Some Varieties of Quasigroups , Loops and their Parastrophes

The role of the parastrophes in the theory of quasigroups and loops is well known. It is our approach to investigate remarkable classes of loops and quasigroups and to relate them to their parastrophes. Some consequences for code loops are presented.


Introduction
Denote by Q l the variety of all quasigroups which have a left identity element 1 l and by Q LF l its subvariety consisting of all quasigroups Q ∈ Q l satisfying the identity (LF) x( yz) = (x y)(e(x)z), where x e(x) = x for all x ∈ Q.
In the variety Q l one has a convenient homomorphism theory, so it is not necessary to consider the general theory of congruence relations which is inevitable in the variety of all quasigroups (see [10], [1, Chapter IV.9], [3, p. 55f.]).More precisely, for Q, R ∈ Q l and a homomorphism η : Q → R one defines the left kernel by It follows from [10, Section 3, p. 104] that ker l η is a normal subquasigroup of Q and that η(Q) is isomorphic to the factor quasigroup Q/ker l η, consisting of the cosets (ker l η)x, x ∈ Q with the multiplication (ker l η)x • (ker l η) y = (ker l η)(x y).
In our note we make use of the well known fact that in an arbitrary L Fquasigroup Q the mapping e : Q → Q arising from the identity (LF) is an endomorphism of Q (see [2, p. 108], [23], [20,Proposition 1.1]).If in addition Q belongs to Q l we use the existence of the left kernel ker l e to study the structure of Q.Note that ker l e coincides with the left nucleus Having shown in [20,Proposition 1.3] that every L F -quasigroup is isotopic to an L F -quasigroup with a left identity element our interest is directed mostly to the variety Q LF l .By [2, p. 108] every L F -quasigroup is isotopic to an L M -loop (see also [20,Proposition 1.4]).There are various answers to the question what L Fquasigroups are isotopic to Moufang loops ( [8], [28], [9]).We would like to add that for a Moufang loop M the (left) parastrophe is isotopic to M .
Our approach gives another way to establish the relation between L Fquasigroups and Moufang loops.In recent publications (see [6]) one can find the parastrophe approach applied to Moufang loops in general.
A comprehensive list of references on this theme can be found in [24].
In Section 3 we give examples showing what smooth L F -quasigroups can be obtained as parastrophes of smooth loops (Theorem 4.5).
In the closing section we describe properties of parastrophes of Code loops.Similar results hold for Chein loops (see [5]).(2.1)

Given a quasigroup
It is easy to see that in the magma In the usual way one defines the multiplication groups of the quasigroup Q • .Define left and right translations of Q • as Then one obtains the groups of left, right and two sided multiplications of Q • as the subgroups of the symmetric group of the set Q generated by the respective sets of translations: Proof.Denoting the multiplication in Proof.Let ϕ be an epimorphism from Q • onto some quasigroup X with a left neutral element 1 l .Then the mapping q → ϕ(q) gives us a homomorpism ϕ * from the parastrophe Q * = P(Q • ) onto P(X ).By construction the quasigroups Q • and Q * have the same left neutral element 1 l , and the left kernel of ϕ * is the parastrophe of the left kernel of ϕ.Putting X = Q • /N • the assertion follows from our remark on the left kernels in the introduction.
Let Q • be a quasigroup and let T = ( f , g, h) be a triplet of bijectons on the set Q.With the definition One says that a loop Q has the left inverse property if there is a mapping (2.8) Consider the isotopism T (I ) = (I , id Q , id Q ).Hence we have the following Proposition 2.5.For a loop Q • with the inverse property the quasigroups Conversely, if x * x = 1 l for all x ∈ Q, then 1 l is left neutral element of the quasigroup P(Q * ), again by Proposition 2.1(ii).Denoting the multiplication in P(Q * ) by • one obtains for all x ∈ Q. Hence 1 l is the two-sided neutral element of P(Q * ).

L F -quasigroups
Definition 3.1.For a quasigroup Q * the mapping Definition 3.2.A quasigroup in which the deviation is an endomorphism will be called a quasigroup with endomorphic deviation [17].Note that every loop is a quasigroup with endomorphic deviation.

Proposition 3.3. An L F -quasigroup is a quasigroup with endomorphic deviation.
A quasigroup Q * is called left square-distributive (see [26]) if the identity holds.One has the following well known result (see [2, p. 67]).

Proposition 3.4. A loop satisfies the left square distributive identity if and only if it is a commutative Moufang loop.
The validity of the following theorem was noted in [26].Later square distributive quasigroups played a great role as parastrophes of L F -quasigroups in [12] -where they are called left semimedial -and in [28].Let G be a group which is not of exponent 2. Since G is an L F -quasigroup, it follows from Theorem 3.5 that the parastrophe P(G) is a quasigroup, satisying the left square distributive identity, but P(G) is not a loop by Propositon 2.6.So we have shown the Corollary 3.8.There are square distributive quasigroups which are not loops.
In contrast to this corollary it was shown in [14] that the validity of one of the Moufang identities in a quasigroup H implies that H is a Moufang loop.
We now use the fact that according to Proposition 3.3 in an L F -quasigroup Q the deviation e, given by e(x) = x\x is an endomorphism.Remark 3.9.We would like to emphasize that a quasigroup (not a loop) with the endomorphic deviation is not necessarily an L F -quasigroup.This is because that the property of e(x) being an endomorphism in a quasigroup is equivalent to the fact that x → x 2 is the endomorphism in its left parastrophe.Such a situation happens in a commutative diassociative loop, which is not necessarily a Moufang loop (see [11]).
We define

Furthermore we call
the hyperimage of e. Proposition 3.10.In an L F -quasigroup Q with a left neutral element 1 l the following statements are true: Proof.Statement (i) and the inclusions in statement (ii) are obvious.The remainder of the Proposition follows from remarks made in the introduction.

Lie groups
In this section we describe some peculiarities of parastrophes of connected abelian Lie groups.
Example 4.1.In the parastrophe Q = P(Z) one has e(x) = x + x.It follows that the mapping e is injective but not surjective.One has The same is true for every torsionfree abelian group which is not divisible by 2.
By Theorem 3.6 the parastrophe ) is an L F -quasigroup.For a subset S of the set Q we denote by 〈S〉 + the subgroup generated by S in the abelian group C 2 ∞ .By what we have shown before, in the quasigroup Q * the following statements are hold: The statements under (a) and (b) hold as well for any vector group R n .
Using the examples above we obtain Theorem 4.5.Let G be a connected analytic commutative Moufang loop.Then the following propositions hold The closure of the hypernucleus of P(G) coincides with P((R/Z) m ).(d) e * is surjective but for m > 0 it is not an automorphism.
Proof.Proposition (a) follows from [18], [25], while (b) is obvious.The remaining propositions are consequences of the examples in this section.

Nuclear L F-quasigroups
We call an L F -quasigroup Q with a left neutral element nuclear if its nuclear series becomes stationary for some index i 0 ∈ N. In this case we call the smallest number

Proof.
To shorten the notation we put N n = N n (Q).As mentioned already in the introduction we know that N 1 = ker l e = N uc l (Q) ⊳ Q and all N i+1 /N i are groups.
First we want to show that N 2 is isotopic to a group.On Q we consider the isotopism T = (R −1 1 l , id, id).The multiplication of Q T is given by x • y = (x/1 l ) y.
we consider on the set Q the multiplication defined by a * b = a\ • b.

Theorem 3 . 5 .Theorem 3 . 6 .Corollary 3 . 7 .
A quasigroup (Q, •, \, /) is an L F -quasigroup if and only if the parastrophe P(Q) satisfies the left square distributive identity.Proof.In order to show that P(Q) satisfies identity (3.1) we denote the multiplication in P(Q) by * .One has x( yz) = (x y)(e(x)z), e(x) = x\x (3.2) and a\b = a * b. (3.3) Setting t = x y and w = (x\x)z gives us y = x\t = x * t, z = (x * x) * w. (3.4) From (3.2) we obtain yz = x\((x y)((x\x)z) (3.5)which implies(x * t)((x * x) * w) = x * (t w) (3.6)We substitute u = t w.Then w = t\u = t * u.Inserting this in (3.6) yields(x * x) * (t * u) = (x * t) * (x * u).(3.7)Thus we have shown that the quasigroup P(Q) is left square distributive.To prove the converse it is sufficient to observe that by Proposition 2.1 (i) the quasigroups P(P(Q)) and Q coincide.Using Propostion 2.6, Lemma 3.5 and Proposition 3.4 one immediately obtains the following The parastrophe P(Q) of an L F -quasigroups Q is a loop if and only if P(Q) is a commutative Moufang loop.The left parastrophe Q * = P(Q) of a commutative Moufang loop Q is an L F -quasigroups with left neutral element 1 l such that the identity x * x = 1 l holds.Moreover, in this case the left Bol identity holds in Q * .

Remark 3 . 11 .
In[20, Corrolary 3.4]  we have shown that in a finite L F -quasigroup Q with the left neutral element 1 l one has a Fitting decomposition where N

Example 4 . 3 .Example 4 . 4 .
Example 4.2  shows that for a compact connected abelian Lie groups T the hypernucleus N ∞ (P(T )) is a dense countable subquasigroup of P(T ).To see this it is enough to consider the 1-dimensional torus R/Z.But this group contains a dense subgroup isomorphic to C 2 ∞ .For the additive group R + and Q * = P(R + ) one easily sees that the following statements are true (a) N 1 (Q