroup

HI ZHI{e} QI ZhR,Li/(R − (1 − L)) 1 CI Z 2 MTS Z[ζ] 1 LSI Z ∗ Z 2 STS F3

Now, 2x − 1 = x + 1 = 0 leads to 0 = −3, accounting for F3 appearing in our coproduct decomposition. It is straightforward to verify Zhxi → Z; x 7→ −1 is a surjective map with kernel (x + 1). Finally, each ring coming from B# is the free group ring on two generators by Lemma 3.3.5. 55

CHAPTER 4. AFFINE MENDELSOHN TRIPLE SYSTEMS AND THE EISENSTEIN INTEGERS

A paper submitted to The Journal of Combinatorial Designs

Alex W. Nowak

4.1 Abstract

McCourt, Oprˇsal, and Stanovský). As a consequence, all entropic MTS and distributive MTS of order coprime with 3 are classified. The classification is accomplished via the representation theory

4.2 Introduction

4.2.1 The isomorphism problem for aﬃne MTS

A quasigroup (Q, ·, /, \) consists of a set Q –in this paper, we will assume Q is nonempty– equipped with three binary operations: · denoting multiplication, / right division, and \ left division; these operations adhere to the following identities:

y\(y · x) = x; (4.2.1)

y · (y\x) = x; (4.2.2) 56

(x · y)/y = x; (4.2.3)

(x/y) · y = x. (4.2.4)

Remark 4.2.1. This paper will adopt the following conventions:

(a) We convey multiplication of quasigroup elements by concatenation, and in some instances,

use both · and concatenation. Concatenation takes precedence over ·. For example, xy · z ≡

(xy)z ≡ (x · y) · z.

(b) Arguments will appear to the left of functions. For example, the image of a point x under

the function g is written xg, or even as xg. Exponentiation of an argument by its function

can help to curb the proliferation of brackets to which works in nonassociative algebra are

susceptible.

Mendelsohn quasigroups are speciﬁed by the additional axioms of idempotence

x2 = x, (4.2.5) and semisymmetry

x · yx = y. (4.2.6)

A ﬁnite Mendelsohn quasigroup Q gives rise to a Mendelsohn triple system (MTS) (Q, B), where B is a collection of blocks, or cyclic triples of points from Q, satisfying the property that for any ordered pair of distinct points (x, y) ∈ Q2, there is a unique block (x y xy) ≡ (xy x y) ≡

(y xy x) ∈ B. Conversely, any Mendelsohn triple system (Q, B) corresponds to a ﬁnite Mendelsohn quasigroup on Q: let x2 = x for any x ∈ Q, and xy be the element in the block containing (x, y) when x 6= y (cf. [37, Th. 1]). Divisions in the Mendelsohn quasigroup are given by the opposite of multiplication: x/y = x\y = yx. That is, every Mendelsohn quasigroup takes the form (Q, ·, ◦, ◦), where x ◦ y = y · x. We use the terms MTS and Mendelsohn quasigroup interchangeably, as all quasigroups under consideration are ﬁnite.

When studying a particular class of idempotent quasigroups, it is often helpful to examine those which are aﬃne over abelian groups. 57

Deﬁnition 4.2.2. Let (M, +) be an abelian group. Fix R ∈ Aut(M, +). The operation

xy = xR + y(1 − R) (4.2.7) endows M with the structure of an idempotent quasigroup, Aff(M,R). We say that Aff(M,R) is affine over (M, +).

The isomorphism problem for idempotent, aﬃne quasigroups is made concrete by the following.

Theorem 4.2.3 ([53]). Let (M, +) be an abelian group. The idempotent, aﬃne quasigroups

Aﬀ(M,R1) and Aﬀ(M,R2) are isomorphic if and only if there is some ψ ∈ Aut(M, +) such that

−1 ψ R1ψ = R2.

It is well-known that all aﬃne, commutative Mendelsohn quasigroups (Steiner triple systems)

n are of the form Aﬀ((Z/3) , −In), where In is the n × n identity matrix over Z/3 [50, Th. 11.5]. A noncommutative version of this comes from Donovan et al. [15].

Proposition 4.2.4 ([15]). An idempotent, aﬃne quasigroup Aﬀ(M,R) is Mendelsohn if and only if R is annihilated by the polynomial X2 − X + 1.

Remark 4.2.5. The polynomial, X2 − X + 1 will be abbreviated to f(X). As a historical note, we think it is appropriate to give credit to Sade, whose work in [46] –to the best of our knowledge– has gone unnoticed in the literature on Mendelsohn triple systems. This paper seems to mark the

ﬁrst instance in which someone recognized the signiﬁcance of f(X) in constructing idempotent, semisymmetric quasigroups.

By the Chinese Remainder Theorem, the isomorphism problem for aﬃne MTS is equivalent to the following:

Given an abelian group (M, +) of prime-power order, describe the automorphisms of (M, +) that are annihilated by f(X), up to conjugacy.

In [15], the authors classify aﬃne Mendelsohn quasigroups of orders p and p2, for any prime p.

They also enumerate, using GAP [22], isomorphism classes for prime powers less than 1000. We 58 will extend the results of Donovan et al. to all prime powers, excluding powers of 3. This latter case will be referred to as the ramiﬁed case. This terminology comes from the prime ideal structure

2 of the ring of Eisenstein integers, Z[X]/(X − X + 1).

4.2.2 Aﬃne MTS, entropicity, and distributivity,

Here, we simply state some universal algebraic interpretations of our results. For a thorough account of the interactions between linearity, the distributive law, and the entropic law in Mendelsohn triple systems, one should consult the introduction of [15].

By the Toyoda-Bruck-Murdoch Theorem [56, 5, 40], a Mendelsohn triple system is aﬃne over an abelian group if and only if it is entropic, meaning that its multiplication adheres to

ux · yz = uy · xz. (4.2.8)

This identity is also referred to as the medial law (cf. [13, 28, 53]). Hence, Problem 4.2.1 is equivalent to the isomorphism problem for entropic Mendelsohn quasigroups.

Because of idempotence, it is readily seen that entropic MTS also satisfy the left and right distributive laws:

x · yz = xy · xz; (4.2.9)

xy · z = xz · yz. (4.2.10)

The Fischer-Smith-Galkin Theorem [20, 48, 21] is a result on the structure of distributive quasigroups. It states that a distributive quasigroup decomposes as a direct product of prime-power order subquasigroups in such a way that any factor that is not aﬃne over an abelian group has order 3n. Therefore, the results of Section 4.5 can be regarded as a solution to the isomorphism problem for distributive Mendelsohn triple systems having order coprime with 3. 59

4.3 Background on the Eisenstein integers

This section collects classical results from number theory and representation theory. For a complete treatment of the structure of ﬁnitely generated modules over a principal ideal domain, see [17]. We use [23, 26] as sources for the prime structure of the Eisenstein integers.

4.3.1 Modules over the Eisenstein integers

2 2πi/6 The Eisenstein integers, Z[X]/(X − X + 1), may also be speciﬁed as Z[ζ], where ζ = e ; that is, it is the free abelian group on the basis {1, ζ}, inheriting Z-algebra structure from the complex numbers.

There is a faithful representation in M2(Z) given by

  a −b   a + bζ 7→   . (4.3.1) b a + b

In fact,   a −b   2 2 (a + bζ)ν := det   = a + ab + b (4.3.2) b a + b gives Z[ζ] the structure of a Euclidean domain [26, Prop. 1.4.2]. Therefore, the Eisenstein integers form a principal ideal domain (PID), and their ﬁnitely generated modules decompose as a direct sum of cyclic modules. We will give the elementary divisors version of the structure theorem for

ﬁnite Z[ζ]-modules.

Theorem 4.3.1. A ﬁnite Z[ζ]-module M is isomorphic to a direct sum n M ri Z[ζ]/(πi ), (4.3.3) i=1

r1 rm where each πi is prime in Z[ζ]. The elementary divisors of M, π1 , . . . , πm , are unique, up to multiplication by units.

The elementary divisors representation breaks up a ﬁnite Z[ζ]-module into indecomposable repre- ∼ sentations. A module (over any commutative ring) M is indecomposable if whenever M = M1 ⊕M2 60

is a module isomorphism, either M1 = 0, or M2 = 0. What’s more, this decomposition is essentially ∼ m unique, meaning that if M = ⊕i=1Ni, and each summand is indecomposable, then m = n, and

∼ ri there is a permutation σ ∈ Sn such that for all 1 ≤ i ≤ n, N(i)σ = Z[ζ]/(πi ). × The group of units of the Eisenstein integers, Z[ζ] = {±1, ±ζ, ±ζ}, consists of the sixth roots × of unity in C . Primes in Z[ζ] belong to, up to association by units, three pairwise disjoint classes [26, Prop. 9.1.4]:

1. π, where πν is prime in Z, and πν ≡ 1 mod 3;

2. p, where p is prime in Z, and p ≡ 2 mod 3;

3.1+ ζ.

If p ≡ 1 mod 3 is a prime integer, we say that it splits in Z[ζ] because there is an Eisenstein prime from class (1), π, such that p = ππ, where π is the complex conjugate of π. These class (1) primes

π are always irrational (nonzero scalar on ζ). Since primes congruent to 2 modulo 3 remain prime in Z[ζ], we say they are inert, and moreover, that they are rational Eisenstein primes. Finally, 3 is −1 ramiﬁed in Z[ζ] because it is associated to the square of an Eisenstein prime via the unit ζ ; that is, 3 = ζ−1(1 + ζ)2. This terminology will appear in later sections.

Theorem 4.3.2 is a combination of results from [6, 38].

Theorem 4.3.2. Let n ≥ 1. We may partition the collection of primary ideals in Z[ζ] into four classes. These, along with their respective quotient rings are summarized below.

(a) Suppose π is an irrational prime. That is, p = πν ≡ 1 mod 3 is prime in Z. Then n ∼ Z[ζ]/(π ) = Z/pn .

n ∼ (b) Suppose p ≡ 2 mod 3 is a rational prime in Z[ζ]. Then Z[ζ]/(p ) = Z/pn [ζ]

n ∼ (c) Suppose n = 2k is even. Then Z[ζ]/((1 + ζ) ) = Z/3k [ζ].

(d) Suppose n = 2k + 1 is odd. Then

n ∼ k+1 k 2 Z[ζ]/((1 + ζ) ) = Z[X]/(3 , 3 X,X − X + 1). 61

4.4 A direct product decomposition for aﬃne MTS

The structure of a Z[ζ]-module is encompassed entirely by an ordered pair (M,R), where (M, +) is an abelian group, and R ∈ Aut(M, +) represents the action of ζ (the action by ζ has to be invertible because ζ is a unit in Z[ζ]). Note that Aff(M,R) is an affine MTS if and only if (M,R) is a Z[ζ]-module. It is the goal of this section to provide more details regarding this correspondence. The main result is that affine MTS have a direct product decomposition that mirrors the elementary divisors decomposition of the corresponding Z[ζ]-module. Recall our definition of indecomposable modules in Section 4.3. We may similarly define an affine ∼ quasigroup Aff(M,R) to be indecomposable if whenever Aff(M,R) = Aff(M1,R1) × Aff(M2,R2), either M1 = 0 or M2 = 0. We will show that any affine MTS decomposes into a direct product of indecomposable quasigroups, and that this factorization is essentially unique.

Lemma 4.4.1. Let g :(M,R) → (P,L) be a Z[ζ]-module homomorphism. Then g is also a quasigroup homomorphism between Aﬀ(M,R) and Aﬀ(P,L).

Proof. Since R and L represent the respective actions of ζ,(xR)g = (xg)L for all x ∈ M. Thus,

(xy)g = (xR + y(1 − R))g

= (xR)g + yg − (yR)g

= (xg)L + yg − (yg)L

= (xg)L + (yg)(1 − L)

= (xg)(yg) whenever x, y ∈ M.

Lemma 4.4.2. Let (M, +) be an abelian group and R,L ∈ Aut(M, +) be annihilated by f(X).

Then (M,R) =∼ (M,L) if and only if Aﬀ(M,R) =∼ Aﬀ(M,L).

Proof. ( =⇒ ) This follows from Lemma 4.4.1. 62

( ⇐= ) By Theorem 4.2.3 there is an abelian group automorphism ψ for which L = ψ−1Rψ.

Then ψ intertwines with the action of ζ: for all x ∈ M, we have (xζ )ψ = ((x)R)ψ = ((x)ψ)L =

ζ (x)ψ . Thus, ψ is an isomorphism of Z[ζ]-modules.

Lemma 4.4.3. Let (M1 ⊕ M2,R1 ⊕ R2) represent the direct sum of Z[ζ]-modules (M1,R1) and

(M2,R2). Then ∼ Aff(M1 ⊕ M2,R2 ⊕ R2) = Aff(M1,R1) × Aff(M2,R2). (4.4.1)

Proof. We show that Aﬀ(M1 ⊕ M2,R2 ⊕ R2) has the universal property of the direct product in the category of quasigroup homomorphisms. By Lemma 4.4.1, the canonical projections πi :

(M1 ⊕ M2,R1 ⊕ R2) → (Mi,Ri) are quasigroup homomorphisms. Suppose Q is a quasigroup, and that for i = 1, 2, gi : Q → Aﬀ(Mi,Ri) are homomorphisms. Since the set M1 ⊕ M2 is just the

g g direct product of sets M1 and M2, there is a unique function h; q 7→ (q 1 , q 2 ) such that hπi = gi, for i = 1, 2. If we show h is a quasigroup homomorphism, we are done. Indeed, if q, r ∈ Q then

(qr)h = ((qr)g1, (qr)g2)

= ((qg1)(rg1), (qg2)(rg2))

g1 g1 g2 g2 = (q R1 + (r )(1 − R1), q R2 + (r )(1 − R2))

g1 g2 g1 g2 = (q R1, q R2) + (r (1 − R1), r (1 − R2))

g1 g2 g1 g2 = (q , q )(R1 ⊕ R2) + (r , r )(1 − (R1 ⊕ R2))

= (qh)(rh).

Lemma 4.4.4. An aﬃne MTS Aﬀ(M,R) is indecomposable if and only if the Z[ζ]-module (M,R) is indecomposable.

∼ ∼ Proof. By Lemmas 4.4.2 and 4.4.3,(M,R) = (M1 ⊕ M2,R1 ⊕ R2) if and only if Aﬀ(M,R) =

Aﬀ(M1,R1) × Aﬀ(M2,R2). The desired result is a straightforward consequence of this observation. 63

Theorem 4.4.5. Let Aﬀ(M,R) be an aﬃne MTS. Suppose the Z[ζ]-module (M,R) has elementary

r1 rn divisors π1 , . . . , πn . Then n ∼ Y Aﬀ(M,R) = Aﬀ(Mi,Ri) (4.4.2) i=1 ri where Mi denotes the abelian group structure on Z[ζ]/(πi ), and Ri is an automorphism of Mi annihilated by f(X). This is an essentially unique factorization into indecomposable subquasigroups.

Proof. Suppose the elementary divisors decomposition of (M,R) is given by (4.3.3). An isomorphism of (M,R) onto (4.3.3) is a quasigroup isomorphism by Lemma 4.4.1. Since the direct sum of modules commutes with the direct product of quasigroups (cf. Lemma 4.4.3), the aﬃne MTS which we associate with (4.3.3) is isomorphic to the direct product in (4.4.2). The factors are indecomposable by Lemma 4.4.4 and the fact that the elementary divisors decomposition yields indecomposable summands for Z[ζ]-modules. Essential uniqueness of the factorization follows from that of the elementary divisors decomposition, and the fact that, by Lemma 4.4.2, module isomorphisms on the summands correspond to quasigroup isomorphisms on the respective factors.

Now, in order to classify aﬃne MTS, it suﬃces to describe the possible isomorphism classes on

n the abelian groups of the quotient rings Z[ζ]/(π ), where π ∈ Z[ζ] is prime. In this framework, there are four classes of abelian groups to consider, each corresponding to the classes of quotient rings in Theorem 4.3.2:

1. Z/pn , where p ≡ 1 mod 3 is prime,

2 2.( Z/pn ) , where p ≡ 2 mod 3 is prime,

2 3.( Z/3n ) , and

4. Z/3n ⊕ Z/3n+1 (unlike the previous three, this case includes n = 0).

In the next section, we will consider the cases when πν ≡ 1, 2 mod 3. These are our non- ramified cases. Accordingly, an affine MTS whose order is not divisible by 3 is a non-ramified affine MTS. 64

4.5 Aﬃne non-ramiﬁed MTS

4.5.1 Direct product decomposition for non-ramiﬁed MTS

Throughout the rest of the paper, we require the following three lemmas. They each concern f(X) over various rings.

Lemma 4.5.1. Let S be a nonzero (not necessarily commutative), unital ring. Suppose that ζ ∈ S is a root of f(X).

(a) The root ζ is a unit in S, and its inverse is also a root of f(X).

(b) The identity ζ = ζ−1 holds if and only if S is a ring of characteristic 3, and in this case,

ζ = −1.

Proof. (a) Note ζ2 − ζ + 1 = 0 if and only if ζ(1 − ζ) = 1. Thus, ζ−1 = 1 − ζ, and it is easily seen that (1 − ζ)2 − (1 − ζ) + 1 = 0.

(b) If ζ = ζ−1 = 1 − ζ, then 1 = 2ζ. By (a), 22 − 2 + 1 = 0, equivalent to 3 = 0. Conversely, if char(S) = 3, then f(X) = (X + 1)2.

Lemma 4.5.2 ([15]). Let p be a prime, n ≥ 1. Then f(X) ∈ Z/pn [X] has

(a) two distinct roots modulo pn if p ≡ 1 mod 3;

(b) no roots modulo pn if p ≡ 2 mod 3;

Lemma 4.5.3. Let p be a prime congruent to 2 modulo 3, and n ≥ 1. Suppose S is any of the following rings: Z, Z/pn , or Z/3n . For any A ∈ M2(S) such that f(A) = 0, we have 1 = det(A) =

Tr(A). Therefore, f(X) is equal to det(I2X − A), the characteristic polynomial of A.

Proof. We treat S = Z/3 as a special case. Since S is a ﬁeld in this scenario, the Cayley-Hamilton theorem dictates that f(X) must be the characteristic polynomial of A, for it is monic and its degree matches the dimension of A. 65

Now, let   a b   A =   . c d We will establish some facts that hold over any commutative, unital ground ring. By Lemma 4.5.1,

A is invertible and det(A) ∈ S×. Moreover,     d −b 1 − a −b −1   −1   (det(A))   = A = I2 − A =   . (4.5.1) −c a −c 1 − d

Then b = (det(A))−1b, c = (det(A))−1c, and 1 − a = d(det(A))−1. If at least one of b or c is a unit in S, then det(A) = 1. If S is an integral domain and at least one of b or c is nonzero, det(A) = 1.

If det(A) = 1, then 1 − a = d =⇒ Tr(A) = 1. We proceed to a case-by-case argument by contradiction.

Suppose S = Z, a domain. If both b = c = 0, then a and d are integral roots of f(X), but f(X) is irreducible over Z, a contradiction.

Assume S = Z/pn . For b and c to both be non-invertible means that p | b, c. Hence, if we take the entries of A modulo p, we get a diagonal matrix annihilated by f(X), contradicting Lemma

4.5.2(b).

2 Let S = Z/3n for some n ≥ 2. By inspecting the top left entry of f(A), we know a +bc−a+1 ≡ 0 mod 3n, and so a2+bc−a+1 ≡ 0 mod 9. However, if neither b nor c are units, 9|bc, and a2−a+1 ≡ 0 mod 9, contradicting Lemma 4.5.2(c).

±1 Proposition 4.5.4. Let p ≡ 1 mod 3 be prime, and n ≥ 1. Let a be the roots of f(X) in Z/pn . −1 If Aff(Z/pn ,R) is an affine MTS, then it is isomorphic to either Aff(Z/pn , a) or Aff(Z/pn , a ), where a±1 also stand for right multiplication by the corresponding roots.

∼ × Proof. It is well-known that Aut(Z/pn , +) = (Zpn ) . Then Aﬀ(Z/pn ,R) is Mendelsohn if and only if R represents right multiplication by a unit which is also a root of f(X).

Next, we generalize the argument of [15, Th. 2.12] from matrices over ﬁnite ﬁelds to matrices over Z/pn . The presence of zero divisors means that extra care must be taken when addressing characteristic and minimal polynomials of matrices. 66

2 Proposition 4.5.5. Suppose p ≡ 2 mod 3 is prime, and n ≥ 1. Any affine MTS Aff((Z/pn ) ,R) 2 is isomorphic to Aff((Z/pn ) ,T ), where

  0 −1   T =   (4.5.2) 1 1 is the companion matrix of f(X) over Z/pn .

Proof. Let A ∈ M2(Z/pn ) be a matrix representation of R. By Theorem 4.2.3, it suﬃces to show A is similar to (4.5.2). By Lemma 4.5.3, f(X) is the characteristic polynomial of A. We invoke

Theorem 1 in [43], which states that A is similar to the companion of its characteristic polynomial if and only if there is some v ∈ M so that the matrix   v     (4.5.3) vA is invertible; note this result applies to matrices with elements from arbitrary commutative, unital

n rings. Recall Z/pn is a local ring with unique maximal ideal m generated by p + (p ). Taking the ∼ 2 entries of A modulo p gives rise to an action on the quotient M/Mm = (Z/p) , a vector space. This automorphism, we’ll call it A, also has f(X) as its characteristic polynomial, and because f(X) does not split in Z/p, it has no eigenvalues. Hence, {v, vA} is a basis for M/Mm whenever v 6= 0. By Nakayama’s lemma (applied to the special case of local rings), such a basis lifts to a minimal generating set {v, vA} of M, and minimal generating sets for free modules over local rings are bases

[36, Th. 2.3]. Conclude that there is some v 6= 0 for which the matrix (4.5.3) is invertible, and that

A is similar to (4.5.2).

Theorem 4.5.6. Let Q be an aﬃne Mendelsohn quasigroup of order

r1 rk s1 sl n = p1 ··· pk q1 ··· ql , where each pi is a prime congruent to 1 modulo 3, and each qi is a prime congruent to 2 modulo 3. Then     k l Y Y Y Y 2 Q ∼ Aﬀ / t , a × Aﬀ ( / uj ) ,T , (4.5.4) =  Z j j   Z q j  pi i i=1 j i=1 j 67

tj where each aj is a root of f(X) modulo pi , each Tj is the companion matrix of f(X) over Z/ uj , qi P P j tj = ri, and j uj = si.

Proof. We know that Q has a direct product decomposition as in Theorem 4.4.5. Propositions 4.5.4 and 4.5.5 specify the choices for indecomposable factors as those which appear in (4.5.4).

4.5.2 Enumeration

For a positive integer n, let l(m) denote the number of isomorphism classes of aﬃne MTS of order m. The determination of l(m), for m arbitrary, reduces to ﬁnding l(pn) for all prime p and n ≥ 1. Donovan et al. have determined l(p) and l(p2) for all primes [15, Th. 2.12], along with l(pn) for pn < 1000. Theorem 4.5.6 allows us to calculate l(pn) for arbitrary powers, as long as p 6= 3.

We shall compare our results to Donovan et al.’s after we state and prove Theorem 4.5.7.

+ We think of integer partitions as multisets (X, µ) of positive integers, where X ⊆ Z , and the + function µ : X → Z determines the multiplicity of an integer in the partition. Write (X, µ) ` n if

(X, µ) is a partition of n. Let P (n) denote the number of integer partitions of n, and PE(n) signify the number of partitions consisting only of even parts.

Theorem 4.5.7. Let p 6= 3 be prime, and n ≥ 1.

(a) If p ≡ 1 mod 3, then X X l(pn) = (µ(r) + 1) . (4.5.5) (X,µ)`n r∈X

n (b) If p ≡ 2 mod 3, then l(p ) = PE(n).

Proof. (a) Let Q be an aﬃne Mendelsohn quasigroup of order pn. By Theorem 4.5.6,

∼ Y Q = Aﬀ(Z/pri , ai), (4.5.6) i P where i ri = n. Use (X, µ) to denote the partition of n associated with this sum. Fix r ∈ X.

Then Z/pr appears exactly µ(r) times as a factor in (4.5.6). With two choices for MTS structure 68 on µ(r) copies, there are

(µ(r) + 2 − 1)! (µ(r) + 1)! = µ(r)!(2 − 1)! µ(r)!

= µ(r) + 1

µ(r) allowable isomorphism classes attached to (Z/pr ) . This count applies to each element of X, verifying the indexing on the innermost sum of (4.5.5), while the outermost indexing corresponds to the fact that we had P (n) choices for the decomposition (4.5.6).

(b) Let Q be aﬃne Mendelsohn quasigroup of order p2n. We have

∼ Y 2 Q = Aﬀ((Z/pri ) ,T ), (4.5.7) i

2 2ri P with Aﬀ((Z/pri ) ,T ) = p for each i, and ri = n. Then there is a one-to-one correspondence between choices of the decomposition (4.5.7) (i.e., isomorphism classes) and integer partitions of

2n consisting solely of even parts.

According to the above formulas, when p ≡ 1 mod 3, l(p) = 2, l(p2) = 5, and l(p3) = 10. These

first two values match Theorem 2.2(a) in [15], while the last coincides with these authors’ value of l(73) given in Table 1 of their paper. For the p ≡ 2 mod 3 case, our results dictate l(p2n+1) = 0 for all n, l(p2) = 1, l(p4) = 2, l(p6) = 3, and l(p8) = 5. These first two are covered by [15, Th. 2.12(b)], while Table 1 confirms l(24) = l(54) = 2. Moreover, 3 and 5 appear as the values of l(26) and l(28), respectively.

4.6 Aﬃne ramiﬁed MTS

Classifying affine MTS whose order is divisible by 3 presents an issue: the fact that 3 is ramified in Z[ζ] complicates the direct product decomposition for affine MTS. This section elaborates this issue and presents partial results towards a resolution. 69

4.6.1 Even powers of 1 + ζ

2 Our goal now is to prove that if Q = Aﬀ((Z/3n ) ,R) is an indecomposable MTS, then Q is 2 isomorphic to Aﬀ((Z/3n ) ,T ), where T is the companion of f(X) over Z/3n The case where n = 1 is known to be true [15, Th. 2.12]. The following lemma is needed for higher values of n.

n Lemma 4.6.1. Let n ≥ 2, A ∈ M2(Z). If A ≡ −I2 mod 3, then f(A) 6≡ 0 mod 3 .

Proof. The last paragraph of the proof of Lemma 4.5.3 furnishes a suﬃcient argument.

2 Proposition 4.6.2. Suppose n ≥ 2. Any affine MTS Aff((Z/3n ) ,R) is isomorphic to 2 Aff((Z/3n ) ,T ), where T is the companion matrix of f(X) over Z/3n .

Proof. Let A denote a matrix representation of R. Because n ≥ 2, f(X) has no roots in Z/3n (cf.

Lemma 4.5.2(c)). Furthermore, det(I2X − A) = f(X) (cf. Lemma 4.5.3). Just as in the proof of Proposition 4.5.5, it suﬃces to show that A is similar to the companion matrix of its characteristic

2 polynomial by producing a v ∈ (Z/3n ) that makes the transpose of (v, vA) invertible. Deﬁne A to be the matrix consisting of the entries of A taken modulo 3. Then A acts on

∼ 2 n M/Mm = (Z/3) , where m = (3 + (3 )) is the unique maximal ideal of Z/3n . Recall that in Z/3, 2 det(XI2 − A) = f(X) = (X + 1) . Let W denote the eigenspace of A associated with −1. Then

⊥ dim(W ) = 1 or 2. By Lemma 4.6.1, A 6= −I2, so dim(W ) = 1. Pick v ∈ W . Then {v, vA} forms a basis of M/Mm. Once again, apply Nakayama’s lemma, and we have {v, vA} as a basis for the free Z/3n -module M. Conclude A is similar to the companion of f(X).

4.6.2 Odd powers of 1 + ζ: mixed congruence

In this section, we consider possible MTS structures on (Z/3n ) ⊕ (Z/3n+1 ). We use the work presented in a terriﬁc note by Hillar and Rhea to translate this problem into terms of matrix algebra [24]. Here we are presented with a mixed congruence representation theory. To explain, let M = Z/3n ⊕ Z/3n+1 , πn : Z → Z/3n and πn+1 : Z → Z/3n+1 be natural projections, and

π = πn ⊕ πn+1. Consider the subset

R3 = {(aij) ∈ M2(Z) : 3 | a21} (4.6.1) 70

of 2 × 2 integral matrices. This set is a subalgebra of M2(Z) [24, Lm. 3.2], and

ψ : R3 → End(G); A 7→ ((x1, x2)π 7→ ((x1, x2)A)π) (4.6.2) is a surjective ring homomorphism [24, Th. 3.3] with kernel

k k+1 {(aij) ∈ M2(Z) : 3 | a11, a12 and 3 | a21, a22} (4.6.3)

[24, Lm. 3.4]. Furthermore, an endomorphism Aψ ∈ End(G) is an automorphism if and only if

(A mod 3) ∈ GL2(Z/3) [24, Th. 3.6]. The enumeration techniques of Donovan et al. classify the n = 0, 1, 2 cases.

Proposition 4.6.3. Let Q = Aﬀ(Z/3n ⊕Z/3n+1 ,R) be an aﬃne MTS, M0 = Z/3, M1 = Z/3 ⊕Z/9,

M2 = Z/9 ⊕ Z/27,   2 −1   T =   , 3 −1 and ψ the ring homomorphism (4.6.2).

∼ (a) If n = 0, then Q = Aﬀ(M0, −1).

∼ ψ (b) If n = 1, then Q = Aﬀ(M1,T ).

∼ ψ (c) If k = 2, then Q = Aﬀ(M2,T ).

Proof. (a) By Lemma 4.5.1, f(X) has −1 as its only root in Z/3.

(b) Table 1 in [15] tells us that there are 3 isomorphism classes of aﬃne MTS of order 33. The possible elementary divisors decompositions are

3 1.( Z[ζ]/(1 + ζ)) ;

2 2. Z[ζ]/(1 + ζ) ⊕ Z[ζ]/(1 + ζ);

3 3. Z[ζ]/((1 + ζ) ); 71

The first must correspond to −I3 ∈ M3(Z/3), and the second, by Proposition 4.6.2, to T ⊕ (−1), ψ where T is the companion of f(X) ∈ Z/3[X]. This leaves us with Aff(M1,T ) as the only possible affine MTS structure on (3).

(c) Donovan et al.’s enumeration relates 7 isomorphism classes of aﬃne MTS of order 35. There are 7 abelian groups of order 35, and each corresponds to a unique elementary divisors decomposition for a Z[ζ]-module. In particular, the there is one module on Z/32 ⊕ Z/33 , and it is represented by ψ Aﬀ(M2,T ).

Based upon the evidence proﬀered by small examples in Proposition 4.6.3, we make the following conjecture.

Conjecture 4.6.4. Let n ≥ 3, and fix M = Z/3n ⊕ Z/3n+1 . Any given affine MTS Aff(M,R) is isomorphic to Aff(M,T ψ) where and T ψ is defined along the lines of Proposition 4.6.3. Moreover, the enumeration of order-3n, affine MTS is given by l(3n) = P (n).

4.6.2.1 Lifting

The following discussion presents techniques for lifting the mixed congruence representation theory into M2(Z). Take n ≥ 1. The projection Z → Z/n extends naturally to a map between matrix groups SL2(Z) → SL2(Z/n), furnishing a short exact sequence

1 → Γ(n) → SL2(Z) → SL2(Z/n) → 1. (4.6.4)

The kernel Γ(n) is referred to as the principal congruence subgroup of level n. A subgroup of SL2(Z) is a congruence subgroup if it contains a principal congruence subgroup of any level. The Hecke congruence subgroup of level n, denoted Γ0(n), is the preimage of upper-triangular matrices under the projection SL2(Z) → SL2(Z/n) [14, Sec 1.2]. Notice that the group of determinant-one units of the matrix subalgebra R3 is precisely Γ0(3).

Lemma 4.6.5. All elements in M2(Z) annihilated by the polynomial f(X) have determinant 1, and they lie in the same SL2(Z)-conjugacy class. 72

Proof. Suppose A ∈ M2(Z), and that f(A) = 0. By Lemma 4.5.3, we have det(A) = Tr(A) = 1.

Thus, A ∈ SL2(Z). In [10, Th. 1], it is shown that the number of SL2(Z)-conjugacy classes whose constituents have trace t is equal to the number of equivalence classes of binary quadratic forms with discriminant t2 − 4. Up to equivalence, there is only one binary quadratic form with discriminant

12 − 4 = −3.

Lemma 4.6.6. Suppose A ∈ Γ0(3), and f(A) = 0. Then there exists some P ∈ Γ0(3) so that   2 −1 −1   P AP =   . (4.6.5) 3 −1

Proof. Lemma 4.6.5 tells of a P ∈ SL2(Z) for which (4.6.5) holds. We prove that any such P must belong to Γ0(3). Our argument relies on the fact that when the entries of A and P are taken modulo 3, we get invertible matrices in SL2(Z/3). Now,   −1 −1 −1   P AP ≡   mod 3. 0 −1

We claim that either   −1 −1   A ≡   mod 3, (4.6.6) 0 −1 or   −1 1   A ≡   mod 3. (4.6.7) 0 −1

Notice that f(A) = 0 ≡ 0 mod 3. The only upper-triangular matrices in SL2(Z/3) annihilated by f(X) are (4.6.6), (4.6.7), and −I2. However, since f(A) = 0 ≡ 0 mod 9 we cannot have

A ≡ −I2 mod 3, for this would contradict Lemma 4.6.1. Thus, (4.6.6) or (4.6.7) holds. A straightforward computation in GAP veriﬁes that the only elements of SL2(Z/3) conjugating (4.6.6) to itself, or (4.6.7) to (4.6.6) contain a zero in the lower left-hand entry. Thus, P is upper-triangular in SL2(Z/3), so P ∈ Γ0(3).

For our next proposition, we deﬁne

ψ 2 C = {α ∈ Aut(G) | ∃A ∈ Γ0(3) (A = α ∧ A − A + I2 = 0)}. (4.6.8) 73

Proposition 4.6.7. All members of C are conjugate.

ψ Proof. Let α ∈ C and A ∈ Γ0(3) so that A = α, and f(A) = 0 over Z. We show α is conjugate to  ψ 2 −1   γ =   . 3 −1

−1 ψ By Lemma 4.6.6, there is some P ∈ Γ0(3) so that (4.6.5) holds. Therefore, γ = (P AP ) = (P ψ)−1αP ψ.

Example 4.6.8. The key to proving Conjecture 4.6.4 may be showing that all members of Aut(G) annihilated by f(X) belong to C. This is the case for G = Z/3 ⊕ Z/9. Matrix representatives for the six automorphisms annihilated by f(X) and corresponding lifts into Γ0(3) are given below:     2 2 2 −1       ≡   ; 3 8 3 −1     2 1 2 1       ≡   ; 6 8 −3 −1     2 2 5 −1       ≡   ; 3 5 21 −4     2 1 5 1       ≡   ; 6 5 −21 −4     2 2 26 −31       ≡   ; 3 2 21 −25     2 1 26 31       ≡   . 6 2 −21 −25

These matrix representations were obtained by implementing the procedure outlined in the proof of [24, Th. 4.1] in GAP. 74

4.7 Combinatorial properties of aﬃne MTS

4.7.1 Purity

Deﬁnition 4.7.1. A Mendelsohn triple system (Q, B) is pure if whenever a, b ∈ Q, we have

(a b ab) ∈ B =⇒ (b a ab) ∈/ B.

Definition 4.7.1 is equivalent to anticommutativity of the underlying quasigroup. The class of anticommutative quasigroups is specified in Q by the quasi-identity (one defined by implication; cf. [7, Def. V.2.24])

xy = yx =⇒ x = y. (4.7.1)

Hence, the class of anticommutative quasigroups forms a quasivariety, or a class of algebras deﬁned by quasi-identities. Because Steiner triple systems are equivalent to MTS whose underlying quasigroup is commutative, purity can be seen as a quality of MTS which are maximally non-Steiner.

Constructing affine MTS over finite fields, Mendelsohn proved the existence of pure MTS of order p for p ≡ 1 mod 3 and of order p2 for p ≡ 2 mod 3 [37, Ths. 6, 7]. The existence spectrum of pure MTS was expanded using linear techniques [13], until Bennett and Mendelsohn showed that the existence spectrum of pure MTS matches (with the exception of order 3) that of all MTS. That is to say, a pure MTS of order n exists, with the exception of n = 1, 3, 6, if and only if n ≡ 0, 1 mod 3 [3].

Lemma 4.7.2. Let Q1 and Q2 be quasigroups. The direct product Q1 × Q2 is anticommutative if and only if each of Q1 and Q2 are anticommutative.

Proof. Quasivarities are closed under direct product [7, Th. V.2.25]. Hence, the product of anticommutative quasigroups is anticommutative. To obtain the other direction, we prove the contrapos- itive. Suppose Q1 is not anticommutative and that a, b ∈ Q1 are distinct elements that commute.

2 2 Then for any x ∈ Q2,(a, x) 6= (b, x), while (a, x)(b, x) = (ab, x ) = (ba, x ) = (b, x)(a, x).

Proposition 4.7.3. Every non-ramiﬁed, aﬃne MTS is pure. 75

Proof. By Lemma 4.7.2, it suffices to show that for p ≡ 1 mod 3, Aff(Z/pn , a) is pure, and that for p ≡ 2 mod 3, Aff(Z/pn [ζ]) is pure. In either case, xy = yx means xR+y(1−R) = yR+x(1−R) =⇒ x(2R − 1) = y(2R − 1), so it will suffice to show 2R − 1 is invertible.

The case of (Z/p, a) is covered in [37, Th. 6]. If 2a − 1 were not invertible in Z/p, then 2a − 1 = 0 =⇒ a = 2−1. By Lemma 4.5.1, f(2) = 3 = 0, a contradiction. Furthermore, by way of

n n contradiction, suppose a ∈ Z, f(a) ≡ 0 mod p , and 2a − 1 is not a unit modulo p , where n > 1. k This means that 2a − 1 = mp for some m ∈ Z and 1 ≤ k ≤ n, whereby f(a) ≡ 0 mod p, and 2a − 1 ≡ 0 mod p, which contradicts the n = 1 case.

Fix p ≡ 2 mod 3. We use   0 −1   A =   1 1 to represent Aﬀ(Z/pn [ζ]). Then   −1 −2   det (2A − I2) = det   2 1

= 3,

n which is a unit modulo p , so 2A − I2 is invertible.

Proposition 4.7.4. Let Aff(M,R) be a ramified affine MTS. Then M is not a pure MTS.

Proof. By Lemma 4.7.2, it suﬃces to show that any MTS of the form

n ∼ Aﬀ(Z[ζ]/(1 + ζ) ,R) is not pure. Note Z[ζ]/(1 + ζ) = Z/3, so if n = 1, the system is Steiner. Henceforth, assume n ≥ 2. According to the discussion of mixed congruence in Section 4.6.2, whether n is even or odd, R can be represented by a 2 × 2 integral matrix; call it A. By Lemma

4.5.3 –dropping the second row of A down a level of congruence if n is odd– there is some k ≥ 1 such that det(A) ≡ Tr(A) ≡ 1 mod 3k. Consider   2a − 1 2a  11 12  P := 2A − I2 =   . 2a21 2a22 − 1 76

We claim that P is not invertible. It suﬃces to show det(P ) ≡ 0 mod 3. In fact,

det(P ) = 4a11a22 − 2a11 − 2a22 + 1 − 4a12a21

= 4(a11a22 − a12a21) − 2(a11 + a22) + 1

= 4 det(A) − 2Tr(A) + 1

≡ 3 mod 3k

≡ 0 mod 3.

With P singular, we may choose some nonzero

v ∈ KerP = {v ∈ M | vA + vA = v}.

Since 2 - |M|, v 6= −v. Moreover, v(−v) = vA − v + vA = v − v = 0 = −v + v = (−v)A + v − vA = (−v)v. Conclude there exist distinct vectors commuting under the quasigroup operation given by

Theorem 4.7.5. Let M be a finite abelian group of order n, and Aff(M,R) an affine MTS. Then

Aff(M,R) is pure if and only if 3 - n. That is, an affine MTS is pure if and only if it is non-ramified.

Q Proof. Let Aff(M,R) = i Qi be the decomposition of Theorem 4.4.5. By Lemma 4.7.2, Aff(M,R) is pure if and only if each Qi is anticommutative. Propositions 4.7.3 and 4.7.4 reveal that a given factor Qi is anticommutative if and only if 3 - |Qi|. Therefore, Aff(M,R) is pure if and only if each

Qi is anticommutative if and only if 3 - |Qi| for all i if and only if 3 - n.

4.7.2 The converse of aﬃne MTS

4.7.2.1 Self-Converse MTS

Let ∗ : Q2 → Q be a binary operation on a set Q, and τ :(x, y) 7→ (y, x) the twist on Q2.

We deﬁne the opposite of ∗ to be the binary operation τ∗ :(x, y) 7→ y ∗ x. Given a quasigroup

(Q, ·, /, \), its opposite quasigroup is the quadruple (Q, ◦, \\, //), where ◦, \\, and // denote the opposite operations of multiplication, left division, and right division respectively. Recall that 77 a Mendelsohn quasigroup has the form (Q, ·, ◦, ◦), so its opposite will be (Q, ◦, ·, ·). This has a combinatorial interpretation.

Deﬁnition 4.7.6. Let (Q, B) be an MTS. The converse of (Q, B) is given by (Q, B−1), where

B−1 = {(b a ab) | (a b ab) ∈ B}.

Deﬁnition 4.7.7. Let (Q, B) be an MTS with corresponding quasigroup (Q, ·, ◦, ◦). We say that

(Q, B) is self-converse if (Q, ·, ◦, ◦) is isomorphic to its opposite (Q, ◦, ·, ·). An affine MTS Aff(M,R) is self-converse if and only if Aff(M,R) =∼ Aff(M, 1 − R) = Aff(M,R−1).

The term self-converse comes from graph theory. The converse of a digraph consists of the same set of points, with arrows reversed. In fact, in order to obtain the converse of an MTS, one may take the Cayley graph with respect to multiplication in the underlying quasigroup and reverse arrows. The examination of self-converse MTS ﬁrst appears in the literature with Colbourn and

Rosa’s survey on directed triple systems [11]. Just as in the case of pure systems, it was shown that the existence spectrum of self-converse MTS matches the existence spectrum of arbitrary MTS [8].

We characterize self-converse aﬃne non-ramiﬁed MTS and conjecture a complete characterization of entropic self-converse MTS.

Lemma 4.7.8. Let Aff(M1,R1) and Aff(M2,R2) be Mendelsohn quasigroups. The direct product Q i Aff(Mi,Ri) is self-converse if and only if each factor Aff(Mi,Ri) is self-converse.

Q ±1 ∼ ±1 ±1 Proof. Begin by noticing i Aff(Mi,Ri ) = Aff(M1 × M2,R1 × R2 ). We can extend isomor- −1 Q phisms fi : Aff(Mi,Ri) → Aff(Mi,Ri ) uniquely to the product f1 × f2 : i Aff(Mi,Ri) → Q −1 Q i Aff(Mi,Ri ); this proves sufficiency. Conversely, we can restrict an isomorphism f : i Aff(Mi,Ri) → Q −1 −1 i Aff(Mi,Ri ) to isomorphisms f |0×Mi : Aff(Mi,Ri) → Aff(Mi,Ri ), verifying necessity.

Theorem 4.7.9. Let M be a finite abelian group of order n, and Aff(M,R) an affine non-ramified

MTS. Then Aﬀ(M,R) is self-converse if and only if for each prime p | n, we have p ≡ 2 mod 3.

Proof. The MTS Aﬀ(M,R) is self-converse if and only if each factor in its direct product decomposition (4.5.4) is self-converse. Propositions 4.5.4 and 4.5.5 reveal that this is possible if and only

2 if each factor is of the form Aﬀ((Z/pn ) ,T ) for some p ≡ 2 mod 3. 78

Conjecture 4.6.4 implies the following:

Conjecture 4.7.10. Let M be a finite abelian group, and Aff(M,R) an MTS. Then Aff(M,R) is self-converse if and only if no prime dividing the order of M is congruent to 1 modulo 3.

4.7.2.2 Self-Orthogonal MTS

Deﬁnition 4.7.11. Two quasigroups on the same set of symbols (Q, ·) and (Q, ∗) are orthogonal if for every a, b ∈ Q the system of equations

x · y = a

x ∗ y = b has a unique solution. A quasigroup Q is self-orthogonal if it is orthogonal to its opposite.

Lemma 4.7.12. Let (Q, ·, /, \) be a quasigroup with distinct, commuting elements. Then (Q, ·, /, \) is not self-orthogonal.

Proof. Choose distinct, commuting elements q, q0 ∈ Q, and ﬁx a = q · q0 = q0 · q. With respect to the system of equations

x · y = a

x ◦ y = a, the ordered pairs (q, q0), (q0, q) constitute distinct solutions.

The above lemma demonstrates that self-orthogonal quasigroups are necessarily anticommutative. In general, the converse does not hold. However, Di Paola and Nemeth showed that any MTS aﬃne over a ﬁeld which is pure (anticommutative) is necessarily self-orthogonal [13, Th. 7]. We prove that their result extends to quotients of Z[ζ] by primary ideals.

Theorem 4.7.13. Let M be a finite abelian group, and Aff(M,R) an affine MTS. Then Aff(M,R) is self-orthogonal if and only if it is non-ramified if and only if it is pure. 79

Proof. Note that a quasigroup is self orthogonal if and only if x1y1 = x2y2 and y1x1 = y2x2 =⇒ x1 = x2 and y1 = y2. With this characterization, one may prove –employing a quasivariety argument identical to that of the proof of Lemma 4.7.2– that the direct product of quasigroups is self-orthogonal if and only if each factor is self-orthogonal. Thus, the fact that ramified affine MTS are not self-orthogonal follows directly from Proposition 4.7.4 and Lemma 4.7.12. This proves that all self-orthogonal affine MTS are non-ramified.

Next, to prove the converse, we have to establish self-orthogonality of Aff(Z/pn , a) whenever 2 p ≡ 1 mod 3, and Aff((Z/pn ) ,T ) whenever p ≡ 2 mod 3. But first, we make a few observations regarding an arbitrary affine MTS Aff(M,R). Suppose x1, x2, y1, y2 ∈ M so that x1y1 = x2y2 and

−1 2 y1x1 = y2x2. Recall 1 − R = R = −R . Then by hypothesis,

2 2 x1R − y1R = x2R − y2R , (4.7.2) and

2 2 y1R − x1R = y2R − x2R . (4.7.3)

Now multiply (4.7.2) and (4.7.3) by R−1 and rearrange terms in order to obtain

x1 − x2 = (y1 − y2)R, (4.7.4) and

y1 − y2 = (x1 − x2)R. (4.7.5)

Substituting the right-hand side of (4.7.5) in for y1 − y2 in (4.7.4) yields

2 x1 − x2 = (x1 − x2)R . (4.7.6)

Similarly,

2 y1 − y2 = (y1 − y2)R . (4.7.7)

2 Now, assume we are working in Aﬀ(Z/pn , a). If x1 − x2 6= 0, in order for (4.7.6) to hold, a − 1 2 must be a zero divisor in Z/pn , so p | a − 1. Thus, a is a square root of 1 modulo p, and if this is the case, then a ≡ ±1 mod p. We also have a2 − a + 1 ≡ 0 mod p. Clearly then, a cannot 80 be congruent to 1, so it must be congruent to −1 modulo p. However, f(−1) ≡ 0 mod p if and only if p = 3 by Lemma 4.5.1(b), contradicting our assumption that p ≡ 1 mod 3. We are left to conclude x1 = x2; similarly y1 = y2.

2 The case of Aﬀ(Z/pn [ζ]) is more concrete. By Proposition 4.5.5, we can choose R so that R is represented by   −1 −1     . 1 0

Suppose x1 − x2 = (a, b) 6= 0. That is,

(a, b) = (a, b)R2

= (b − a, −a).

Then b = −a = 2a, so that 0 = 3a. Since, p does not divide 3, it is invertible in Z/pn . Conclude a = −b = 0, a contradiction. Thus, x1 − x2 = 0 = y1 − y2. We have thus proven the equivalence of aﬃne MTS being non-ramiﬁed and self-orthogonal. The equivalence of the former with purity was established in Theorem 4.7.5.

Acknowledgements

The author would like to thank his doctoral advisor, Jonathan D. H. Smith, for helping him to prepare the manuscript, and for pointing out the connection between the ramiﬁed classiﬁcation problem of Section 4.6 and congruence subgroups.

Thanks are also due to two anonymous referees, whose comments greatly clariﬁed our presentation of the results. Our proof of Lemma 4.5.3 was improved by their suggestions. 81

CHAPTER 5. TRIALITY IN THE THEORY OF QUANTUM QUASIGROUPS

An extension of results from the paper “Algebraic properties of quantum quasigroups” submitted

to The Journal of Pure and Applied Algebra1

Bokhee Im, Alex W. Nowak, and Jonathan D. H. Smith.

In the aforementioned paper, the authors introduce conjugates for quantum quasigroups. As we shall see, triality does not seem to be baked into the deﬁnition, but does, in many cases occur.

The transfer of algebraic theories into symmetric monoidal categories is trivial2 when working with cocommutative, coassociative structures (v.i. Section 5.2). Therefore, since triality is an equational theory, in order for our deﬁnition to be an honest generalization, it has to exist in noncoassociative and noncocommutative settings. This chapter extends the work of [25] in two ways: 1.) proving that H-bialgebras exhibit conjugate triality (Prop. 5.2.5), and 2.) showing that the quantum couple of a group acting by automorphisms on a ﬁnite, nonassociative, noncommutative quasigroup is to borrow a phrase of Majid [33, Ch. 1], “truly quantum” (Th. 5.3.6).

5.1 Quantum quasigroups and their conjugates

Throughout the chapter, we use K to denote a commutative, unital ring. We recall the deﬁnition of quantum quasigroup from our introduction:

Deﬁnition 5.1.1. Let A be a K-module, and ∇ : A ⊗ A → A and ∆ : A → A ⊗ A be K-linear

⊗2 maps such that ∆ is a homomorphism from (A , (1A ⊗ τ ⊗ 1A)(∇ ⊗ ∇)) into (A, ∇) We say that (A, ∇, ∆) is a quantum quasigroup if G, the left composite

∆⊗1A 1A⊗∇ A ⊗ A / A ⊗ A ⊗ A / A ⊗ A, (5.1.1)

1Results taken directly from the paper will be cited as such. All others are the author’s own 2We mean trivial in the sense that group algebras and their duals are trivial in the theory of Hopf algebras. 82 and a, right composite

1A⊗∆ ∇⊗1A A ⊗ A / A ⊗ A ⊗ A / A ⊗ A (5.1.2) are invertible.

Example 5.1.2. Suppose (Q, ·, /, \) is a ﬁnite quasigroup. We write KQ for the free K-module with basis Q, and use K(Q) = K-Mod(KQ, K) to denote the dual space of functionals on KQ.

The elements of the dual basis for K(Q) are denoted using Kronecker-delta notation: δq : p 7→ δp,q.

(a) The linear extension of · : Q2 → Q to KQ⊗2 → KQ gives us the quasigroup algebra of Q

over K,(KQ, ∇). With ∆ : q 7→ q ⊗ q and ε : q 7→ 1, (KQ, ∇, ∆, ε) is a cocommutative,

coassociative, counital quantum quasigroup, known as the quantum quasigroup algebra of Q

over K. Indeed,

−1 G = (∆ ⊗ 1KQ)(1KQ ⊗ ∇l),

where ∇l is the linear extension of \. Similarly,

−1 a = (1KQ ⊗ ∆)(∇r ⊗ 1KQ),

where ∇r is the linear extension of /.

(b) The dual quantum quasigroup algebra of Q over K is (K(Q), ∆∗, ε∗, ∇∗). Given two dual basis

∗ P elements δq, δp, the product is (a)δqδp = a(δp,q) for all a ∈ Q, ε selects the unit p∈Q δp, ∗ P P while ∇ : δq 7→ p∈Q δq/p ⊗ δp = p∈Q δp ⊗ δp\q. This describes a commutative, associative, unital quantum quasigroup.

5.1.1 Quantum conjugates

Given a multiplication ∇ : A ⊗ A → A in K-Mod, we let ∇t denote the opposite multiplication,

τ∇. For a comultiplication ∆ : A → A⊗A, use ∆t = ∆τ. Typically, the opposite of a multiplication ∇ (comultiplication ∆) is denoted ∇op (∆co−op). The t-subscript notation, however, will be used to swiftly convey triality between (co)multiplications of conjugates. The goal is to evoke the presentation

ht, l, r|t2 = l2 = 1, tlt = ltl = ri 83

of S3.

Deﬁnition 5.1.3 ([25]). Let A stand for a quantum quasigroup (A, ∇, ∆) with respective left and right composites G and a.

(a) We use At to denote the structure (A, ∇t, ∆t), referred to as the opposite, or transpose, of the quantum quasigroup A = (A, ∇, ∆).

(b) A quantum quasigroup Al = (A, ∇l, ∆l) is a quantum left conjugate of (A, ∇, ∆) if the left

composite Gl of (A, ∇l, ∆l) is inverse to the left composite G of (A, ∇, ∆).

composite ar of (A, ∇r, ∆r) is inverse to the right composite a of (A, ∇, ∆).

The following is an immediate consequence of Deﬁnition 5.1.3.

Lemma 5.1.4 ([25]). Let A be a quantum quasigroup.

(a) The transpose of At is A.

(b) Suppose that A has a quantum left conjugate Al. Then A is a quantum left conjugate of Al.

Ar.

Lemma 5.1.5 ([25]). Let A be a quantum quasigroup, with composites G and a. Let Gt and at be the respective left and right composites of the opposite. Then τG = atτ and τGt = aτ.

Proof. Consider the commutative diagram

x ⊗ yR ⊗ yL r8 OO 1⊗∆t r OO ∇t⊗1 rrr OO rr OOO rr OO' 2rr at x ⊗ y / yRx ⊗ yL _ _ τ τ G y ⊗ x / yL ⊗ yRx LL o7 LL ooo LLL oo ∆⊗1 LL oo1⊗∇ L& /ooo yL ⊗ yR ⊗ x 84 for the ﬁrst equation, working with the Jay calculus [27].

The second equation follows from the ﬁrst equation as applied to the opposite quantum quasigroup (A, ∇t, ∆t), using Lemma 5.1.4(a).

Proposition 5.1.6 ([25]). If (A, ∇, ∆) is a quantum quasigroup, then its opposite is a quantum quasigroup.

Proof. The bimagma condition for the opposite of (A, ∇, ∆) is veriﬁed along the lines of [29,

−1 −1 Prop. III.2.3] or [44, Ex. 5.1.2]. Since Gt = τaτ by Lemma 5.1.5, one has Gt = τa τ. Similarly, −1 −1 at = τG τ. Thus (A, ∇t, ∆t) is a quantum quasigroup.

Deﬁnition 5.1.7. A quantum quasigroup A is said to exhibit quantum conjugate triality if

(a) A has a quantum left conjugate Al,

(b) A has a quantum right conjugate Ar,

(c)( Al)t is a quantum left conjugate for Ar, and

(d)( Ar)t is a quantum right conjugate for Al.

Proposition 5.1.8. Let A be a quantum quasigroup exhibiting conjugate triality. There is an action of S3 on the set {Ag | g ∈ S3}.

Proof. By Lemma 5.1.4,(As)s = A for all s ∈ {l, r, t}. Condition 5.1.7(c) means that ((Al)t)l exists and coincides with Ar, while 5.1.7(d) means the same for ((Ar)t)r and Al. Following the

−1 proof of Lemma 5.1.5, Gt = τaτ, so Gt = τarτ = (Gr)t = (Gl)r. Which is to say, At has a left conjugate, and thus, ((At)l)t = ((Al)r)t = ((Al)l)r = Ar = ((Al)t)l, reproducing the braid relation ltl = tlt.

5.2 Triality of quantum conjugates for H-bialgebras

The H-bialgebra (no relation to H-symmetry) construction has been a cornerstone for much of the work done in nonassociative Lie theory for the past decade. 85

Deﬁnition 5.2.1 ([42]). An H-bialgebra (H, ∇, ∇r, ∇l, ∆, ε) consists of a counital (possibly non associative and noncoassociative) bialgebra (H, ∇, ∆, ε) over K accompanied by linear divisions: a right division ∇r : H ⊗ H → H; x ⊗ y 7→ x/y and a left division ∇l : H ⊗ H → H; x ⊗ y 7→ x\y. These data satisfy

(∆ ⊗ 1H )(1H ⊗ ∇)∇l = ε ⊗ 1H = (∆ ⊗ 1H )(1H ⊗ ∇l)∇, (5.2.1) and

(1H ⊗ ∆)(∇ ⊗ 1H )∇r = 1H ⊗ ε = (1H ⊗ ∆)(∇r ⊗ 1H )∇. (5.2.2)

Remark 5.2.2. In terms of elements, (5.2.1) translates to

X X xL\(xRy) = xεy = xL(xR\y), (5.2.3) and (5.2.2) to X X (yxL)/xR = xεy = (y/xL)yR. (5.2.4)

They amount to linearizations of the quasigroup axioms.

We will show that the quantum quasigroup reducts of coassociative, cocommutative H-bialgebras exhibit conjugate triality, and that these conjugates are themselves H-bialgebras. The proof is not hard, but we feel it is important to have the result on record as the theory of quantum quasigroup triality advances. The heavy lifting is done by P´erez-Izquierdoan Mostovoy’s theory of linearizing identities (cf. [42, Sec. 3]). We avoid a formal discussion of the theory, but it may be summarized as such: any identity expressed in the words of a free quasigroup has a corresponding expression in H-bialgebra operations. For example, the semisymmetric identity y · (xy) = x becomes

P L R ε (x ⊗ y)(1H ⊗ ∆)(τ ⊗ 1H )(1H ⊗ ∇)∇ = y (xy ) = y x = (x ⊗ y)(1H ⊗ ε). The payoﬀ is that if u = v is a consequence of a set Σ of quasigroup identities, then the linearization of u = v is a consequence of the linearization of each identity in Σ, as long as comultiplication is assumed to be coassociative and cocommutative.

Lemma 5.2.3 ([52]). Given an H-bialgebra (H, ∇, ∇r, ∇l, ∆, ε), the bimagma reduct (H, ∇, ∆) is a quantum quasigroup. 86

Lemma 5.2.4. Suppose (H, ∇, ∇r, ∇l, ∆, ε) is a coassociative, cocommutative H-bialgebra. Then for any x, y ∈ H

xL/(y\xR) = xεy = (xL/y)\xR. (5.2.5)

Proof. The condition (5.2.5) is the linearization of (x/y)\x = y = x/(y\x), and since ∆ is assumed to be coassociative and cocommutative, linearization carries through.

Proposition 5.2.5. Let (H, ∇, ∇r, ∇l, ∆, ε) be a coassociative, cocommutative H-bialgebra. The bimagma reduct (H, ∇, ∆) exhibits conjugate triality; its quantum left conjugate is (H, ∇l, ∆) and its quantum right conjugate is (H, ∇r, ∆). Furthermore, for each

σ ∈ ht, l, r|t2 = l2 = 1, tlt = ltl = ri, (5.2.6)

the conjugate (H, ∇σ, ∇σr, ∇σl, ∆, ε) is itself an H-bialgebra.

Proof. Propositions 6 and 7 of [42] prove that (H, ∇l, ∆, ε) and (H, ∇r, ∆, ε) are counital bialgebras.

Let Gl and al denote, respectively, the left and right composites of (H, ∇l, ∆). Then (x⊗y)GGl = L R LL LR R (x ⊗ x y)Gl = x ⊗ x \(x y). By coassociativity,

xLL ⊗ xLR\(xRy) = xL ⊗ xRL\(xRRy)

= xL(xR)ε ⊗ y

= x ⊗ y.

The second equality above is an expression of the left-hand side of (5.2.1). The right-hand side of

−1 (5.2.1) proves (x ⊗ y)GlG = x ⊗ y for all x, y ∈ H. Conclude G = Gl. Thus, (H, ∇l, ∆) is a left quantum quasigroup whose left composite is the inverse of G. Denote art := (1H ⊗ ∆)(∇rt ⊗ 1H ). −1 We’ll prove (H, ∇l, ∆) is a quantum left conjugate by showing al = art, thus making it a two- L R RL L RR sided quantum quasigroup. Indeed, note that (x ⊗ y)alart = (x\y ⊗ y )art = y /(x\y ) ⊗ y . 87

Moreover,

yRL/(x\yL) ⊗ yRR = yLR/(x\yLL) ⊗ yR

= yLL/(x\yLR) ⊗ yR

= x ⊗ (yL)εyR

= x ⊗ y, where the ﬁrst equality results from coassociativity, the second from cocommutativity, and the third (5.2.5). A similar argument, employing the right-hand side of (5.2.5) rather than the left, yields artal = 1H⊗H . −1 A dual argument proves (H, ∇r, ∆) is a quantum right conjugate for (H, ∇, ∆) via ar = a , and −1 Gr = Glt . This latter equality proves that (H, ∇lt, ∆) is a quantum left conjugate for (H, ∇r, ∆.). −1 Above, we showed al = art , proving (A, ∇rt, ∆) is a quantum right conjugate for (A, ∇l, ∆). Thus, (A, ∇, ∆) exhibits conjugate triality.

Recall that classical quasigroup triality is an expression of the symmetry in the quasigroup axioms (IL)-(SR). That each (H, ∇σ, ∇σr, ∇σl, ∆, ε) is an H-bialgebra results from this symmetry in their linearizations. For example, we can make the identiﬁcation ∇lr = ∇rt because (5.2.5) tells us

(1H ⊗ ∆τ)(∇l ⊗ 1H )∇rt = 1H ⊗ ε = (1H ⊗ ∆)(∇rt ⊗ 1H )∇l, (5.2.7) but by cocommutativity,

(1H ⊗ ∆)(∇l ⊗ 1H )∇rt = 1H ⊗ ε = (1H ⊗ ∆)(∇rt ⊗ 1H )∇l. (5.2.8)

That is, ∇rt is a right division with respect to ∇l. Moreover, the axiom (5.2.1) itself shows that ∇ is a left division with respect to ∇l. Hence, (H, ∇l, ∇rt, ∇, ∆, ε) is an H-bialgebra, and the notation

“∇lr is a right division for the operation ∇l, and ∇ll is a left division for ∇l” is consistent with the S3-relations. Similar arguments verify that (H, ∇σ, ∇σr, ∇σl, ∆, ε) is an H-bialgebra for each

σ ∈ S3. 88

5.3 Conjugates of the quantum couple

The purpose of this section is to exhibit a class of noncocommutative, noncoassociative quantum quasigroups possessing conjugate triality. We work with the quantum couple of a group G acting on a ﬁnite quasigroup (Q, ·, /, \) by automorphisms. We recall Smith’s construction of the quantum couple (cf. [52, Sec. 3.6]), a nonassociative generalization of Drinfeld’s quantum double arising from the group algebra of a ﬁnite group [16].

Deﬁnition 5.3.1. Let G be a group, with identity e, acting (from the right) on a nonempty, ﬁnite quasigroup (Q, ·, /, \) by automorphisms. That is, (p ∗ q)g = pg ∗ qg whenever g ∈ G, p, q ∈ Q, and

∗ ∈ {·, /, \}. Let GQ denote the tensor product of the free K-module on G with the free K-module on Q. Denote GQ-basis elements by f|p for f ∈ G and p ∈ Q. The linear extensions of the maps

∇ :(f|p ⊗ g|q) 7→ δpg,q (fg|q) , (5.3.1)

X ∆ : (f|p) 7→ f|p/q ⊗ f|q, (5.3.2) q∈Q X η : 1 7→ e|q ⊗ e|q (5.3.3) q∈Q deﬁne a unital quantum quasigroup (GQ, ∇, η, ∆) which we refer to as the quantum couple of the action.

Remark 5.3.2. In the language of Hopf algebras, the algebra structure of the quantum couple is the Hopf smash product KG#K(Q) (cf. [29, Sec. IX.2]), where the commutative, associative, unital algebra of K-valued functions on Q, K(Q), is regarded as a right Hopf KG-module algebra

(no appeal to the quasigroup structure is needed here, as any G-set leads to such a module-algebra).

This product is associative.

The coalgebra on GQ is simply the tensor square coalgebra of the setlike comultiplication g 7→ g ⊗ g and the comultiplication of the dual quantum quasigroup algebra K(Q) from Example

5.1.2. Because the tensor product coalgebra is cocommutative if and only if each component 89 is cocommutative and coassociative if and only if each component is coassociative, we have the following result.

Proposition 5.3.3. Let (GQ, ∇, ∆) denote the quantum couple of a group G acting on a nonempty,

ﬁnite quasigroup (Q, ·, /, \) by automorphisms.

(a) The comultiplication ∆ is cocommutative if and only if (Q, ·, /, \) is a commutative quasigroup.

(b) The comultiplication ∆ is coassociative if and only if (Q, ·, /, \) is associative, i.e., if and only

if (Q, ·) is a group.

Thus, Theorem 5.3.6 will ensure the existence of noncocommutative, noncoassociative quantum quasigroups exhibiting conjugate triality.

Proposition 5.3.4. Let (GQ, ∇, ∆) denote the quantum couple of a group acting on a nonempty,

ﬁnite quasigroup by automorphisms. With

−1 ∇l :(f|p ⊗ g|q) 7→ δp(f −1g),q f g|q , (5.3.4) and X ∆l :(f|p) 7→ f|pq ⊗ f|q, (5.3.5) q∈Q

(GQ, ∇l, ∆l) is a quantum left conjugate for (GQ, ∇, ∆).

Proof. First, we show that ∆l is a homomorphism of ∇l: 90

 ∇l⊗∇l f −1g (∆ ⊗∆ )(1 ⊗τ⊗1 )(∇ ⊗∇ ) X f −1g (f|p ⊗ g|p ) l l GQ GQ l l =  f|pq ⊗ g|p r ⊗ f|q ⊗ g|r q,r∈Q

X −1 f −1g −1 = δ(pq)f−1g,pf−1gr f g|p r ⊗ δqf−1g,r f g|r q,r∈Q

X −1 −1 −1 = f −1g|pf gqf g ⊗ f −1g|qf g q∈Q

X −1 = f −1g|pf gq ⊗ f −1g|q q∈Q

−1 f −1g = f g|p ∆l

f −1g = (f|p ⊗ g|p )∇l∆l.

The left composite of the quantum couple transforms basis elements of GQ ⊗ GQ as such:

G : f|p ⊗ g|q 7→ f|p/(qg−1) ⊗ fg|q, (5.3.6) and that the inverse of this map is

G−1 : h|r ⊗ k|s 7→ h|r · s(k−1h) ⊗ h−1k|s. (5.3.7)

It is readily veriﬁed that (5.3.7) is the left composite (∆l ⊗ 1GQ)(1GQ ⊗ ∇l) := Gl of the bimagma

(GQ, ∇l, ∆l).

To conclude (GQ, ∇l, ∆l) is a quantum left conjugate for (GQ, ∇, ∆), we need to show that the right composite al := (1GQ ⊗ ∆l)(∇l ⊗ 1GQ) is invertible. We have

−1 −1 −1 al : f|p ⊗ g|q 7→ f g|p(f g) ⊗ g|q\(p(f g)), (5.3.8) from which it is easy to see that

h|r ⊗ k|s 7→ kh−1|rh−1 ⊗ k|r/s (5.3.9) extends linearly to the inverse of al. 91

Proposition 5.3.5. Let (GQ, ∇, ∆) denote the quantum couple of a group acting on a nonempty,

ﬁnite quasigroup by automorphisms. With

−1 −1 ∇r :(f|p ⊗ g|q) 7→ δp,q fg |pg (5.3.10) and X ∆r :(f|p) 7→ f|q ⊗ f|qp, (5.3.11) q∈Q

(GQ, ∇r, ∆r) is a quantum right conjugate for (GQ, ∇, ∆).

Proof. We begin by showing ∇r is a homomorphism of ∆r:

X (f|p ⊗ g|p)(∆r⊗∆r)(1GQ⊗τ⊗1GQ)(∇r⊗∇r) = (f|q ⊗ g|r ⊗ f|qp ⊗ g|rp)∇r⊗∇r q,r∈Q

X −1 g−1 −1 g−1 = δq,r fg |q ⊗ δqp,rp fg |(qp) q,r∈Q

X −1 −1 −1 = fg−1|qg ⊗ fg−1|qg pg q∈Q

X −1 = fg−1|q ⊗ fg−1|qpg q∈Q

−1 g−1 = fg |p ∆r

= (f|p ⊗ g|p) ∇r∆r.

The right composite of (GQ, ∇, ∆) is

a : f|p ⊗ g|q 7→ fg|pg ⊗ g|(pg)\q, (5.3.12) while the right composite of (GQ, ∇r, ∆r) is

−1 −1 ar : h|r ⊗ k|s 7→ hk |rk ⊗ k|r · s. (5.3.13)

−1 It is straightforward to verify ar = a . The linear extension of the map

h|r ⊗ k|s 7→ h|r\(s(k−1h)) ⊗ k−1h|s(k−1h). (5.3.14) 92 inverts

−1 −1 Gr : f|p ⊗ g|q 7→ f|q/p ⊗ fg |qg , (5.3.15) the left composite of (GQ, ∇r, ∆r). Thus, (GQ, ∇r, ∆r) is a quantum quasigroup, and it is a quantum right conjugate of (GQ, ∇, ∆).

Theorem 5.3.6. The quantum couple of a group acting on a nonempty, ﬁnite quasigroup by automorphisms exhibits conjugate triality.

Proof. Propositions 5.3.4 and 5.3.5 conﬁrm items (a) and (b) in Deﬁnition 5.1.7. The left composite of (GQ, ∇lt, ∆lt) is given by

∆lt⊗1GQ 1GQ⊗∇lt h|r ⊗ k|s / h|rR ⊗ h|rL ⊗ k|s / h|r\(s(k−1h)) ⊗ k−1h|s(k−1h),

−1 coinciding with (5.3.14). That is, Glt = Gr so that Condition 5.1.7(c) is met. Next, we have

1 ⊗∆ 1 ⊗∇ GQ rt R L GQ rt −1 −1 art : h|r ⊗ k|s / h|r ⊗ k|s ⊗ k|s / kh |rh ⊗ k|r/s, which matches (5.3.9), the inverse of al. In other words, GQrt is a right quantum conjugate of

GQl. 93

CHAPTER 6. CONCLUSIONS AND FUTURE WORK

The action of S3 on Latin squares leads to an algebraic triality of quasigroups. The less faithful the action —i.e., the larger the kernel— the more equational symmetry a given quasigroup has.

The “H-symmetry” classes coming from the subgroup lattice of S3 have equational representations that give rise to universal algebraic varieties, and, thus, to bicomplete categories. Beck’s theory of modules for objects in categories with pullbacks was adapted by Smith to provide a method for

ﬁnding rings of representation associated with any category of quasigroups.

6.1 Rings of representation

Chapter3 was highly computational, culminating in presentations for the aforementioned rings of each H-symmetry class. We saw that the addition of idempotence imposed a tremendous amount of structure, as the theory of abelian groups in CI, MTS, and STS is equivalent to the module theory of certain PIDs (in the latter case, vector spaces over a ﬁeld).

6.1.1 Future work

The cumbersome nature of the case analysis begs the question of whether or not there is a specialized Fundamental theorem for the H-symmetric varieties. More ambitiously, is there a module theory more conscious of the inherently S3-symmetric nature of quasigroups. Smith’s hyperquasigroups [51] included triality in a way that cut the case analysis required to establish

Evans’ free quasigroup normal form [18] substantially. The module theory of hyperquasigroups may do the same for our ring presentations. 94

6.2 Aﬃne Mendelsohn triple systems

We extended results in the classification and enumeration of affine, h(123)i-symmetric, idempotent quasigroups. Whereas previous attempts stalled out at orders p and p2, the emphasis on the module theory of this variety of quasigroups led to a complete solution of the isomorphism problem for orders coprime with 3. The linear algebra also gave simple characterizations of the block-structural properties of affine MTS.

6.2.1 Future work

Conjecture 4.6.4 certainly requires follow-up attention, with the lifting techniques of Subsection

4.6.2.1 tracing a promising line of attack.

6.3 Conjugates of quantum quasigroups

We described a theory of conjugates for quantum quasigroups introduced by the author, Im, and Smith in [25]. To justify the theory’s suitability for symmetric monoidal categories, we proved that a noncocommutative, noncoassociative quantum quasigroup structure on the smash product

Aut(Q)#K(Q) led to a collection of conjugates on which S3 acts.

6.3.1 Future work

Follow-up work on [25] with Im and Smith is underway. The object of this work will be the

H-symmetry classes of quantum quasigroups which exhibit conjugate triality. Questions which persist regarding conjugate triality include:

Question 6.3.1. Does every cocommutative, coassociative quantum quasigroup in (K-Mod, ⊗,K) exhibit conjugate triality? Are there quantum quasigroups with conjugates which do not exhibit conjugate triality? 95

Question 6.3.2. The quantum couple comes from a standard crossed product construction. Is there a generalized crossed product that can account for the algebra structures of the conjugates of the quantum couple? 96

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