ON THE LATTICE OF QUASIVARIETIES OF COMMUTATIVE MOUFANG LOOPS WITH NILPOTENCY CLASS ≤ 2. I

OF COMMUTATIVE MOUFANG LOOPS WITH NILPOTENCY CLASS ≤ 2. I

VASILE I. URSU

We describe all finite lattices of quasivarieties of commutative Moufang loops with nilpotency class≤2.

AMS 2000 Subject Classification: 20N05.

Key words: loop, nilpotency, quasivariety, associator.

0. INTRODUCTION

The Birkhoff-Mal’cev problem of the structure of lattices of quasivarieties (see [1], [2]) is an important issue in the theory of quasivarieties. This problem is very important for the theory of quasivarieties as a number of quasivariety issues consists in studying the lattices of quasivarieties. Besides the main Birkhoff-Mal’cev problem, there is the local problem of the structure of the lattice of the subquasivarieties of concrete quasivarieties of algebric systems.

In particular, this problems was posed by Gr¨atzer [3] for the variety of pseudo- complected distributive lattices, by Kargapolov [4], for the variety of nilpotent groups of class 2 by Petrich [5] for the variety of idempotent semigroups.

Though the main and the local problems are interrelated, their settlement may be different. A similar problem can be posed for commutative Moufang loops.

The present paper is devoted to the investigation of the problem of the description of the lattice of the quasivarieties of commutative Moufang loops with nilpotency class ≤ 2. The author [6] investigated this problem for the commutative Moufang loops with nilpotency class 2 with exponent 3. It was shown there that the lattice of subquasivarieties of any variety M of commu- tative Moufang loops is finite or continuous, and it is finite if and only if M is generated by a finite group. In this paper, in the class N₂ of commutative Moufang loops with nilpotency class ≤ 2, these are described all quasivari- eties which possess a finite number of subquasivarieties. We present in a clear form the list of all loops (which are cyclic groups or are non-associative loops generated by three elements) with the property that K⊆N₂ has only finite

REV. ROUMAINE MATH. PURES APPL.,54(2009),1, 33–51

many subquasivarieties if and only if K is generated by a finite set of loops of this list. In particular, if a finitely generated commutative Moufang loop L with nilpotency class 2 is not approximated by these described loops, then the quasivariety generated by Lcontains a continuum set of distinct quasiva- rieties. There are also described finite lattices and some of them are presented in detail. It is noticed that in the simplest cases these lattices are not modular.

1. NOTATION AND PRELIMINARY RESULTS

A commutative Moufang loop (see [7]) is an algebra L with the unary operation ⁻¹ and a binary operation·in which for any x, y, z∈L we have

xy =yx, xy·zx=x(yz)·x, x⁻¹·xy=y.

We introduce the following notation:

F_n = F_n(x₁, . . . , x_n) is the free commutative Moufang loop of order n generated by the free elements x1, . . . , xn;

N₂ is the class of all commutative Moufang loops with nilpotency class

≤2 defined by the identity

(xy·z)(x·yz)⁻¹·uv = ((xy·z)(x·yz)⁻¹·u)v;

N_2,3k is the variety defined inN₂ by the identity x^3k = 1,

where k6= 0 is a natural number;

Fn(M) = Fn(M;x1, . . . , xn) is the free commutative Moufang loop of rang nof the variety M⊆N₂;

Q(L) is the quasivariety generated by the commutative Moufang loopL;

LqMis the lattice of subquasivarieties of the quasivarietyM⊆N2; T =Q(F₃), T_k=Q(F₃(N_2,3k));

A ∗

MB is the M-free product of the loops A, B which belong to a quasi- variety M⊆N₂;

Z_p^k is the cyclic group of order p^k, wherep is a prime.

Usually, the elements of the commutative Moufang loop Fn(x1, . . . , xn) are called the words of the variables x₁, . . . , x_n and the elements of F_n⁰ are called associator words.

We shall say that the loopLof the varietyM⊆N2has the representation L =lp(x₁, . . . x_nkR = 1), in M if L ∼= F_n(M;x₁, . . . , x_n)/R, where ¯¯ R is the normal subloop in Fn(M) generated by the set R⊆Fn(M).

A loop is referred to as monolite if it is finitely generated and is not decomposable in a direct product of two nonunit subloops of it.

The exponent of a commutative Moufang loop is the least common mul- tiple of the orders of all its elements.

LetMbe a quasivariety ofN2andLan arbitrary commutative Moufang loop. The least normal subloop H of L for which L/H ∈ M is denoted by M(L) and is called the quasiverbal subloop of the loopLcorresponding to the quasivariety M.

Let us agree that the ohrase “the elementx6= 1 of the loopLis approxi- mated by the loop K” is equivalent to the ohrase “there exists a morphism of loops ϕ:L→K such that x^ϕ 6= 1”.

We shall need below the following belonging criterian, the proof of which can be found in [8].

A finitely generated commutative Moufang loop L belongs to the quasi- variety generated by the class K of commutative Moufang loops if and only if the nonunit elements of L are approximated by the loops of K.

Let us recall some notions and results from [1] which will be further, used without citing them sometimes.

Let L be a commutative Moufang loop. The associator [x, y, z] of the elements x, y, z ∈L is defined as [x, y, z] = (xy·z)(x·yz)⁻¹.The associator L⁰ of loop L is the subloop generated by all associators of L. If X, Y, Z are nonvoid subsets of L, then the notation [X, Y, Z] stands for the subloop of L generated by all associators of the form [x, y, z], wherex ∈X,y ∈Y,z∈Z. The set Z(L) = {x ∈ L |[x, y, z] = 1 for all y, z ∈ L} is called the center of the loopL. A subloopH of a commutative Moufang loopLis called normal if

x·yH =xy·H

for any x, y ∈ L. The subloop and the normal subloop in L generated by given elements a₁, . . . , a_n are denoted bylp(a₁. . . a_n) and lp(a₁, . . . a_n)^L, res- pectively. It is easy to see that the associator L⁰ and the center Z(L) of the loop L are normal subloops while the factor-loop L/L⁰ is an Abelian group.

We can easily check that the subloop L^m inL generated by the mth powers of all elements of L is normal.

In any commutative Moufang loop of classN₂ the identities (1) [xy, z, t] = [x, z, t]·[y, z, t],

(2) [x, y, z]³ = 1,

(3) [x, y, z] = [y, z, x] = [y, x, z]⁻¹. do hold.

Theorem of Moufang. If the associator of three elements a, b, c of a commutative loop Moufang Lis equal to a unit, then the subloop generated by

the elements a, b, c is a group. In particular, any two elements ofL generates an associative subloop.

We formulate below some lemmas from the papers [6] and [8].

Lemma 1.1([6]). The nontrivial quasi-identity

&^m_i=1u_i(x₁, . . . , x_n) = 1→u(x₁. . . , x_n) = 1,

which is valid in the commutative Moufang loop F3(N2,3), is equivalent in the variety N_2,3 to the quasi-identity

&^m_l=1([y3l−2, y3l−1, y_3l]a_l= [y1, y2, y3]a1)

&H(y₁, . . . , y_n) = 1→[y₁, y₂, y₃]a₁= 1,

where 3m ≤n, F3(N2,3) e(K(y₁, . . . yn) = 1 → [y1, y2, y3]a1 = 1), lp(a1, . . . , am) ∪H ⊆ lp([yi, yj, yj] : 1 ≤ j < k ≤ n,(i, j, k) 6= (3l− 2,3l −1,3l), l= 1, . . . , n).

Lemma 1.2 ([6, 8]). Let M be a non-associative quasivariety from the arbitrary variety N∈ {N₂, N_2,3k, k = 1,2, . . .} and A, B loops of M repre- sented in N such that

A=lp(x₁, . . . , x_nkM(x₁, . . . , x_n) = 1), B=lp(y₁, . . . , y_mkN(y₁, . . . , y_m) = 1),

where M, N are totalities of associator words. If H is the normal subloop of C =A ∗

N2

Bgenerated by some associators of the form[x_i, x_j, y_k]or[x_i, y_j, y_k], then C/H ∈M.

Lemma 1.3 ([8]). F3n(N_2,3^k (resp., N2); x1, . . . , x3n)/lp([x1, x2, x3][x4, x5, x6]. . .[x3n−2, x3n−1, x3n])∈T_k (resp.,T).

The following lemma can be proved in the same way as Lemma 4 of [6].

Lemma 1.4. Let L be the N2-free product of the loops F_3nⁱ (N_2,3^k, res- pectively, N₂; xⁱ_l, . . . , xⁱ_3n), i= 1, . . . , l, with glued elements

a=

n

Y

i=1

[x¹_3i−2, x¹_3i−1, x¹_3i] =· · ·=

n

Y

i=1

[x^l_3i−2, x^l_3i−1, x^l_3i].

Then a∈L can be represented as a product of a number < n of associators.

We now introduce the notation to be used in the sequel:

M∞∞∞=F₃(x, y, z), Mr∞∞ =lp(x, y, z)kx^3r = 1, Hrs∞=lp(x, y, zkx^3r=y^3s= 1),

Hrst=lp(x, y, zkx^3r=y^3s=z^3t= 1), H00t=Z₃^t, H00∞=Z, H000={1}, where r, s, tare integers such that 0≤r≤s≤t;

Amk (respectively, Am) =lp(aij, 1≤i≤m, 1≤j≤3m+ 3) is a loop of the variety N_2,3k (respectively, N₂) whose determinant relations are (4)

m+1

Y

i=1

[a13i−1, a13i−1, a13i] =· · ·=

m+1

Y

i=1

[am3i−2, am3i−1, am3i], (5) [aij, a_kl, apr] = 1, i6=k∨i6=p∨k6=p, 3< j, l, r <3m+ 3;

B_lmk = A¹_mk× · · · ×A¹_mk (respectively, B_lm = A¹_m × · · · ×A¹_m) is the Cartesian product of l copies of the loop A_mk (respectively, A_m); C_lmk = Blmk/lp aⁱ(aⁱ)⁻¹, 1≤i≤l

(respectively, Clmk = Blm/lp(aⁱ(aⁱ)⁻¹, 1 ≤i≤ l), whereaⁱ is a copy in the loopAⁱ_mk (respectively,Aⁱ_m) of the element

(6) a=

m+1

Y

i=1

[a13i−2, a13i−1, a_13i] of the loop A_mk (respectively, A_m).

As was shown in [6], the subquasivarieties of N2,3 are characterized by associator quasivarieties. For k ≥2, we shall emphasize in N₂ and in N_2,3k, those quasivarieties which have the same structure and are also characterized by associator quasivarieties. We denote by N_(2,3k) the quasivariety of N_2,3k

defined by the quasi-identities

(7) x^3k−1 = 1→[x, y, z] = 1

and

(8) x³=

n

Y

i=1

[x3i−2, x3i−1, x3i]→x³ = 1

for any natural number n. ByN₍₂₎ we denote the quasivariety of N₂ defined by the quasi-identities

(9) x⁹= 1→x³= 1,

(10) x³ = 1→[x, y, z] = 1,

and

(11) x^p = 1→x= 1,

for all primes p6= 3.

Lemma 1.5. For any l ≥1, the lattices of the non-associative quasiva- rieties N_(2,3l) and N₍₂₎ are isomorphic.

Proof. According to the quasi-identities (7), (8) (respectively (9)–(11)) the determinant relations of any monolite non-associative loop of N_(2,3l) (res- pectively, N₍₂₎) are equalities of some associator words to unit elements (see

Lemma 5 of [8]). It is clear that the non-associative subquasivarieties of the quasivarieties N_(2,3l) and N₍₂₎ are generated by monolite loops.

We establish a reciprocal correspondence between non-associative mono- lite loops of N_(2,3l) and monolite non-associative loops of N₍₂₎ in the fol- lowing way. We consider that to the monolite non-associative loop A^l ∈ N_(2,3l) with generators a^l₁, . . . , a^l_n and the determinant relations (in N_(2,3l)) R(a^l₁, . . . , a^l_n) = 1, there corresponds the monolite non-associative loop A ∈ N₍₂₎ with generatorsa1, . . . , anand determinant relations (in N₍₂₎) R(a1, . . . , an) = 1. We now prove the following statements:

1^◦. If the monolite non-associative loopL=lp(x₁, . . . , x_n)belongs to the quasivariety N_(2,3^l₎ respectively, N₍₂₎, then Q(L/L⁰) = Q(Z₃^l) (respectively, Q(L/L⁰) =Q(Z)).

2^◦. Let A^l =lp(a^l₁, . . . , a^l_n), B^l=lp(b^l₁, . . . , b^l_n) be monolite non-associa- tive loops ofN_2,3landA=lp(a1, . . . , an),B =lp(b1, . . . , bn)the corresponding loops of N₍₂₎. If A^l∈Q(B^l) thenA∈Q(B).

The proof of statement 1^◦ follows from the fact thatL/L⁰ =lp(x₁L⁰)×

· · · × lp(xnL⁰) and every cyclic subgroup lp(xiL⁰) is isomorphic to Z₃^l (res- pectively, Z). So Q(L/L⁰) =Q(lp(x_iL⁰)) or Q(L/L⁰) = Q(Z₃l) (respectively, Q(L/L⁰) =Q(Z)).

To prove 2^◦, let u^l be an arbitrary element of the loop A^l. If the corres- ponding element 1 6=u∈A^l, then u is approximated by the loop B, because Q(Z) ⊆Q(B) and, by 1^◦, Q(A/A⁰) =Q(Z). Let 16=u∈ A⁰. Then u can be represented in the form

u= Y

1≤i<j<k≤n

[ai, aj, a_k]^αijk,

where 0 ≤ α_ijk < 3. Suppose the element u^l = Q

1≤i<j<k≤n[ai, aj, a_k]^αijk is approximated by the loop B^l via a morphism of loops ϕ : A^l → B^l defined by the relations (a^l_i)^ϕ = (b^l₁)^β1i. . .(b^l_m)^βmic^l_i, 1 ≤ i≤ n, where 0 ≤β_ji < 3^l, j = 1, . . . , m,c^l_i∈(B^l)⁰. We investigate the morphism of the loopsψ:A→B defined as

a^ψ_i =b^γ₁¹ⁱ. . . b^γ_m^mici, 1≤i≤n,

where 0 ≤ γ_ji and γ_ji = β_ji mod 3^l, for j = 1, . . . , m, and to the element ci ∈ B⁰ there corresponds the element c^l_i ∈(B^l)⁰. Let us verify that u^ψ 6= 1.

First, we observe that (u^l)^ϕ = Y

1≤i<j<k≤n

h

(a^ϕ_i)^ϕ,(a^l_j)^ϕ,(a^l_k)^ϕ iαijk

, u^ψ = Y

1≤i<j<k≤n

h

a^ψ_i, a^ψ_j, a^ψ_k iαijk

.

We substitute in these equalities (a^l_i)^ϕ, a^ϕ_i by their expressions in terms of b^l_i, b_i and using identities (2) and (3) we obtain

(u^l)^ϕ = Y

1≤i<j<k≤n

h

b^l_i, b^l_j, b^l_kiδ_ijk

, u^ψ = Y

1≤i<j<k≤n

[b_i, b_j, b_k]^θijk. On accont of the equations γij =βij mod 3 and identity (1), we obtain δ_ijk= θijk mod 3. Because B^l and B have in their varieties the same determinant relations and (u^l)^ϕ,u^ψ are written with the same words of the corresponding generators, we deduce that u^ψ 6= 1. The statement 2^◦ is thus proved.

Now, we establish a reciprocal correspondence between non-associative subquasivarieties of N_(2,3l) and non-associative subquasivarieties of N₍₂₎ in the following way. Let N^l be an arbitrary quasivariety of N_(2,3^l₎, so N^l =

Q

A^l_i, i∈I , where A_i is the monolite non-associative loop ofN_(2,3l). Then we consider that the corresponding quasivariety N ⊆ N₍₂₎ is generated by the corresponding loops, i.e., N = Q{A_i, i∈I}, where A_i is the monolite non-associative loop of N₍₂₎ corresponding to the loopA^l_i. It follows from 2^◦ that the correspondence established is independent of the choice of generated loops, is reciprocal and conserves the inclusion. The proof is complete.

Lemma 1.6. Form= 1,2, . . . we have A_mk ∈T_k, A_m ∈T.

Proof. An element a∈A_mk (respectively, Am) is approximated by the morphism of loopsϕ:A_mk→F₃(N_2,3k;x, y, z) (respectively,A_m→F₃(x, y, z)) defined by the equalities

a^ϕ₁₁=x, a^ϕ₁₂=y, a^ϕ₁₃=z, a^ϕ₂₁=x, a^ϕ₂₂=y, a^ϕ₂₃=z, . . . . a^ϕ_m1 =x, a^ϕ_m2 =y, a^ϕ_m3 =z, a^ϕ_ij = 1, ∀i, ∀j >3.

Now, we show that A_mk/lp(a) ∈T_k (respectively, Am/lp(a) ∈T). By Lem- ma 1.3, the loops K_i represented inN_2,3k (respectively,N₂) as

Ki=lp

ai1, . . . , ai3m+3||

m+1

Y

j=1

[ai3j−2, ai3j−1, ai3j] = 1

are contained inTk(respectively,T). Next,Amk/lp(a) (respectively,Am/lp(a)) is a free product in the variety N_2,3k (respectively, N₍₂₎) of the loops K_i fac- tored over equations (5) and by Lemma 1.2 it belongs to the quasivariety T_k (respectively, T). The proof is complete.

2. THE FIRST AUXILIARY RESULT

We introduce the notation for some loops of the variety N_2,3^k (respec- tively, N2) as follows:

B =B(n, V, k) (respectively, B(n, V)) = lp

x, x₁, . . . , x_3n || [x, x^α_i¹, . . . , x^α_3n³ⁿ, x^β₁¹, . . . , x^β_3n³ⁿ] = 1, α1, . . . , α3n

β₁, . . . , β_3n

!

∈V

;

H = H(n, V, k) (respectively, H(n, V)) is the factor-loop of the loop B by the relation

x³ =

n

Y

i=1

[x3i−2, x3i−1, x3i];

H_m is the factor-loop of the N₂-free product of the loops B and A_mk (respectively, Am) by the relation

x³=

n

Y

i=1

[x3i−2, x3i−1, x3i]a, where ais defined by (6).

Lemma 2.1. Hm ∈Q(H).

Proof. An elementa∈Hmis approximated by the loopF3(N_2,3k;u, v, w) (respectively, F₃(u, v, w)) via the morphism of loops ϕ : H_m → F₃(N_2,3k) (respectively, H_m→F_s) defined by the equations

x^ϕ₁ =u⁻¹, x^ϕ₂ =v, x^ϕ₃ =w, x^ϕ₄ = 1, . . . , x^ϕ_3n= 1, x^ϕ = 1, a^ϕ_l

1 =u, a^ϕ_l

2 =v, a^ϕ_l

3 =w, l= 1, . . . , m, a^ϕ_ij = 1, i= 1, . . . , m, j= 4, . . . ,3m+ 3.

Now, we show that H_m/lp(a)∈Q(H). Indeed, H_m/lp(a) ∼=H∗(A_mk/lp(a)) (respectively, H∗(Am/lp(a))), whereAmk/lp(a)∈Tk (respectively,Am/lp(a)

∈ T). But T_k ∈ Q(H) (respectively, T ∈ Q(H)). Then by Lemma 1.2, we have Hm/lp(a)∈Q(H). The proof is complete.

Lemma 2.2. An element x^3α ∈ Hm, where α 6= 0 mod 3, cannot be represented as a product of m−1 associators.

Proof. Let N =lp

x1, x2, . . . , x3n, a11, a12, a13, a21, a22, a23, . . . am1, am2, am3, [x, aij, a_kl], 1≤i, k≤m, 4≤j, l ≤3m+ 3

Hm

⊆Hm.

ThenHm/N is a direct product of the union of the central subloopsx³N =aN of the cyclic group A = lp(xH) and the subloop D =lp(a_ijN, i= 1, . . . , m, j = 4, . . . ,3m+3). By Lemma 1.4, the elementaN ∈Dcannot be represented as a product of m−1 associators. Since the subloop A is contained in the center of the loop H_m/N, aN cannot be represented as a product of m−1 associators in the whole loopHm/N. Butx^2α=a^αmodN, so thatx^3αcannot be written as a product of m−1 associators.

Lemma 2.3. LetH_m³ be the set of the cubes of all elements ofHm. Then H_m³ ∩H_m⁰ ={x^3α, 0≤α <3}.

Proof. Since the Moufang loop is diassociative and in any commutative Moufang loop identity (1) is valid, it is easy to see thatH_m³ is a subloop inHm

that is contained in the center ofZ(H_m), so that it also is normal inH_m. From this and the construction of the loopHm, it is clear that the nonunit elements of the associatorH_m⁰ which are contained in the subloop H_m³ arex^±3.

Lemma 2.4. Forr >3^3m6, m≥n, we have Hm∈/ Q(Hr).

Proof. Let ϕ be an arbitrary morphism of loops from Hm into Hr. By Lemma 2.3, x³ ∈H_m³ and x³ ∈ H_m⁰ , hence (x³)^ϕ ∈H_r³, (x³)^ϕ ∈ H_r⁰, so that (x³)^ϕ∈h³_r∩H_r⁰. According to the construction of the loopHm, the elementx³ and, subsequently, (x³)^ϕ can be represented as a product ofm+n+ 1< r−1 associators. But, by Lemma 2.2, the nonunit elements of H_r³ ∩H_r⁰ cannot be represented as a product of r−1 associators. Thus, (x³)^ϕ = 1, so that Hm ∈/ Q(Hr).

Lemma 2.5. If m >3^3r6, r ≥n, then H_m∈/ Q(H_r).

Proof. Let us denote ai=

m+1

Y

i=2

[ai3j−2, ai3j−1, ai3j], i= 1, . . . , m.

Assume that the statement is not true. Then for x³ there exists a morphism of loops ϕ:Hm→Hr such that (x³)^ϕ6= 1. Since the number of generators of the loop H_r is 3r(r+ 1) +n+ 2 and n≤r, we have

H_r⁰ ≤

F_t⁰

= 3¹³t(t−1)(t−2) ≤3^r6.

Subsequently, on account of the assumption m >3^3r6, for some generators of the loop Hr we have 3r(r+ 1) + 3n+ 1 =tand for n≤r−2 we get

H_r⁰

≤ F_t⁰

= 3¹³t(t−1)(t−2) ≤3^72r6.

Thus, for m >3^72r6, we have

lp(ai1, . . . , ai3m+3)^ϕ ⊆lp(aj1, . . . , aj3m+3, j= 1, . . . , i−1, i+ 1, . . . , m)^ϕ. for some i≤m. From this, using (5), the equality a^ϕ_i = 1 follows. Then

(x³)^ϕ= ([x1, x2, x3]. . .[x3n−2, x3n−1, x3n]a)^ϕ

= ([x₁, x₂, x₃]. . .[x3n−2, x3n−1, x_3n][a_i1, a_i2, a_i3])^ϕ,

so that the element (x³)^ϕ ∈ H_r³ ∩H_r⁰ is represented as a product of n+ 1 associators. Since the nonunit elements of H_r³∩H_r⁰ cannot be represented as a product of r−1≥n+ 1 associators, we have (x³)^ϕ = 1, what cannot hold.

This contradiction completes the proof.

Lemma 2.6(First auxiliary result).The latticeLqQ(H)is continuous.

Proof. By Lemma 2.1, Q(H) contains infinitely many loops Hm, where m takes values in the set of natural numbers. Construct the infinite sequence m_i, i= 1,2, . . ., of natural numbers as follows: m₁ =n+ 2,m_i+1 = 3^3m6i + 1 fori >1. We shall now show that different subsets of loops of the set (M_mi), i= 1,2, . . ., generate different quasivarieties.

Let M= Q{M_mi, i∈I}, N =Q

M_mj, j∈J , I 6= J. Suppose that i∈ I and i /∈J, so M_mi ∈M. We show that M_mi ∈/ N. Indeed, if this does not hold, then the element x³ ∈Mm is approximated by a loop Mmj, j ∈J. According to the choice of the sequence{m_i, i= 1,2, . . .}and by Lemmas 2.4, 2.5, we reach a contradiction.

3. THE SECOND AUXILIARY RESULT

Let L be a loop of the variety N2,3 generated by a finite number of elements x₁, . . . , x_n, and T₁(L) its quasiverbal subloop which corresponds to the quasivariety T1 with elements u1, . . . , ut.

3.1. The construction of the loops Lⁱ_m

By Lemma 1.1, for anyui ∈ {u₁, . . . , ut}there are generatorsz1i, . . . , zni

of the loop L such that its defining relations have the form [z_1i, z_2i, z_3i]v_1i =

· · ·= [z_3li−2,i, z_3li−1,i, z_3li,i]v_l1i=ui, Hi(z1i, . . . , zni) = 1,whereHi,v1i, . . . , v_lii are contained in the subloop generated by the associators [z_ri, z_si, z_pi] for all triples (r, s, p) not contained in the set{(1,2,3),(4,5,6), ...,(3li−2,3li−1,3li)}.

In Section 2 we have defined the commutative Moufang loopB_lmk. Take the loop Blin1 =A^l_m1ⁱⁱ × · · · ×A^l_m1ⁱⁱ and denote the generators of the subloop A^j_m1, 1 ≤ j ≤ li, by a^ji₁₁, a^ji₁₂, . . . , a^ji_m,3m+3, respectively. Now, define the

loop Lⁱ_m as the factor-loop of the N2,3-free product of the loops Blin1 and F_n(N_2,3;z_1i, . . . , z_ni) by the relations

(11) H_i(z_1i, . . . , z_ni) = 1,

(12) [a^ji_rs, a^ki_pq, z_li] (j=k→l /∈ {3j−2,3j−1,3j}), (13) [a^ji_rs, zpi, zqi] = 1 (p∨q /∈ {3j−2,3j−1,3j}), (14) a¹ⁱ[z_1i, z_2i, z_3i]v_1i=· · ·=a^lii[z_3li−2,i, z_3li−1,i, z_3li,i]v_lii,

wherea¹ⁱ, . . . , a^lii are elements of the loopsA¹ⁱ_m, . . . , A^l_m1ⁱⁱ corresponding to the element a(see formula (4)) of the loopA_m1. Denote x_i =a¹ⁱ[z_1i, z_2i, z_3i]v_1i.

3.2. The properties of the loops Lⁱ_m We start with

Lemma 3.1. Lⁱ_m∈/ T1.

Proof. Let ϕ :Lⁱ_m → F₃(N_2,3) be a morphism of loops. Two cases are possible:

(1) (a¹ⁱ)^ϕ=· · ·= (a^lii)^ϕ= 1, so

([z1i, z2i, z3i]v1i)^ϕ =· · ·= ([z_3li−2,iz_3li−1,iz_3li,i]v_lii)^ϕ.

Denote N = lp(a¹ⁱ, . . . , a^lii) ⊂ Lⁱ_m. Obviously, N ⊂ Kerϕ and lp(z_1i, . . . , zlii)N/N∼=L. Hence, also using the fact that [z1i, z2i, z3i]v1i∈T1(L), we deduce ([z1i, z2i, z3i]v1i)^ϕ= 1. This means thatx^ϕ_i = (a¹ⁱ₁)^ϕ·([z1i, z2i, z3i]v1i)^ϕ= 1.

(2) (a^ji)^ϕ 6= 1 for some j, 1≤j≤li.

Assume for simplicity j = 1. Since a¹ⁱ = Qm+1

i=1 [a¹ⁱ_13i−2, a¹ⁱ_13i−1, a¹ⁱ_13i] (see for- mula (4)), we can suppose that [a¹ⁱ₁₁, a¹ⁱ₁₂, a¹ⁱ₁₃]^ϕ6= 1. In the commutative Mou- fang loop F₃(N_2,3) the universal formula

τ = ([x1, x2, x3]6= 1 & [x₁, x2, x4] = 1 & [x1, x3, x4] = 1

& [x₂, x₃, x₄] = 1→[x₄, x₅, x₆] = 1)

is valid. In relations (13) that define the loopsLⁱ_m and in the similar relations for the loop B_lim1 (see the construction of this loop), there are the corres- ponding equations

(15) [a¹ⁱ₁₁, a¹ⁱ₁₂, zpi] = 1, [a¹ⁱ₁₁, a¹ⁱ₁₃, zpi] = 1, [a¹ⁱ₁₂, a¹ⁱ₁₃, zpi] = 1 for all p /∈ {1,2,3}, and

(16) [a¹ⁱ₁₁, a¹ⁱ₁₂, a²ⁱ_1r] = 1, [a¹ⁱ₁₁, a¹ⁱ₁₃, a³ⁱ_1r] = 1, [a¹ⁱ₁₂, a¹ⁱ₁₃, a²ⁱ_1r] = 1 for all r, 1≤r≤3m+ 3.

From the inequality [a¹ⁱ₁₁, a¹ⁱ₁₂, a¹ⁱ₁₃]6= 1 and equations (15), by the formula τ, we obtain [z_4i, z_5i, z_6i]^ϕ = 1 and u^ϕ_2i = 1 (u_2i is a product of associators [x, y, z], where at least one of the variables x, y, z is a generator of the form zpi for some p /∈ {1,2,3}). It follows from the inequality [a¹ⁱ₁₁, a¹ⁱ₁₂, a¹ⁱ₁₃]^ϕ 6= 1, equations (16) and the formula τ that

[a²ⁱ₁₁, a²ⁱ₁₂, a²ⁱ₁₃]^ϕ = 1, . . . ,[a²ⁱ_13m+1, a²ⁱ_13m+2, a²ⁱ_13m+3]^ϕ= 1.

Finally, we obtain that in the first case x^ϕ_i = 1. This means that x^ϕ_i = 1 for any morphism of loops ϕ:Lⁱ_m →F₃(N_2,3). The proof is complete.

From relations (12)–(15) that define the loopLⁱ_m we see that the subloop lp a^si_jr, 1≤s≤l_i, 1≤j≤m, 1≤r≤3m+ 3

is a subloop of the typeB_lim1. This subloop in the lemmas below will be just the loop B_lim1.

Lemma 3.2. The elements of the normal subloop B_l^Lim

im1 generated by the loop B_lim1 are approximated in Lⁱ_m by the loop F₃(N_2,3).

Proof. WriteN =lp(x_i,[z_si, z_ri, z_pi] for any triple (s, r, p)∈ {(1,/ 2,3), . . . , (3li−2,3li−1,3li)} and any j >3li)⊂Lⁱ_m. It follows from the definition of the loop Lⁱ_m thatN∩B_l^Lim

im1 6=∅. Then it is enough to show thatLⁱ_m/N ∈T1. It follows from the definitions of the loops Lⁱ_m and Lⁱ_m/N that Lⁱ_m/N is a Cartesian product of li isomorphic copies of the loop

A=

A_m1 ∗

N2,3

F₃(N_2,3;z₁, z₂, z₃)

/lp(a[z₁, z₂, z₃]). It remains to prove that A∈T1. We show thatA/lp(a)∈T1. Indeed,

A/lp(a) =Am1/lp(a) ∗

N2,3

F3(N2,3;z1, z2, z3)/lp([z1, z2z3]).

Clearly, F₃(N_2,3;z₁, z₂, z₃)/lp([z₁, z₂, z₃]) ∈ T₁ and, from the proof of Lem- ma 1.6, Am1/lp(a)∈T1. Then, by Lemma 1.2,A/lp(a)∈T1. Now, we show that the elements of lp(a) ⊂A are approximated by the loopF₃(N_2,3;x, y, z) via the morphism of loops ϕ:A→F₃(N_2,3;x, y, z) defined by

a^ϕ₁₁=x, a^ϕ₁₂=y, a^ϕ₁₃=z, a^ϕ₂₁=x, a^ϕ₂₂=y, a^ϕ₂₃=z, . . . . a^ϕ_m1=x, a^ϕ_m2=y, a^ϕ_m3=z, z₁^ϕ =x, z^ϕ₂ =y, z₃^ϕ=z⁻¹,

a^ϕ_ij = 1, i= 1, . . . , m, j= 4,5, . . . ,3m+ 3.

So,A/lp(a)∈T1and the elements oflp(a) are approximated byF3(N2,3;x, y, z).

Therefore, A∈T₁ and the proof is complete.

Lemma 3.3. Lⁱ_m∈Q(L).

Theprooffollows from the isomorphismLⁱ_m/B_l^Lim

im1

∼=Land Lemma 3.2.

Lemma 3.4. The element x_i ∈Lⁱ_m is approximated by the element c_i = a¹ⁱ =· · ·=a^lii of the loop C_lim1.

Proof. Obviously,

Lⁱ_m/lp(zli, . . . , zni)^Lim∼=Clim1. Then, as a result of composition of morphisms of loops,

Lⁱ_m →Lⁱ_m/lp(zli, . . . , zni)^Lim →Clim1, which completes the proof.

Lemma 3.5. The elementx_i ∈Lⁱ_m cannot be represented as a product of m−1 associators.

Proof. According to Lemma 3.4, it is sufficient to show this for the element

ci =a¹ⁱ =· · ·=a^lii of the loop C_lim1. Let

N =lp

a¹ⁱ₁₁, a¹ⁱ₁₂, a¹ⁱ₁₃, a¹ⁱ₂₁, a¹ⁱ₂₂, a¹ⁱ₃₂, . . . , a¹ⁱ_m1, a¹ⁱ_m2, a¹ⁱ_m3, . . . , a^l₁₁ⁱⁱ, a^l₁₂ⁱⁱ, a^l₁₃ⁱⁱ, a^l₂₁ⁱⁱ, a^l₂₂ⁱⁱ, a^l₂₃ⁱⁱ, . . . , a^l_m1ⁱⁱ , a^l_m2ⁱⁱ , a^l_m3ⁱⁱ

C_lim1.

By Lemma 1.3, the element c_iN of the loop C_lim1/N cannot be represented as a product of m−1 associators. So, this is also true forc_i inC_lim1 and x_i in Lⁱ_m, thus completing the proof.

3.3. The construction of the loops Lm

Let the loop L ∈ N_2,3 be generated by the elements z₁, . . . , z_n, and let {u₁, . . . , ut} be the set of nonunit elements ot the quasiverbal subloopT1(L).

As was shown in Section 3.1, for any element u_i, 1 ≤ i ≤ n, a system of generators z1i, . . . , zni of the loop Lcan be chosen such that

[z1i, z2i, z3i]v1i=· · ·= [z3li−2,i, z3li−1,iz3li,i]vlii =ui, Hi(z1i, . . . , zni) = 1.

If the loops L(z1, . . . , zn) andL(z1i, . . . , zni) are different, then there exists an isomorphismϕ_i :L(z₁, . . . , z_n)→L(z_1i, . . . , z_ni) such thatu^ϕ_i = [z_1i, z_2i, z_3i]v_1i=

· · · = [z3li−2,i, z3li−1,i, z3li−2]vlii, H(z1, . . . , zn)^ϕi = H(z1i, . . . , zni). Let ψ :

Lⁱ_m/B_l^Lim

im1 →L(z_1i, . . . , z_ni).Since the elementsz₁^ϕi, . . . , z_n^ϕi generate the loop L(z1i, . . . , zni), the cosetsz_1i^ϕiψ−1, . . . , z_ni^ϕiψ−1 generate the factor-loopLⁱ_m/B_l^Lim

im1. In each of these sets we choose an element zⁱ₁, . . . , z_nⁱ such that z₁ⁱ, . . . , z_nⁱ ∈ lp(z_1i, . . . , z_ni) ⊂ Lⁱ_m. Then Lⁱ_m = lp(zⁱ₁, . . . , z_nⁱ, B_lim1). Now, for simplicity denote

Bi =B_lim1, xj =z_j¹, . . . , z_j^s∈L¹_m× · · · ×L^s_m.

At the beginning, for an arbitrary labelling of the elementsu1, . . . , ut∈T1(L), we define the commutative Moufang loop Lm(1, . . . , s) as

Lm(1, . . . , s) =lp(x1, . . . , xn, Bi, i= 1, . . . , s)⊂L¹_m× · · · ×L^s_m. It is clear that L_m(1, . . . , s) is the subdirect product of the loopsL¹_m, . . . , L^s_m. Now, we label the elements u1, . . . , ut and choose the numbers such that

(a)Lm(1), Lm(1,2), . . . , Lm(1, . . . , s)∈/T1; (b)L_m(1, . . . , s)∈T₁ for allk, s < k≤t.

Such a choice is possible because, by Lemma 3.1,Lm(1) =L¹_m ∈/T1. The loop L_m is just the loop L_m(1, . . . , s) that verifies conditions (a) and (b) above.

3.4. Properties of the loops Lm

We start with

Lemma 3.6. Lm∈Q(L).

The proof follows from the definition of the loop L_m (it is a subdirect product of the loops Lⁱ_m) and Lemma 3.3.

Lemma 3.7. T₁(L_m)∩(B₁× · · · ×B_s)^Lm = 1.

Proof. We have to show that every nonunit element of (B1× · · · ×Bs)^L1m =B^L₁^1m× · · · ×B_s^Lm

is approximated by the loopF₃(N_2,3). To do this, it is sufficient to investigate the nontrivial projection of the element onto the ith component and to use Lemma 3.2.

Lemma 3.8. T₁(L_m) ⊆ {1, u₁f₁, . . . , u_sf_s} for some f₁, . . . , f_s ∈ (B₁×

· · · ×B_s)^Lm⊂L_m, andu_i=u_i(x₁, . . . , x_n)∈T₁(L(x₁, . . . , x_n)).

Proof. Leta∈T1(Lm). The correspondencexj(B1×· · ·×B_s)^Lm↔xj ↔ z_j is extended up to the isomorphism of the loop L_m/(B₁× · · · ×B_n)^Lm ' L(x₁, . . . , x_n) ' L(z₁, . . . , z_n). That is why, according to Lemma 3.7, a = u_k(x₁, . . . , x_n)·f, whereu_k∈ {u₁, . . . , u_t},f ∈(B₁×· · ·×B_s)^Lm. Suppose that the assertion is not true, i.e. k > s, and construct the commutative Moufang

loop Lm(1, . . . , s, k) = lp(y1, . . . , ym, Bi, i= 1, . . . , s, k), where yi = xjz_j^k. Now, consider that the initial generators z₁, . . . , z_n of the loop L correspond to the loop L^k_m, i.e.,

(17) u_k(z₁, . . . , z_n) = [z₁, z₂, z₃]v₁ =· · ·=

= [z_3lk−2, z_3lk−1, z_3lk]v_3lk, H_s(z₁, . . . , z_n) = 1.

Then in the projections of the loop Lm(1, . . . , s, k) onto each component Lⁱ_m we have the equations

(18) [z₁ⁱ, z₂ⁱ, z₃ⁱ]vⁱ₁≡ · · · ≡[zⁱ_3lk−2, zⁱ_3lk−1, z_3lⁱ _k]vⁱ_lkmodB_i^Lim.

In particular, in the projection on the kth component, according to the defi- nition, we have

(19) [z^k₁, z₂^k, z₃^k]v1a^1k=· · ·= [z^k_3lk−2, z^k_3lk−1, z_3l^k_k]v_lka^lk,k,

where a^1k, . . . , a^lk,k ∈ Bm^Lkm. In the projections of the first s components, by multiplying equations (18), we obtain

(20) [x₁, x₂, x₃]v₁b₁=· · ·= [x_3lk−2, x_3lk−1, x_3lk]v_lkb_lk, where b₁, . . . , b_lk ∈(B₁× · · · ×B_s)^Lm(1,...,s).

It is clear that asb₁we can take every element of (B₁× · · · ×B_s)^Lm(1,...,s), and then a = [x1, x2, x3]v1b1. Finally, in the loop Lm(1, . . . , k) we have the equation

(21) [y1, y2, y3]v1b1a^1k=· · ·= [y_3lk−2, y_3lk−1, y_3lk]v_lkb_lka^lk,k.

Setx= [y₁, y₂, y₃]v₁b₁a^1k ∈L_m(1, . . . , s, k) and let us show thatxis not appro- ximated by the commutative Moufang loop F3(N2,3). Let λ:Lm(1, . . . , s, k)

→F3(N2,3) be some morphism of loops. Two cases are possible:

(1) (a^jk)^λ 6= 1 for some j ≤ l_k. For simplicity, let j = 1. Then by formula τ (see the proof of Lemma 3.1) and the definition of the loop L¹_m we have (a^2k)^λ = 1. From the definition of the loop Lm(1, . . . , s, k) we deduce that v^λ₂ = 1, [y4, y5, y6]^λ = 1, b^λ₂ = 1. This means thatx^λ= 1.

(2) (a^jk)^λ = 1 for allj = 1, . . . , lk. Denote

N =lp(a^jk, j= 1, . . . , l_k), A=lp(y₁, . . . , y_n, B₁, . . . , B_s)⊂L(1, . . . , s, k).

Then, obviously, the application θ:y_jN →x_j, j = 1, . . . , n,bN →¯b,b∈B_i,

¯b ∈ B¯_i, where ¯B_i is the same loop B_i, but from L(1, . . . , s), ¯b is the element corresponding to b, can be extended up to an isomorphism between AN/N and L(1, . . . , s).

Consider an application ¯λ:AN/N →F3(N2,3) such that (yN)^¯λ =y^N for any y∈A; ¯λis a morphism of loops, sinceN ⊂Ker¯λ. This means that ¯λθ⁻¹ :