LEFT DISTRIBUTIVE LEFT QUASIGROUPS. PhD Thesis. Charles University in Prague. July 2004

Transcription

1 LEFT DISTRIBUTIVE LEFT QUASIGROUPS DAVID STANOVSKÝ PhD Thesis Charles University in Prague July

2 2 David Stanovský: Left distributive left quasigroups Acknowledgement I wish to thank to everybody I learned anything from during past twenty six years. Above all, to my advisor, Jaroslav Ježek, for his support and insightful advices. To my inofficial coadvisor, Ralph McKenzie, for numerous inspirative conversations and for financial and other support during my stay at Vanderbilt University in Nashville. To Věra Trnková, Aleš Drápal, Tomáš Kepka, Přemysl Jedlička and other people from the Department of Algebra at Charles University in Prague, for their interest and useful advices. To people from the Department of Mathematics at Vanderbilt University in Nashville, who made my stay in the United States nice and fruitful. To Anna Romanowska and Barbara Roszkowska from the Technical University in 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 under grants 201/02/0594 and 201/02/0148 is greatfully acknowledged.

3 David Stanovský: Left distributive left quasigroups 3 Contents Acknowledgement 2 Contents 3 I. Introduction 5 II. Preliminaries 9 1. Basic facts 9 2. Normal form of terms Definable sets are left ideals Quasigroups and loops 14 III. Left symmetric left distributive idempotent groupoids Normal form of terms Exponent Representation by Bruck loops Applications Cycles Group of displacements Left ideal decomposition and BL-sum Cores 27 IV. On loops isotopic to left distributive quasigroups 29 V. Equational theory of group conjugation The variety generated by conjugation The equational theory of conjugation On the role of idempotency 36 VI. Subdirectly irreducible non-idempotent left distributive left quasigroups Description Examples 42 VII. On varieties of left distributive left idempotent groupoids Varieties satisfying x n+1 x Varieties satisfying x m+n x m 47 VIII. Miscellaneous remarks and open problems Miscellaneous remarks Open problems 50 Appendix: Homomorphic images of subdirectly irreducible algebras The necessary condition Groupoids Algebras with rich signature Monounary algebras 62

4 4 David Stanovský: Left distributive left quasigroups 5. Unary algebras Particular varieties 65 References 66

5 David Stanovský: 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 look at the recent monography [11] of P. Dehornoy. We consider a particular subclass: left distributive left quasigroups, known also as racks, wracks, quandles, automorphic sets, pseudo-symmetric sets, crystals, etc. (some of the names refer to the idempotent case, some of them don t). A groupoid (i.e. a set equipped with a binary operation) is called left distributive, if it satisfies the identity (LD) x(yz) (xy)(xz). It is called left quasigroup, if (LQ) for every a, b there is a unique c with ac = b; such c is denoted a\b. A groupoid is called idempotent, if it satisfies (I) xx x. There is a very natural construction of LDI left quasigroups: on a group G, define a new operation by a b = aba 1. 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 left quasigroups. We do not build a compact theory, the chapters are rather independent. We are concerned with two types of problems, quite different in their nature: idempotent problems and non-idempotent problems. Problems in general (non-idempotent) case are usually reduced to the idempotent case, which is usually more complicated. For instance, we find all non-idempotent subdirectly irreducible left distributive left quasigroups, but we have no idea how the idempotent ones look like. We describe part of the lattice of subvarieties, modulo the lattice of idempotent subvarieties (which is probably very complicated). A short overview of results follows. Chapter II contains notation, terminology and basic properties of left distributive left quasigroups used throughout the thesis. A necessary introduction to quasigroups and loops is also there. Except for these preliminaries, the other chapters are intended to be more or less self-contained. Chapters III, IV and V deal with idempotent problems. In Chapter III, we study the variety of left symmetric left distributive idempotent groupoids (LDI left quasigroups satisfying the identity x(xy) y) in full detail. Several structural results are obtained, mainly for quasigroups. The main result establishes a correspondence between LSLDI groupoids of odd exponent and the well known class of Bruck loops of odd exponent.

6 6 David Stanovský: Left distributive left quasigroups In Chapter IV, we show a solution to the eighth Belousov s problem from his book [1]. We find a smallest example of a left distributive quasigroup such that it is isotopic to a loop which is not a Bol loop. The equational theory of conjugation groupoids is investigated in Chapter V. We show 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 cannot find it, but we conjecture, that the theory is not finitely based. The results of this chapter were published in [53]. Chapter VI and VII deal with non-idempotent problems. Subdirectly irreducible non-imdepotent left distributive left quasigroups are described in Chapter VI. We generalize the results of the paper [18]. In Chapter VII we study varieties of left distributive left idempotent groupoids of finite exponent. LDLI groupoids are not necessarily left quasigroups, however, LD left quasigroups are the most important examples of LDLI groupoids and the results of this chapter are well applicable to 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 whole material appeared in a contest thesis (SVOČ 2001). The thesis consists of original achievements of the author. Results of other authors are explicitly quoted. Parts of the thesis appeared in author s Master s thesis, author s thesis for the SVOČ contest and in papers [52], [53] and [54]. Historical remarks. The first explicit allusion to selfdistributivity is, perhaps, in the work of C. S. Peirce [39] from He discusses various forms of a distributive law and he pleads also that selfdistributivity, which has hitherto escaped notice, is not without interest. Another early work, which mentions an example of a nonassociative distributive structure, is [51] of E. Schröder (1887). The first known article fully pursued to selfdistributive structures is [8] of C. Burstin and W. Mayer from They investigated two-sided distributive idempotent quasigroups. The first 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-sided case. 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 theory of 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 of LDI left quasigroups was the discovery of the so called knot quandle (independently by D. Joyce [22] and S. V. Matveev [33] in 80 s), an LDI left quasigroup assigned to a knot, which turns out to be a full knot invariant. The construction works as follows. A knot is a subspace of a (3-dimensional) sphere S 3 homeomorphic to a circle. Two knots K 1 and K 2 are equivalent, if there is an (auto)homeomorphism of the sphere such that its restriction to K 1 is a homeomorphism of K 1 and K 2. One of the most important problems in the knot theory is, given two knots, to decide, whether

7 David Stanovský: Left distributive left quasigroups 7 they are equivalent. For this reason, invariants (with respect to the equivalence) of knots are searched. Usually only tame knots are studied (tame means equivalent to a close polygonal curve). A classical invariant of tame knots is the fundamental group. It is defined for a knot K to be the fundamental group of the topological space S 3 K. It is a full invariant, i.e. two tame knots are equivalent, iff their fundamental groups are isomorphic. The problem is that fundamental groups are usually not very well computable, so one would like to look for more simple invariants. The knot quandle 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 threefold crossing. This projection divides the knot to arcs (by an arc we mean a segment from one underpass through some overpasses to the next underpass); denote the set of arcs A. Choose an orientation of the knot and define the knot quandle to be an LDI left quasigroup generated by A and presented by relations (1) ab = c for every underpass, where a coming under b yields c and where b is going over a in the left-right direction (in a sense of the orientation of the knot), and (2) a\b = c in the situation, where b is going over a in the right-left direction. The involutory knot quandle is an LSLDI groupoid (i.e. an LDI left quasigroup satisfying x\y xy) defined analogously regardless the orientation. Examples: (i) a b c trefoil The trefoil quandle is presented by a, b, c; ab = c, bc = a, ca = b. (ii) One can easily compute that the involutory trefoil quandle is the core of the cyclic group Z 3. (The core of an abelian group G is the groupoid G( ) defined by a b = 2a b.) c a d b figure-eight The the figure-eight quandle is presented by a, b, c, d; ab = d, b\c = a, cd = b, d\a = c.

8 8 David Stanovský: Left distributive left quasigroups One can easily compute that the involutory figure-eight quandle is the core of the cyclic group Z 5 (recall that it is generated by arcs, not the set of arcs). Joyce and Matveev proved that two tame knots are equivalent, iff their knot quandles are isomorphic. (Thus, in particular, the knot quandle does not depend on the chosen regular projection.) The proof is based on investigation of the role of conjugation in fundamental groups and it is rather difficult. Involutory knot quandles are also invariant, however, there are two non-equivalent tame knots with isomorphic involutory knot quandles; on the other hand, known examples are rather complicated.

9 David Stanovský: Left distributive left quasigroups 9 II. Preliminaries In the present chapter, we summarize basic properties of left distributive left quasigroups used throughout the thesis. We assume only very basic preliminary knowledge of universal algebra. An introduction to the theory can be found for instance in the textbooks [7] or [34]. Our notation and terminology is rather standard. We denote F V (X) the free groupoid over a set X in a variety V. We consider the standard representation of free groupoids by terms modulo equivalence within the theory. The letters x, y, z usually refer to term variables. A subgroupoid generated by a set X is denoted X. 1. Basic facts In the present thesis, we consider mostly binary algebras; they are called shortly groupoids. The binary operation is denoted implicitly multiplicatively, unless other notation is introduced (such as or ). In this section, we summarize some terminology regarding groupoids. First, we list some of the most often used groupoid identities: (LD) x(yz) (xy)(xz) left distributivity (RD) (zy)x (zx)(yx) right distributivity (M) (xy)(uv) (xu)(yv) mediality (also entropy) (n-ls) x(x( (x y) y }{{} left n-symmetry n (LI) (xx)y xy left idempotency (I) xx x idempotency Left and right distributive groupoids are called distributive (D). Note that medial idempotent groupoids are distributive. Left symmetry refers implicitly to left 2- symmetry. Let G be a groupoid and a G. The left translation by a in G is a mapping L a : G G, x ax; the right translation is defined dually and denoted R a. For operations other than, for instance, we use the notation L a, Ra. It is often useful to translate identities into the language of translations. A groupoid G is left (right) distributive, iff all left (right) translations are endomorphisms. It is left n-symmetric, iff L n a = id for every a G. And it is left idempotent, iff L a = L aa for every a G. Also, note that a groupoid G is medial, iff the mapping G G G, (a, b) ab, is a homomorphism. We say that a groupoid G is left cancellative, if all left translations are injective, and left divisible, if all left translations are onto. It is called a left quasigroup, if all left translations are permutations; or equivalently, if the equation ax = b has a unique solution x for any a, b G. Such x is usually denoted a\b. Left quasigroups do not form a groupoid variety (i.e. they cannot be axiomatized by identities in

10 10 David Stanovský: Left distributive left quasigroups the language of multiplication). However, they are axiomatized by the identities x(x\y) y and x\(xy) y in the language {,\}. For a left quasigroup G, we define the left multiplication group LMlt(G), to be a subgroup of the symmetric group over G generated by all L a, a G. Left n-symmetric groupoids are indeed left quasigroups with a\b = a(a(... (a b))). }{{} n 1 On the other hand, any finite left quasigroup is left n-symmetric for some n Lemma. (1) A groupoid is a left distributive left quasigroup, iff all its left translations are automorphisms. (2) If G is a left distributive left quasigroup, then L ϕ(a) = ϕl a ϕ 1 for every a G and every automorphism ϕ of G. In particular, L ab = L a L b L 1 a for every a, b G (3) The mapping λ : a L a is a homomorphism of an LD left quasigroup G into the conjugation groupoid of the left multiplication group of G. Proof. (1) follows immediately from the definition. (2) Since ϕ(ab) = ϕ(a)ϕ(b) for every a, b G, we have ϕl a = L ϕ(a) ϕ and thus also L ϕ(a) = ϕl a ϕ 1 for every a G. (3) It follows from (2) that λ(ab) = L ab = L a L b L 1 a = λ(a) λ(b) Lemma. Let G be a left distributive left quasigroup. Then (1) G is left idempotent; (2) if G is right cancellative, it is idempotent; (3) if G is right distributive, it is idempotent. Proof. (1) xy x(x(x\y)) LD (xx)(x(x\y)) (xx)y. (2) x(xx) LD (xx)(xx) and use right cancellativity. (3) (xx)(xx) RD (xx)x and use left cancellativity. There is a recent important structural result on non-idempotent LDLI groupoids, due to P. Jedlička [17]. Let ip G denote the smallest equivalence on a groupoid G containing the set {(a, aa) : a G} Lemma. (P. Jedlička) Let G be an LDLI groupoid. Then (a, b) ip G iff there are k, l such that a k = b l, the equivalence ip G is a congruence, G/ip G is idempotent and ip G is the smallest congruence such that the corresponding factor is idempotent. Moreover, for any (a, b) ip G, ac = bc holds for every c G. Proof. See [17]. Consequently, any block of ip G is a subalgebra of G and it is term equivalent to a connected monounary algebra. We will thus use the terminology of unary algebras (and directed graphs) for ip G -blocks. By a circle of length n we mean a groupoid isomorphic to the groupoid C n, defined on the set {0,..., n 1} by ab = b+1 mod n. An infinite path is a groupoid isomorphic to the groupoid C of integers with the operation ab = b + 1. It is easy to see that any ip-block of an LD left quasigroup is either a circle, or an infinite path. And any ip-block of an n-lsld groupoid is a circle of length k for some k n. Example. Let G be a group and put a b = aba 1 for every a, b G. Then G( ) is an LDI left quasigroup, called the conjugation groupoid of G. Moreover, if A is

11 David Stanovský: Left distributive left quasigroups 11 a subset of G such that a n = 1 for every a A, the subgroupoid A of G( ) is n-lsldi. Example. Let G be a group, e a central element in G and put a e b = aba 1 e for every a, b G. Then G( e ) is an LD left quasigroup and it is idempotent, iff e = 1. All ip-blocks of G( e ) are cycles of length equal to the order of e. Again, if A is a subset of G such that a n = 1 for every a A and e n = 1, the subgroupoid A of G( e ) is n-lsld. Example. Let G be a group, e a central involution in G and put a e b = ab 1 ae for every a, b G. Then G( e ) is an LSLD groupoid. It is idempotent, iff e = 1; in such a case it is called the core of G. Cores play quite important role in the theory of LSLDI groupoids. See more in Chapter III. There are also other constructions of LSLD groupoids from groups. We refer to the paper [54]. Example. Let R be a commutative ring with a unit, M a module over R and k R such that the mapping x kx is a permutation of M. For all a, b M, put a k b = (1 k)a + kb. Then M( k ) is a medial idempotent left quasigroup. More examples of LD left quasigroups (mostly from geometry) can be found, for instance, in the paper [13]. Example. There are two isomorphism types of left distributive left quasigroups on a 2-element set (one of them is idempotent, the other is not): a b a a b b a b a b a b a b b a There are six isomorphism types of left distributive left quasigroups on a 3- element set (three of them idempotent, three not): a b c a a b c b a b c c a b c a b c a a c b b a b c c a b c a b c a a c b b c b a c b a c a b c a b c a b b c a c b c a a b c a a b c b a c b c a c b a b c a a c b b a c b c a c b The third idempotent example, the core of the cyclic group Z 3, is a commutative medial quasigroup and it forms the smallest Steiner triple system. It appears as an example of a non-associative distributive structure already in the paper [51] from Example. According to D. Joyce [23], finite simple LDI left quasigroups can be described in terms of triples (G, C, m), where G is a finite simple group, C a conjugacy class in G and m a positive integer. It follows from Lemma 1.3 that cycles

12 12 David Stanovský: Left distributive left quasigroups of prime length are the only simple non-idempotent LD left quasigroups (see also [17]). 2. Normal form of terms In order to decrease the number of parentheses in terms, we assume implicitly that letters in terms are right associated, i.e. xyz = x yz = x(yz). We define the n-th power by x n = x } {{ xx }. (Note that other possible definitions of powers n do not really make sense in left idempotent groupoids: one can check by an easy induction that any unary term t(x) is LI-equivalent to the term x n, where n is the depth of the rightmost variable in t.) Also, x [n] y will stand for x } {{ x } y = L n x(y). There should be no confusion, because it follows from left idempotency by an easy induction that x k y xy for any k. For instance, the left n-symmetry takes the form x [n] y y in this notation. It is important to observe that in any LD left quasigroup In case of left n-symmetry we get and similarly by induction xy z xy x(x\z) LD xy(x\z). xy z xyx [n 1] z (x 1 x 2 x m ) z x 1 x 2 x m 1 x m x [n 1] m 1 x[n 1] 2 x [n 1] 1 z. Consequently, using left n-symmetry and left distributivity, one can write any term over an alphabet X in the form x 1 x 2 x n 1 x m, for some m and x 1,..., x m X. The following theorem shows a useful normal form of terms in the variety of n- LSLDI and n-lsld groupoids. It is a generalization of a similar result of the author obtained for 2-LSLD(I) groupoids in [54] Theorem. Let n 2 and X be a non-empty set. For every x X, let C x be a cyclic group of order n generated by x and denote G the free product of all C x, x X. (1) The free n-lsldi groupoid over X is isomorphic to the subgroupoid X of the conjugation groupoid of the group G. For every term over the alphabet X there is a unique n-lsldi-equivalent term of the form x [k1] 1 x [k2] 2 x [km 1] m 1 x m, where x 1,..., x m X, x i x i+1 and k i {1,..., n 1} for every i = 1,..., m 1. (2) The free n-lsld groupoid over X is isomorphic to the subgroupoid X of the groupoid H( (1,e) ), where H denotes the direct product G C e and C e is a cyclic group of order n generated by e. For every term over the alphabet X there is a unique n-lsld-equivalent term of the form x [k1] 1 x [k2] 2 x [km 1] m 1 xkm m, where x 1,..., x m X, x i x i+1 for every i = 1,..., m 1 and k i {1,..., n 1}. n

13 David Stanovský: Left distributive left quasigroups 13 Proof. We have seen that any term is n-lsld-equivalent to a term x 1 x 2 x m. Collect equal neighbours and use left n-symmetry to decrease the exponents bellow n (resp. idempotency in (1) for the rightmost variable). (1) Denote F the subgroupoid of the conjugation groupoid of G generated by X. Clearly, F is left n-symmetric, because it is generated by elements of order n. We show that different terms in the described form represent different elements in F. Let x 1,..., x m X such that x i x i+1, i = 1,..., m 1, and k i {1,..., n 1}. The word x [k1] 1 x [k2] 2... x [km 1] m 1 x m = x k1 1 xk xkm 1 x mx km 1 m 1... x k2 2 x k1 1 is irreducible in the free product, so such words are pairwise different. (2) Proceed similarly. Since x [k1] 1 x [k2] 2... x [km 1] m 1 x km m = x k1 1 xk xkm 1 x mx km 1 m 1... x k2 2 x k1 1 e 1+ i ki, different terms represent different elements in the groupoid. Remark. The free n-lsld groupoid over X is isomorphic to the product of the free n-lsldi groupoid over X and the cycle of length n. It can be checked directly from the representation proven above or one can apply Theorem VII.1.3. Remark. According to [12], the free groupoid over X in the variety generated by LDI left quasigroups is isomorphic to the subgroupoid X of the conjugation groupoid of the free group over X (see also Chapter V). Consequently, in the language {, \}, there is a normal form for terms in the variety of LD(I) left quasigroups similar to that from Theorem 2.1 (in fact, one can use the representation of terms by words in free groups). 3. Definable sets are left ideals Let G be a groupoid. A non-empty subset I G is called a left ideal, if GI I. In other words, if ab I for every a G, b I. Clearly, if G is a left quasigroup and I G is a left ideal, then G I is also a left ideal. A subset S G is called definable in G if there exists a formula Φ (in the language Λ consisting of the multiplication symbol) with a single free variable such that S = {a G : Φ(a)}. A relation α on G is called definable, if there exists a formula Φ (in the language Λ) with two free variables such that α = {(a, b) G G : Φ(a, b)}. A relation α is called right stable, if (a, b) α implies (ac, bc) α for every c. A variant of the following proposition can be found in Jeřábek, Kepka, Stanovský [18] for 2-LSLD groupoids Theorem. Let G be a left distributive left quasigroup. Then (1) every definable subset in G is either empty, or a left ideal; (2) every definable right stable equivalence on G is a congruence of G. Proof. Let S = {a G : Φ(a)} for a formula Φ(x). We need to prove that Φ(a) implies Φ(ba) for every a, b G. Let Φ(x) be a formula of the form (Q 1 x 1 )(Q 2 x 2 )... (Q n x n )Ψ(x, x 1,..., x n ), where the letters Q 1,..., Q n stand for quantifiers and Ψ is a formula with n + 1 free variables without quantifiers. Since all left translations are permutations, it is enough to prove the following statement: for all a, a 1,..., a n, b G ( ) Ψ(a, a 1,..., a n ) implies Ψ(ba, ba 1,..., ba n ).

14 14 David Stanovský: Left distributive left quasigroups By induction. Assume Ψ is an atomic formula of the form t s, t, s terms. If t(a, a 1,..., a n ) s(a, a 1,..., a n ), then t(ba, ba 1,..., ba n ) = bt(a, a 1,..., a n ) = bs(a, a 1,..., a n ) = s(ba, ba 1,..., ba n ) (the first and third equalities follow from the fact that L b is an automorphism) and thus ( ) holds. Assume Ψ is of the form Ψ 1 Ψ 2, where {, }. It is clear that if Ψ 1, Ψ 2 satisfy ( ), then so does Ψ. Finally, assume that Ψ is of the form Ψ 1 and Ψ 1 satisfies ( ). Suppose that Ψ(a, a 1,..., a n ) is true and Ψ(ba, ba 1,..., ba n ) is not. Then Ψ 1 (ba, ba 1,..., ba n ) is true and so is Ψ 1 (L 1 b (ba), L 1 b (ba 1 ),..., L 1 b (ba n )) Ψ 1 (a, a 1,..., a n ), since L 1 b is an automorphism of G. Hence, a contradiction with Ψ = Ψ 1. For right stable equivalences proceed analogously. It follows from Lemma 1.3 that every equivalence bellow ip G is right stable, hence one can use Theorem 3.1 in such a situation. Remark. Note that Theorem 3.1 holds also for sets and equivalences definable in the language {, J 1,..., J n }, where J 1,..., J n are left ideals and J 1,..., J n the corresponding unary relational symbols. (We have no use for this more general statement.) 4. Quasigroups and loops In this section, we summarize some terminology and basic facts regarding quasigroups and loops. We will use them mainly in Chapters III and IV. A quasigroup is a groupoid, where all left and right translations are permutations (i.e. which is both left and, dually, right quasigroup). A loop is a quasigroup with a unit element 1 (i.e. x 1 1 x x holds). An alternative definition says that quasigroups are groupoids with left and right cancellation, it means ab = ac b = c and ba = ca b = c holds for any a, b, c, and left and right division, it means for every a, b the equations ax = b and ya = b have a solution. By cancellativity, the solutions are unique and they are denoted a\b and b/a. Quasigroups and loops can be, indeed, axiomatized by indentities in the language {, /, \}. An introduction to the theory of quasigroups and loops can be found, for instance, in the book [40]. We say that quasigroups Q 1 and Q 2 are isotopic, if there are bijective mappings α, β, γ : Q 1 Q 2 such that α(ab) = β(a)γ(b) for every a, b Q 1. It means, Q 2 is obtained from Q 1 by permuting rows, columns and renaming entries in the multiplication table. A left quasigroup Q is said to have the left inverse property (LIP), if for every a L there is a 1 Q such that a\b = a 1 b for every b Q. The RIP is defined dually. Loops may be regarded as non-associative groups. Weaker forms of associativity are extensively studied. A loop is called Moufang, if the following equivalent identities hold: xyxz (xy x)z, (zx y)x z(xyx), xy zx (xyz)x x(yz x). Commutative Moufang loops are usually denoted additively. They can be based (relatively to the axioms of loops) by the identity A loop is called left Bol, if (x + x) + (y + z) (x + y) + (x + z). xyxz (xyx)z

15 David Stanovský: Left distributive left quasigroups 15 holds (the word left is usually dropped). A loop is Moufang, iff it is left and right Bol. Bol loops have LIP, thus Moufang loops have both LIP and RIP. Moufang loops are diassociative, i.e. any 2-generated subloop is a group (hence all terms in two variables are associative). Bol loops are power-associative, i.e. any 1-generated subloop is a cyclic group. Consequently, powers are well-defined and we can speak about the order of an element and about the exponent of a Bol loop. Also, the Bol identity implies xxy x 2 y (left reflexivity). It is well-known that any loop isotopic to a Bol loop is a Bol loop. More information about Moufang and Bol loops can be found for instance in [9], [29] or [1]. Later we will need the following two lemmas (they can be found e.g. in [29]) Lemma. Let L be a left quasigroup with a unit satisfying LIP and the Bol identity. Then L is a Bol loop, a\b = a 1 b and a/b = b 1 (ba b 1 ) for every a, b L. Proof. Since a(a 1 b) = b, it is the unique solution of ax = b. To prove right cancellation, suppose ac = bc, compute ca = cacc 1 = Bol (cac)c 1 = (cbc)c 1 = Bol cbcc 1 = cb and use left cancellation. Finally, we need to check that (b 1 (ba b 1 ))b = a. Indeed, by the Bol identity, the left side is equal to b 1 (ba)b 1 b = b 1 ba = a. A loop with LIP is said to have the automorphic inverse property (AIP), if it satisfies (xy) 1 x 1 y Lemma. Let L be Bol loop. Then the following properties of L are equivalent. (1) AIP. (2) (xy) 2 xy 2 x. (3) L 2 ab = L al 2 b L a for every a, b L. Proof. (1) (2). xy xy 2 xx 1 y 1 AIP xy 2 x(xy) 1 Bol (xy 2 x)(xy) 1. Hence by 4.1 xy 2 x xy/(xy) 1 (xy)((xy) 1 (xy) (xy)) (xy) 2. (2) (3). To show (ab) 2 c = abbac for every a, b, c L, compute (ab) 2 c = (2) (ab 2 a)c = Bol ab 2 ac = LR abbac. (3) (1). For any a, b L we have ab = ab 2 aa 1 b 1 = L a L 2 b L a(a 1 b 1 ) = (3) L 2 ab (a 1 b 1 ) = (ab) 2 (a 1 b 1 ). Left cancellation by ab yields (ab)(a 1 b 1 ) = 1, hence AIP holds. Bol loops satisfying the properties of the previous lemma are called Bruck loops (also known as K-loops or gyrocommutative gyrogroups) 1. A Bruck loop is Moufang, iff it is commutative. Indeed, we derive commutativity from (2) by diassociativity. On the other hand, commutative Moufang loops obviously satisfy AIP. There are two recent books on Bruck loops: H. Kiechle s [29] and A. A. Ungar s [59]. The first one is written from a purely algebraic point of view, whereas the second one is motivated by relativistic physics: the key example (pointed out by Ungar) which led to the fast growth of the theory of Bruck loops is the set of all admissible velocities in relativistic physics with the Einstein addition. A remarkable paper [16] of G. Glauberman deals with Bruck loops of odd order (called B-loops there). For historical notes see e.g. the appendix of [29]. 1 The name is chosen in honor of R. H. Bruck, who first explicitly used the notion in his book [5]. It seems to be the oldest name for this structure, see for instance [44].

16 16 David Stanovský: Left distributive left quasigroups III. Left symmetric left distributive idempotent groupoids In the present chapter we consider left 2-symmetric left distributive idempotent left quasigroups; we call them shortly LSLDI groupoids. In literature, they appear under a bunch of other names, such as kei, symetric sets, symmetric groupoids or involutory quandles. The first paper on LSLDI groupoids seems to be [56] of M. Takasaki (1943), however, most results come from 70 s and 80 s. 2 Our aim is to continue investigations of this interesting variety, mainly in direction of representation of LSLDI groupoids by other structures. We also summarize some of the older results and shortly reprove some of them. We start with a normal form of terms in LSLDI and LSMI (M refers to mediality) and define the exponent. Then we proceed with the main theorem, a representation of LSLDI groupoids of odd exponent by Bruck loops of odd exponent. In the next section, we investigate cycles, an important structural feature of LSLDI groupoids. Further, we present a construction, called BL-sum, of LSLDI groupoids from Bol loops. Finally, the last section is devoted to cores. The origin of the theory of LSLDI groupoids lies in the book [32] of Ottmar Loos (1969). He studied so called symmetric spaces and he found that they can be described equivalently as differentiable manifolds equipped with a smooth LSLDI operation satisfying the following condition: for any point a, there is a neighborhood U of a such that a is the only fixed point of the left translation L a on U. A basic example of a symmetric space comes as follows: on any manifold with metric, one can define a product x y to be the image of y by the symmetry through x. For instance, on the real line x y = 2x y, on the n-sphere x y = 2 x, y x y. Actually, the 3-sphere gives a reason why the fixed point condition is defined locally the point symmetric to the south pole through the north pole is the south pole. A different example is the set of all n n positive definite symmetric matrices over the reals with the operation A B = AB 1 A. Grassman manifolds and Jordan algebras are also symmetric spaces. And, of course, any LSLDI groupoid can be considered as a discrete symmetric space. Motivated by the above natural example, Nobuo Nobusawa [37] and others started to investigate LSLDI groupoids from a purely algebraic point of view in 70 s. The most remarkable references are [37], [24], [41], [42], [22], [23]. The results are surveyd in [55] Lemma. Let G be a groupoid. Then (1) G is LSLD, iff every left translation is either the identity, or an involutory automorphism of G; (2) if G is LSLD, then L ab = L a L b L a for every a, b G. 2 Actually, the content of the rather long Takasaki s paper is a big puzzle it is in Japanese. It seems that it contains some basic properties and many examples. The journal is available in the library of the Mathematical Institute of the Polish Academy of Sciences in Warsaw.

17 David Stanovský: Left distributive left quasigroups 17 (3) if G is LSLD, the mapping λ : a L a is a homomorphism of G into the core of the left multiplication group of G. Proof. Easy (cf. II.1.1). A groupoid is called effective, if its left translations are pairwise different. An effective LSLDI groupoid thus can be embedded into the core a group. The kernel of the homomorphism λ was called a central congruence by R. S. Pierce and it was extensively studied in his papers [41], [42]. Let L be a Bol loop. We define the core of L, denoted by Core(L), to be the groupoid L( ) with a b = ab 1 a. It is easy to check that cores of groups are LSLDI. It is not that easy to check that cores of Bol loops are also LSLDI (a different proof can be found in [1], for instance) Lemma. The core of a Bol loop is LSLDI. Proof. For idempotency, a a = aa 1 a = a. For left symmetry, a a b = a(ab 1 a) 1 a = a(ab 1 a) 1 ab 1 aa 1 b = Bol a(ab 1 a) 1 (ab 1 a)a 1 b = aa 1 b = b. For left distributivity, a (b c) = a(bc 1 b) 1 a = a(bc 1 b) 1 aa 1 bc 1 bb 1 cb 1 a = Bol and similarly we get a(bc 1 b) 1 (bc 1 b)b 1 cb 1 a = ab 1 cb 1 a (a b) (a c) = (ab 1 a)(ac 1 a) 1 (ab 1 a) = ab 1 a(ac 1 a) 1 (ab 1 a)(ab 1 a) 1 ac 1 aa 1 ca 1 (ab 1 a) = ab 1 cb 1 a. Cores will be discussed in the last section in detail. 1. Normal form of terms We recall from the introduction that LSLD groupoids satisfy and similarly by induction also xy z LS xy xxz LD xyxz (x 1 x 2 x n ) z x 1 x 2 x n x 2 x 1 z. Consequently, using left symmetry and left distributivity, one can write any term over X in the form x 1 x 2 x n 1 x n, x i X. Note that mediality takes the form xyzt zyxt. Using these observations, one can derive a useful normal form of terms in LSLDI. (The first part of the following theorem is a particular case of Theorem 2.1.) 1.1. Theorem. Let X be a non-empty set. ( ) (1) The free LSLDI groupoid over X is isomorphic to the subgroupoid X of the core of the free group over X and any term over X is LSLDI-equivalent to a unique term of the form x 1 x 2 x n 1 x n, x 1,..., x n X and x i x i+1, i = 1,..., n 1.

18 18 David Stanovský: Left distributive left quasigroups ( ) (2) The free LSMI groupoid over X is isomorphic to the subgroupoid X of the core of the free abelian group over X and any term over X is LSMIequivalent to a unique term of the form x 1 x 2 x n 1 x, x 1,..., x n 1, x X, {x 1, x 3,... } {x 2, x 4,... } =, x 1 x 3..., x 2 x 4..., x / {x n 1, x n 3,... }, where (X, ) is a fixed linear order. Proof. We have seen that any term over X can be written in LSLDI in the form x 1 x 2 x n. If x i = x i+1, we use left symmetry to cancel them out. Using mediality, we can reorder the variables to the form ( ). Let F (G resp.) be the subgroupoid X of the core of free (abelian, resp.) group over X. We show that different terms in the form ( ) (( ) resp.) represent different elements in F (G resp.). For x 1,..., x n X with x i x i+1, i = 1,..., n 1, put ε = ( 1) n. The word x 1 x 2 x n 1 x n = x 1 x 1 2 x ε n 1x ε n x ε n 1 x 1 2 x 1 is irreducible in the free group, so such words are pairwise different. Similarly, in the free abelian group (denoted additively) x 1 x 2 x n 1 x = 2x 1 + 2x 3 + 2x 2 2x 4 + εx. When {x 1, x 3,... } {x 2, x 4,... } = and x / {x n 1, x n 3,... }, the value uniquely determines the term up to the order of x 1, x 3,... and x 2, x 4,... A groupoid G is called n-medial, if every n-generated subgroupoid of G is medial. Clearly, 4-medial groupoids are medial. It follows from 1.1 that every LSLDI groupoid is 2-medial, since F LSLDI (2) = F LSMI (2). Ježek, Kepka, Němec proved in [21], section IV.2, that an idempotent groupoid is (left and right) distributive, iff it is 3-medial. The backward implication is indeed easy: zx yx zy xx zy x. Using 1.1, one can prove rather easily the forward implication for left symmetric groupoids: using right distributivity in the form xyzx zyx, one can transform any term in three variables into its medial normal form (the details are omitted). The following two lemmas will be used later Lemma. Let G be an LSLD groupoid. Then G is medial, iff L a L b L c L d = L c L d L a L b for every a, b, c, d G. Proof. Using the normal form, it is easy to see that abcdx = cbadx = cdabx for every a, b, c, d, x G in LSMI groupoids. On the other hand, if the latter is true, (xy)uv LSLD xyxuv xuxyv LSLD (xu)yv Lemma. Let G be a LSMI quasigroup. Then L a L b L c = L (a/b)bc for all a, b, c G. Proof. Using several times Lemmas 0.1 and 1.2, we get L a L b L c = L (a/b)b L b L c = (0.1) L a/b L b L a/b L b L c = L a/b L b L a/b L c L c L b L c = (1.2) L a/b L b L c L b L a/b L c L c = (0.1) L a/b L bc L a/b = L (a/b)bc.

19 David Stanovský: Left distributive left quasigroups Exponent We define a sequence of binary terms w n by w 1 (x, y) = x and w n+1 (x, y) = xw n (y, x) for every n 1. In other words, w n (x, y) is xy yx }{{} n for n odd and xy xy }{{} n for n even. It is easy to see that F LSLDI (x, y) = F LSMI (x, y) = {w n (x, y), w n (y, x) : n N} and they are isomorphic to the core of the additive group of integers. Under the isomorphism ϕ determined by x 1, y 0 one has ϕ(w n (x, y)) = n and ϕ(w n (y, x)) = 1 n Lemma. Let w m (x, y) y and w n (x, y) y hold in a variety V of LSLDI groupoids. Then w k (x, y) y holds in V, where k is the greatest common divisor of m, n. Proof. Suppose m < n. For m, n of the same parity we have Otherwise, y w n (x, y) = xy xyw m (x, y) xy xyy = w m n (x, y). y w n (x, y) = xy yxw m (y, x) xy yxx = w m n (x, y). Proceed further by the Euclid s algorithm. By the exponent of an LSLDI groupoid G we mean the least positive integer n for which w n (x, y) y holds in G, if such exists, and otherwise. It is denoted exp(g). The exponent of a variety V of LSLDI groupoids is defined to be the least n such that all its elements have exponent at most n (in other words, the least n such that w n (x, y) y holds in V, if such exists) Proposition. (1) Every locally finite variety of LSLDI groupoids has a finite exponent. (2) Every finite LSLDI groupoid has a finite exponent. Proof. (1) Since F V (x, y) is finite, there are m, n such that w m (x, y) w n (x, y) or w m (x, y) w n (y, x) holds in the variety. Using left multiplication and left symmetry, we can derive w k (x, y) y for certain k. Take the smallest then. (2) The variety generated by a finite groupoid is indeed locally finite, hence both the variety and the groupoid have a finite exponent Proposition. Let G be an LSLDI groupoid of finite exponent. Then the following statements are equivalent. (1) G is a quasigroup; (2) G is right cancellative; (3) G has odd exponent. Proof. (1) (2) is trivial. (2) (3) Suppose G is an LSLDI quasigroup of exponent 2n. Then w 2n (x, y) y and there is a, b G such that w n (a, b) b. However, w n (x, y)y w 2n (x, y) y yy, hence, by right cancellativity, w n (x, y) y. A contradiction with minimality of the exponent. (3) (1) Let n be the exponent of G and a, b G. Then w (n+1)/2 (a, b)b = w n+1 (a, b) = aw n (b, a) = aa = a, so w (n+1)/2 (a, b) is a solution of the equation xb =

20 20 David Stanovský: Left distributive left quasigroups a. Note that w (n+1)/2 (xy, y) w n+1 (x, y) xw n (y, x) xx x. Consequently, whenever ac = bc, we have a = w (n+1)/2 (ac, c) = w (n+1)/2 (bc, c) = b Proposition. Let G be a finite LSLDI quasigroup. Then G is odd. Proof. Fix some e G and assume all sets G a = {a, ea}, a G. Clearly, G e = 1, G a = 2 for all a e (since G has odd exponent, ea a) and for every a, b either G a = G b or G a, G b are disjoint. Consequently, G is a disjoint union of one 1-element and several 2-element sets, thus it has an odd number of elements. The converse is not true, groupoids satisfying xy y may have arbitrary number of elements, though they have exponent 2. However, a finite Bruck loop L has odd exponent, iff L is odd (see [16], cf. next section). 3. Representation by Bruck loops Let L be a Bol loop. Put for all a, b L a b = a 2 b 1 = a ab 1. The resulting groupoid L( ) is denoted DG(L) Lemma. Let L be a Bruck loop. Then DG(L) Core(L) via a a 2. Proof. We need to verify that (a b) 2 = (a 2 b 1 ) 2 = a 2 b 2 a 2 = a 2 b 2 for every a, b L. However, this follows directly from 4.2(3). Let G be an LSLDI quasigroup of an odd exponent n. Choose e G and put for all a, b G a b = (L a L e ) (n+1)/2 (b) = ae } {{ ea } eb. n The resulting algebra G(, e) is denoted BL e (G). The definition of the operation appears, for instance, in the paper [46] of B. Roszkowska, where it is used to establish a correspondence of LSMI quasigroups of odd exponent and abelian groups of odd exponent. We extend the corresponence to LSLDI quasigroups and Bruck loops Theorem. Let G be an LSLDI quasigroup of an odd exponent n and e G, let L be a Bruck loop of an odd exponent n. Then (1) BL e (G) is a Bruck loop of exponent n; (2) DG(L) is an LSLDI quasigroup of exponent n; (3) DG(BL e (G)) = G and BL 1 (DG(L)) = L. Proof. (1) Put m = (n + 1)/2. Note that w n (x, y) y implies that (L a L e ) n = id for any a, e. Indeed, (L a L e ) n (b) = w n (a, e)eb = eeb = b. Now, we are ready to check the axioms of Bruck loops. Let a, b, c G. e a = (L e L e ) m (a) = a. a e = (L a L e ) m (e) = aw n (e, a) = aa = a. a (ea) = (L a L e ) m (ea) = (L a L e ) m 1 (aeea) = (L a L e ) m 1 (a) = w n (a, e) = e. (ea) a = (L ea L e ) m (a) = (L e L a L e L e ) m (a) = (L e L a ) m (a) = ew n (a, e) = ee = e. (ea) (a b) = (L ea L e ) m (L a L e ) m (b) = (L e L a ) m (L a L e ) m (b) = (L e L a ) m (L e L a ) m (b) = b.

21 David Stanovský: Left distributive left quasigroups 21 Consequently, a 1 = ea. the Bol identity: (a b a) c = (L a b a L e ) m (c) = ((L a L e ) m (L b L e ) m L a (L e L b ) m (L e L a ) m L e ) m (c) = (L a L e ) m (L b L e ) m [L a (L e L b ) m (L e L a ) m L e (L a L e ) m (L b L e ) m ] m 1 L a (L e L b ) m (L e L a ) m L e (c) and we need to prove that this is equal to a b a c = (L a L e ) m (L b L e ) m (L a L e ) m (c). Hence, it remains to show that the mapping ϕ = [L a (L e L b ) m (L e L a ) m L e (L a L e ) m (L b L e ) m ] m 1 L a (L e L b ) m L e is the identity. Indeed, we have L a (L e L b ) m (L e L a ) m L e (L a L e ) m (L b L e ) m = L a (L e L b ) m (L e L a ) n+1 L e (L b L e ) m = and so L a (L e L b ) m L e L a L e (L b L e ) m = (L a (L e L b ) m L e ) 2 ϕ = (L a L e (L b L e ) m ) n 1 L a L e (L b L e ) m = (L a L e (L b L e ) m ) n = L e (L e L a L e (L b L e ) m L e ) n L e = L e (L a e L b e b... ) n L e = L 2 e = id. AIP: a 1 b 1 = (L ea L e ) m (eb) = (L e L a ) m (eb) = e(l a L e ) m (b) = e(a b) = (a b) 1. It follows from 4.1 that G( ) is a Bruck loop. We claim that k-th power of a in BL e (G), denoted by a k, is w k (a, e). Proceed by induction: a = w 1 (a, e) and a k = a a k 1 = (L a L e ) m (w k 1 (a, e)) = w n+k (a, e) = w k (a, e) for k 2. (2) According to Lemmas 3.1 and 0.2, DG(L) Core(L) is LSLDI. We show that w k (a, b) = b in DG(L), iff (ab 1 ) k = 1 in L. For k, we have w k (a, b) = a b b a = a 2 b 2... a 2 b 2 a = a 2 b 2... aab 2 a = Bruck a 2 b 2... a 2 b 2 a(ab 1 ) 2 = Bol a 2 b 2... (a 2 b 2 a)(ab 1 ) 2 = Bruck a 2 b 2... a(ab 1 ) 4 = = a(ab 1 ) k 1. In order to get b, one needs k divisible by the order of ab 1. (3) The first statement follows from a (a b 1 ) = (L a L e ) m (L a L e ) m (eb) = (L a L e ) n+1 (eb) = (L a L e ) n (aeeb) = ab. For the second, a 1 1 a 1 b = a a b = } a 2 {{ a } 2 b = a n+1 b = ab. (n+1)/ Corollary. Let n be odd. The variety of pointed LSLDI groupoids satisfying w n (x, y) y is term equivalent to the variety of Bruck loops satisfying x n 1. (A pointed groupoid is an algebra of type (2, 0), without any additional condition on the constant.) 3.4. Corollary. (B. Roszkowska [46]) In the notation of 3.2, if G is medial, then BL e (G) is an abelian group, and if L is an abelian group, then DG(L) = Core(L) is medial. Consequently, LSMI quasigroups of finite exponent are cores of abelian groups of the same exponent.

22 22 David Stanovský: Left distributive left quasigroups Proof. For the first assertion, it is enough to prove asociativity. Write (a b) c as (L a b L e ) m (c) = ((L a L e ) m L b (L e L a ) m L e ) m (c) = ((L a L e ) m L b L e (L a L e ) m ) m (c), use 1.2 to get ((L a L e ) m (L a L e ) m L b L e ) m (c) = ((L a L e ) n L a L e L b L e ) m (c) = (L a L e L b L e ) m (c) and, again by 1.2, conclude that (L a L e ) m (L b L e ) m (c) = a (b c). Commutativity follows from AIP. For the second statement, observe that a b c d = 2a 2b+2c d = c b a d for any a, b, c, d L Corollary. In the notation of 3.2, if G is distributive, then BL e (G) is a commutative Moufang loop, and if L is a commutative Moufang loop, then DG(L) = Core(L) is distributive. Consequently, LSDI quasigroups of finite exponent are cores of commutative Moufang loops of the same exponent. Proof. For the first part, it is enough to prove that BL e (G) is commutative. According to Ježek, Kepka, Němec [21], distributive groupoids are trimedial, i.e. every 3-generated subgroupoid of a distributive groupoid is medial. Since the proof of commutativity of BL e (H) for H medial involves only 3-generated subgroupoids (a b is in the subgroupoid generated by a, b, e), we get commutativity in the distributive case too. To prove that cores of commutative Moufang loops are distributive, use for instance the Belousov s theorem ([1], theorem 9.7) saying that the core of a Moufang loop is distributive, iff the loop satisfies xy 2 x yx 2 y. This is certainly satisfied in the commutative case. The direct proof is also easy: (x z) (y z) 2(2x z) (2y z) 2(2x z) + (z 2y) CML (2x z + z) + ((2x z) 2y) 2x + (( 2y) + (2x z)) Bol (2x 2y + 2x) z 2(2x y) z (x y) z using diassociativity and commutativity several times. A particularly interesting case is the exponent 3. The identity w 3 (x, y) y is indeed equivalent to commutativity (and also to right symmetry) and commutative LSLDI groupoids are traditionally called distributive symmetric quasigroups. They correspond to commutative Moufang loops of exponent 3 via the operation a b = aeaeb = eab (see Belousov [1]). The operation can be defined equivalently in the following way: a b = Re 1 (a)l e (b), where R e is the right translation by e. Consequently, the loop BL e (G) is isotopic to G and the above theorems say that every LSLDI (LSMI, LSDI) quasigroup is isotopic to a Bruck (abelian, commutative Moufang) loop (group). Also, we can define the operation for any LSLDI quasigroup (not necessarily with infinite exponent) in this way. It is proven independently in [1], [36], [30] and maybe also elsewhere that Q( ) is a Bruck loop for any LSLDI quasigroup Q. However, none of the authors observed that the operation can be defined polynomially (for finite exponent) and that there is a there-and-back correspondence as described in Theorem 3.2.

23 David Stanovský: Left distributive left quasigroups Applications Example. Let L be a Bol loop of an odd exponent n. Then Core(L) has exponent n and the operation of BL 1 (Core(L)) is a b = a (n+1)/2 ba (n+1)/2 = ab a. Hence L( ) is a Bruck loop. In particular, every Bol loop is isotopic to a Bruck loop. Theorem 3.2 puts together the results of G. Glauberman [16], Kano, Nagao, Nobusawa [24] and V. M. Galkin [14], [15]. They proved that both Bruck loops of odd order (Glauberman) and finite LSLDI quasigroups (K.N.N. and independently Galkin) are solvable, possess the Lagrange property and satisfy a Sylow theorem. (A Bruck loop is said to be solvable, if it can be constructed from the trivial loop by a chain of extensions by abelian groups. An LSLDI quasigroup is said to be solvable, if it can be constructed from the trivial quasigroup by a chain of extensions by medial LSLDI quasigroups (note that the definitions of solvability are in accordance with the correspondence 3.2). A groupoid G is said to have the Lagrange property, if H divides G for every subgroupoid H of G. A groupoid G satisfies a Sylow theorem, if for every prime divisor p of G and every subgroupoid H with H = p k there is a subgroupoid K of G containing H such that K is a maximal prime power divisor of G, i.e. K = p l so that p l+1 does not divide G.) The approach of the first two papers is rather similar. K.N.N. use the natural embedding of an LSLDI quasigroup G into the core of the symmetric group over G, while Glauberman embeds a Bruck loop L into the group under the operation ab a (cf. the previous example). Then they translate properties of groups into those of the embedded structures. Galkin s approach is more general. He proves that an LSLDI quasigroup G (not necessarily finite) is solvable, iff for every ϕ in the commutant of the left multiplication group of G and for every a G the mapping L a ϕ has a unique fixpoint. An example of a non-solvable (infinite) LSLDI quasigroup is provided: on the set {(x 1, x 2, x 3 ) R 3 : x x 2 2 x 2 3 = 1, x 3 > 0} take the operation x y = 2 x, y x y, where x, y = x 1 y 1 x 2 y 2 + x 3 y Cycles In this section we study cycles, introduced by N. Nobusawa in [37]. They seem to be an important structural feature of LSLDI groupoids. We gather partially from the paper [37], but our approach is inovative. A sequence w 1 (a, e), w 2 (a, e), w 3 (a, e)... is called a cycle generated by a over e. If no confusion may happen, we denote the elements a 1, a 2, a 3,.... Let ord e (a) denote the order of the cycle, that is the least n such that w n (a, e) = e, if such exists, or otherwise. A cycle is said to be k-periodic, if w i (a, e) = w i+k (a, e) for every i. Let G be an LSLDI groupoid and a, e G. Put n = ord e (a) and m = ord(l a L e ) (i.e. m is the order of L a L e in the symmetric group over G). For m finite, we have a 2m = e, because w 2m (a, e) = (L a L e ) m (e) = id(e) = e. Hence n 2m. On the other hand, for n finite, (L a L e ) n (t) = w n (a, e)et = eet = t for any t, so (L a L e ) n = id and m divides n. Consequently, (C1) m =, iff n =. If m, n are finite, n = m or n = 2m. Let us discuss periodicity. Since w k+2m (a, e) = (L a L e ) m (w k (a, e)) = w k (a, e), we see that