# LEFT DISTRIBUTIVE LEFT QUASIGROUPS - stanovsk/math/ left distributive left quasigroups,

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LEFT DISTRIBUTIVE LEFT QUASIGROUPS

DAVID STANOVSKY

PhD Thesis

Charles University in Prague

July 2004

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2 David Stanovsky: Left distributive left quasigroups

Acknowledgement

I wish to thank to everybody I learned anything from during past twenty sixyears.

Above all, to my advisor, Jaroslav Jezek, for his support and insightful advices.To my inofficial coadvisor, Ralph McKenzie, for numerous inspirative conversa-

tions and for financial and other support during my stay at Vanderbilt Universityin Nashville.

To Vera Trnkova, Ales Drapal, Tomas Kepka, Premysl Jedlicka and other peoplefrom the Department of Algebra at Charles University in Prague, for their interestand useful advices.

To people from the Department of Mathematics at Vanderbilt University inNashville, who made my stay in the United States nice and fruitful.

To Anna Romanowska and Barbara Roszkowska from the Technical Universityin Warsaw, for providing useful materials and for an invitation to visit Warsaw.

And, especially, to my parents for their support during long years of my studies.

A partial financial support of the Grant Agency of the Czech Republic undergrants 201/02/0594 and 201/02/0148 is greatfully acknowledged.

David Stanovsky: Left distributive left quasigroups 3

Contents

Acknowledgement 2

Contents 3

I. Introduction 5

II. Preliminaries 91. Basic facts 92. Normal form of terms 123. Definable sets are left ideals 134. Quasigroups and loops 14

III. Left symmetric left distributive idempotent groupoids 161. Normal form of terms 172. Exponent 193. Representation by Bruck loops 204. Applications 235. Cycles 236. Group of displacements 257. Left ideal decomposition and BL-sum 268. Cores 27

IV. On loops isotopic to left distributive quasigroups 29

V. Equational theory of group conjugation 311. The variety generated by conjugation 312. The equational theory of conjugation 343. On the role of idempotency 36

VI. Subdirectly irreducible non-idempotent left distributive leftquasigroups 381. Description 382. Examples 42

VII. On varieties of left distributive left idempotent groupoids 451. Varieties satisfying xn+1 x 452. Varieties satisfying xm+n xm 47

VIII. Miscellaneous remarks and open problems 491. Miscellaneous remarks 492. Open problems 50

Appendix: Homomorphic images of subdirectly irreducible algebras 521. The necessary condition 522. Groupoids 543. Algebras with rich signature 584. Monounary algebras 62

4 David Stanovsky: Left distributive left quasigroups

5. Unary algebras 646. Particular varieties 65

References 66

David Stanovsky: Left distributive left quasigroups 5

I. Introduction

Selfdistributive groupoids arise on borders of several mathematical disciplines,such as algebra, geometry and set theory. An interested reader is encouraged to lookat the recent monography [11] of P. Dehornoy. We consider a particular subclass:left distributive left quasigroups, known also as racks, wracks, quandles, automor-phic sets, pseudo-symmetric sets, crystals, etc. (some of the names refer to theidempotent case, some of them dont).

A groupoid (i.e. a set equipped with a binary operation) is called left distributive,if it satisfies the identity

x(yz) (xy)(xz).(LD)

It is called left quasigroup, if

for every a, b there is a unique c with ac = b;(LQ)

such c is denoted a\b. A groupoid is called idempotent, if it satisfies

xx x.(I)

There is a very natural construction of LDI left quasigroups: on a group G,define a new operation by a b = aba1. It turns out that the groupoid G(),called the conjugation groupoid of G, is an LDI left quasigroup.

In the present thesis we study several problems regarding left distributive leftquasigroups. We do not build a compact theory, the chapters are rather inde-pendent. We are concerned with two types of problems, quite different in theirnature: idempotent problems and non-idempotent problems. Problems in gen-eral (non-idempotent) case are usually reduced to the idempotent case, which isusually more complicated. For instance, we find all non-idempotent subdirectly ir-reducible left distributive left quasigroups, but we have no idea how the idempotentones look like. We describe part of the lattice of subvarieties, modulo the lattice ofidempotent subvarieties (which is probably very complicated). A short overview ofresults follows.

Chapter II contains notation, terminology and basic properties of left distribu-tive left quasigroups used throughout the thesis. A necessary introduction to quasi-groups and loops is also there. Except for these preliminaries, the other chaptersare intended to be more or less self-contained.

Chapters III, IV and V deal with idempotent problems. In Chapter III, westudy the variety of left symmetric left distributive idempotent groupoids (LDI leftquasigroups satisfying the identity x(xy) y) in full detail. Several structuralresults are obtained, mainly for quasigroups. The main result establishes a corre-spondence between LSLDI groupoids of odd exponent and the well known class ofBruck loops of odd exponent.

6 David Stanovsky: Left distributive left quasigroups

In Chapter IV, we show a solution to the eighth Belousovs problem from hisbook [1]. We find a smallest example of a left distributive quasigroup such that itis isotopic to a loop which is not a Bol loop.

The equational theory of conjugation groupoids is investigated in Chapter V. Weshow that it coincides with the equational theory of left distributive left quasigroups(and some other classes), it contains the well known class of groupoid modes and,mainly, we are concerned about finding a basis of the equational theory. We cannotfind it, but we conjecture, that the theory is not finitely based. The results of thischapter were published in [53].

Chapter VI and VII deal with non-idempotent problems. Subdirectly irre-ducible non-imdepotent left distributive left quasigroups are described in ChapterVI. We generalize the results of the paper [18]. In Chapter VII we study varieties ofleft distributive left idempotent groupoids of finite exponent. LDLI groupoids arenot necessarily left quasigroups, however, LD left quasigroups are the most impor-tant examples of LDLI groupoids and the results of this chapter are well applicableto them.

Finally, Chapter VIII contains miscellaneous remarks and open problems.The Appendix consists of a completely independent topic: homomorphic images

of subdirectly irreducible algebras. The main result was published in [52], the wholematerial appeared in a contest thesis (SVOC 2001).

The thesis consists of original achievements of the author. Results of otherauthors are explicitly quoted. Parts of the thesis appeared in authors Mastersthesis, authors thesis for the SVOC contest and in papers [52], [53] and [54].

Historical remarks. The first explicit allusion to selfdistributivity is, perhaps, inthe work of C. S. Peirce [39] from 1880. He discusses various forms of a distributivelaw and he pleads also that selfdistributivity, which has hitherto escaped notice,is not without interest. Another early work, which mentions an example of a non-associative distributive structure, is [51] of E. Schroder (1887). The first knownarticle fully pursued to selfdistributive structures is [8] of C. Burstin and W. Mayerfrom 1929. They investigated two-sided distributive idempotent quasigroups. Thefirst articles dealing with non-idempotent distributive groupoids were [49] of J.Ruedin (1966) in the two-sided case and [26] of T. Kepka (1981) in the one-sidedcase. A comprehensive survey for study of two-sided selfdistributive systems is [21],regarding one-sided distributivity we recommend a recent book [11] of P. Dehornoy.There are several rather new papers that develop and summarize an algebraic theoryof left distributive left quasigroups, mainly from a geometrical point of view: e.g.[13] of R. Fenn and C. Rourke, [50] of H. Ryder or [4] of E. Brieskorn.

The knot quandle. The main reason that led to the development of the theory ofLDI left quasigroups was the discovery of the so called knot quandle (independentlyby D. Joyce [22] and S. V. Matveev [33] in 80s), an LDI left quasigroup assignedto a knot, which turns out to be a full knot invariant. The construction works asfollows.

A knot is a subspace of a (3-dimensional) sphere S3 homeomorphic to a circle.Two knots K1 and K2 are equivalent, if there is an (auto)homeomorphism of thesphere such that its restriction to K1 is a homeomorphism of K1 and K2. One of themost important problems in the knot theory is, given two knots, to decide, whether

David Stanovsky: Left distributive left quasigroups 7

they are equivalent. For this reason, invariants (with respect to the equivalence) ofknots are searched.

Usually only tame knots are studied (tame means equivalent to a close polygonalcurve). A classical invariant of tame knots is the fundamental group. It is definedfor a knot K to be the fundamental group of the topological space S3 r K. It isa full invariant, i.e. two tame knots are equivalent, iff their fundamental groupsare isomorphic. The problem is that fundamental groups are usually not very wellcomputable, so one would like to look for more simple invariants. The knotquandle is a possibility.

Consider a regular projection of a tame knot, i.e. a mapping to a (2-dimensional)plane such that there are only finitely many crossings and none of them is a three-fold crossing. This projection divides the knot to arcs (by an arc we mean a segmentfrom one underpass through some overpasses to the next underpass); denote the setof arcs A. Choose an orientation of the knot and define the knot quandle

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