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

OF COMMUTATIVE MOUFANG LOOPS WITH NILPOTENCY CLASS ≤ 2. II

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: associator, quasiidentity, lattice, commutative loop.

1. THE MAIN RESULT We shall prove the following

Theorem. A lattice K of subquasivarieties of commutative Moufang loops with nilpotency class ≤ 2 is finite if and only if any quasivariety in K is generated by a finite set of loops of the types H∞∞∞, Hr∞∞, Hrs∞, H_rst, Z_p^m, where p6= 3 is a prime number.

Proof. Sufficiency. Let Σ be the set of all loops of the types indicated and L the direct product of a finite number of loops of Σ. Obviously, K = Q(L) contains only a finite number of loops. Each of them is generated by three elements. So, it is sufficient to show that every subquasivariety in Kis determined by loops generated by three elements. According to the belonging criteria, this will be true if every finitely generated loopB ∈Kis approximated by its subloops generated by three elements.

We show that it is indeed the case. Ifb∈B and b /∈B⁰ then, obviously, b is approximated by a cyclic subgroup. Let b ∈ B⁰. Since H_rst, H∞st, H∞∞t, H∞∞∞ are approximated respectively by the loops Z₃^t ×F3(N2,3), Z₃^t×Z×F₃(N_2,3), Z₃^t×Z×F₃(N_2,3) andZ×F₃(N_2,3),b is approximated by the loop F₃(N_2,3). Let ϕ: B → F₃(N_2,3;x, y, z) be a morphism of loops such thatb^ϕ = [x, y, z]. ThenB/B⁰Kerϕis a product of 3 cyclic subgroups of order 3. According to Theorem 8.1.1 in [1] there are generatorsb₁B⁰, . . . , b_mB⁰ of the loop B/B⁰ such that

B/B⁰ = lp(b1B⁰)× · · · ×lp(bmB⁰),

REV. ROUMAINE MATH. PURES APPL.,54(2009),2, 161–169

(Kerϕ·B⁰)/B⁰ = lp(b³₁B⁰)×lp(b³₂B⁰)×lp(b³₃B⁰)×lp(b4B⁰)× · · · ×lp(bmB⁰).

We put C= lp(b₃, . . . , b_m)⊂B and show that lp(b₁, b₂, b₃)∩C^B = 1.

Obviously, lp(b₁, b₂, b₃)^ϕ =F₃(N_2,3) and (C^B)^ϕ ⊆F₃(N_2,3)⁰. Then lp(b₁, b₂, b₃)

∩C^B ⊆Kerϕ·lp(b1, b2, b3)⁰. This means that every elementx∈lp(b1, b2, b3)∩ C^B can be represented in the form x =y[b₁, b₂, b₃]^ϕ for some y ∈ Kerϕ and α < 3. Since x ∈ lp(b1, b2, b3) and lp(b1, b2, b3)∩Kerϕ = lp(b³₁, b³₂, b³₃), we have y =z³ for a fixed z∈lp(b₁, b₂, b₃). So,x=z³[b₁, b₂, b₃]^ϕ. Besides, since the commutative group B/B⁰ is the direct product of the cyclic subgroups lp(b_iB⁰), i= 1, . . . , m, we have lp(b₁, b₂, b₃)∩C^B⊆B⁰. In particular,x∈B⁰. Then, by quasiidentities (8) from [2] (obviously, they hold in any loop of the set Σ), we have z³ = 1 and consequently, [b1, b2, b3]^ϕ ∈B⁰∩C^B⊂Kerϕ. But [b1, b2, b3]^ϕ ∈Kerϕjust when α = 0 since as it was observed, [b1, b2, b3]6= 1.

So, lp(a1, a2, a3)∩C^B= 1.

Sinceb^ϕ= [x, y, z] and [b₁, b₂, b₃]^ϕ= [x, y, z]^β,β 6= 0 mod 3, we haveb= [b1, b2, b3]^γ mod Kerϕfor someγ 6= 0 mod 3. Thenb⁻¹[b1, b2, b3]^γ∈Kerϕ∩B⁰. But, it follows from the representation B/B⁰ = lp(b₁B⁰)× · · · ×lp(b_nB⁰) and the inequality [b1, b2, b3]^ϕ6= 1 that

Kerϕ∩B⁰= lp ([bi, bj, bk], 1≤i < j < k ≤m, (i, j, k)6= (1,2,3))⊂C^B. Hence

b⁻¹[b1, b2, b3]^γ∈C^B and b≡[b1, b2, b3]^γ modC^B.

It remains to investigate the natural morphism of loopsφ:B →B/C^B. We have

b^φ= ([b₁, b₂, b₃]^γ)^φ= [b₁, b₂, b₃]^γC^B6= 1C^B,

i.e., the elementbis approximated by the loop lp(b₁, b₂, b₃)⊂B, which is what had to be shown.

Necessity. Let Kbe a quasivariety which contains only a finite number of subquasigroups. Suppose for a contradiction that Kis not generated by a finite set of finite nonisomorphic loops of Σ. It is clear thatKcontains only a finite number of nonisomorphic loops of the set Σ. This means that there is a finitely generated loopL∈Ksuch that the quasivarietyQ(L) is not generated by loops of Σ only. We shall show thatQ(L)⊂Kcontains continuously many different subquasivarieties. So, we obtain the contradiction.

Assume that L chosen in K is such that there are no other loops with a number of generators less than its number of generators. So, if L is finite, it can be considered as having an exponent equal to a power of 3 (i.e., Lis a 3-loop). We investigate two cases.

1. For some elementsa, b1, . . . , b3n inL, a³ =

n

Y

i=1

[b3i−2, b3i−1, b_3i]6= 1.

Denote by V the set of matrices of the form α1 α2 . . . αn

β₁ β₂ . . . β_n

such that

[a, b^α₁¹. . . b^α_3n³ⁿ, b^β₁¹. . . b^β_3n³ⁿ] = 1,

where 0 ≤ α_i < 3 and 0 ≤ β_i < 3. Let as investigate the commutative Moufang loop

B =B(n, V, k) (respectivelyB(n, V))

= lp(x, y₁, . . . , y_3nkx³ =

n

Y

i=1

[y3i−2, y3i−1, y_3i], [x, y^α1, . . . , y^α_3n³ⁿ, y₁^β1, . . . , y_3n^β3n] = 1,

α1, . . . , α3n

β₁, . . . , β_3n

∈V,

where 3^k (respectively 0) is the exponent of the loop L (and the relations are given in the variety N_2,3k (respectively, N₂)). We show that the loop B ∈Q(L). Denote

T = lp x³, [x, y, z] for all elementsy, z ∈B

⊂B.

Any element of the subloopT is approximated byLvia the morphism of loops of B inLdefined by the maps

x→a, y_i →b_i, i= 1, . . . ,3n.

We have B/T ∼=Z3×D, whereZ3 ∼= lp(xT) and D=F3n N_2,3k (respectivelyN2)y1, . . . , y3n

/lp

n

Y

i=1

[y3i−2, y3i−1, y3i]

! .

Obviously, Z3 ∈ Q(L), and by Lemma 1.3 in [2], we have D ∈ Q(L). This means that B/T ∈ Q(L). Then, by the criterion of belonging we have B ∈ Q(L), which is what had to be proved. According to Lemma 2.6 in [2], there is a continuum of different subquasivarieties inQ(B), so this also holds forQ(L).

2. InL the quasiidentities x³=

n

Y

i=1

[x3i−2, x3i−1, x_3i]→x³ = 1, n= 1,2, . . .

do hold. The elements of the associator L⁰ are not approximated by the loop F₃(N_2,3). If this is not so then, according to the belonging criterion,

L∈Q(L/L⁰×F3(N2,3)),

which contradicts our assertion. Consequently, not all elements of L⁰ are ap- proximated by the loop F3(N2,3).

According to Theorem 8.1.1 in [1], the loop L in the varietyN₂ can be represented as

L= lp (x₁, . . . , x_nkx^s_iⁱ,=r_i, i= 1, . . . , l, r_j = 1, j=l+ 1, . . . , n), whereri ∈Fn(x1, . . . , xn)⁰, andsi are suitable integers. We shall show that for everyi≤lthe numbers_iis divisible by 3 and, according to the quasiidentities of our case 2, we have x^si = 1 and ri = 1. Indeed, assume the contrary, i.e., that for some i ≤ l, s_i is not divisible by 3. If r_i = 1 then according to identity (1), xi ∈ Z(L) and lp(xi)∩L⁰ = 1. It follows that L = lp(xi)× lp(x1, . . . , xi−1, xi+1, . . . , xn), which is impossible. Let now ri 6= 1. Then, according to identities (1) and (3) from [2], we have

[xi, y, z] = [x^s_iⁱ, y, z] or [xi, y, z] = [x^s_iⁱ, y, z]⁻¹

for any y, z ∈L. Hence x_i ∈Z(L). According to Dik’s theorem (see [3]), the mapxi →xi, xj →xj, j6=i, j = 1, . . . , n, can be extended up to a morphism of loops ϕ:L→ L. Sincex_i ∈Z(L), we have Kerϕ= lp(x³_i). But x_i can be excluded from the set of generators of the loop L^ϕ. Since lp(x³_i)∩L⁰ = 1 and L/L⁰ = lp(x1L⁰)× · · · ×lp(x1L⁰) ∈ Q(L), we deduce that L is approximated by the loop L^ϕ×lp(x_iL⁰), which contradicts the choice of the loopL.

So, thes_i,i= 1, . . . , l, are divisible by 3, thus the loop L has the repre- sentation

L= lp (x₁, . . . , x_nkx^s_iⁱ = 1, v_i = 1, i= 1, . . . , l, v_j = 1, j=l+ 1, . . . , n). We now investigate the commutative Moufang loop represented in the variety N_2,3k (respectively, N₂) as

B = lp (x1, . . . , xnkvi = 1, i= 1, . . . , n).

It is clear thatB ∈Q(L) and that not all elements ofB⁰ are approximated by the loop F3(N2,3). According to Lemma 1.5 in [2], the lattice of subquasivari- eties ofQ(B) is isomorphic to the lattice of subquasivarieties ofQ(B/B³). Ac- cording to Lemma 3.16 in [2],Q(B/B³) contains continuously many different subquasivarieties. Consequently, Q(B) and also Q(L) contain continuously many different subquasivarieties. The theorem is proved.

According to the belonging criterion, from the theorem there follows the following

Corollary. LetLbe a finite generated commutative Moufang loop with nilpotency class 2. Then the lattice of the subqvasivarieties ofL_qQ(L)is either finite or continuous; LqQ(L) is finite if and only if L is the subdirect product of some loops belonging to the same finite set of loops of Σ.

2. THE DESCRIPTION OF THE LATTICE L_qQ(Z₃k×F₃(N_2,3)) As was observed, any quasivarietyM generated by a commutative Mo- ufang loop with exponent 3^k, which contains finitely many subquasivarieties, is contained in the quasivariety Q(Z₃k, F3(N2,3)) and M = Q(Hr1,s1,t1, . . . , H_rn,sn,tn) for some r_i, s_i, t_i. Let us denote

r₁ r₂ . . . r_n

s1 s2 . . . sn

t1 t2 . . . tn

=Q(Hr1s1t1, . . . , Hrnsntn),

ϕ_rst= x^3r= 1 &y^3s = 1 &z^3t= 1→[x, y, z] = 1

, 1≤r≤s≤t.

Proposition 1. The quasiidentityϕrstis true in the commutative Mou- fang loop H_r⁰_s⁰_t⁰ if and only if at least one of the inequalities r < r⁰, s < s⁰, t < t⁰ does hold.

Proof. Letϕ_rst be true inH_r⁰_s⁰_t⁰ and suppose thatr ≥r⁰, s≥s⁰, t≥t⁰. Then the identityz^3t = 1 is true inH_r⁰_s⁰_t⁰, so it also is the identity [x, y, z] = 1, that cannot hold.

Let now one of the inequalitiesr < r⁰, s < s⁰, t < t⁰ be true, for instance r < r⁰. For x=a, y=b, z =c, where the elementsa, b, c belong to the loop H_r⁰_s⁰_t⁰ = lp x, y, z k x^3r

0

= 1 y^3s

0

= 1 z^3t

0

= 1

, assume that the left side of the quasiidentity ϕ_rstis true, so a^3r = 1, b^3s = 1, c^3t = 1. Sincea, b, c can be represented in the form

a=x^α1·y^β1z^γ1a⁰, b=x^α2 ·y^β2z^γ2b⁰, c=x^γ3 ·y^β3z^α3c⁰, where

0≤αi <3^r0, 0≤βi <3^s0, 0≤γi <3^t0, a, b, c∈H_r⁰⁰_s⁰_t⁰, we obtain the equalities

x^α13r = 1, x^α23r = 1, x^α33r = 1.

Hence

α13^r= 0 mod 3^r0, α23^r= 0 mod 3^r0, α33^r = 0 mod 3^r0.

On account of the inequality r < r⁰ we conclude at the same time that α₁, α₂, α₃ are divisible by 3, so that the elements x^α1, x^α2, x^α3 belong to the central subloop H_r³0s⁰t⁰ ⊂Hr⁰s⁰t⁰. Then we have

[a, b, c] = [x^α1y^β1z^α1, x^α2y^β2z^α2, x^α3y^β3z^α3] = [y^β1z^α1, y^β2z^α2, y^β3z^α3] = 1.

Consequently, ϕrst is true inHr⁰s⁰t⁰. The proof is complete.

Proposition 2.If the subquasivarietyMis contained inQ(Z₃k, F₃(N_2,3)), then

M=

r1 r2 . . . rn

s1 s2 . . . sn

t₁ t₂ . . . t_n

for some ri, si, ti ≤n, r1 ≤r2 ≤ · · · ≤rn and one of the conditions below is verified:

(a) ri ≤(≤) ri+1, si ≤(<) si+1, ti > ti+1; (b) r_i < r_i+1, s_i >(≥) s_i+1, t_i ≥(>) t_i+1; (c) ri <(≤) ri+1, si > si+1, ti ≤(<) ti+1.

Proof. As was observed,

M =

r₁ r₂ . . . r_n

s₁ s₂ . . . s_n

t₁ t₂ . . . t_n ,

where r_i, s_i, t_i are not uniquely determined. Consider the system H_ri,si,ti, i = 1, . . . , n, which contains the least number of non associative loops. It is clear that we can suppose that r₁ ≤r₁ ≤ · · · ≤r_n. Observe that the system contains a single cyclic groupH_00s=Z₃^s if the exponent of the quasivarietyM is greater than the exponent of each non associative loop while in the opposite case the system does not contain groups.

To prove thatri, si, ti verify conditions (a), (b), or (c), it is sufficient to show that in each of these conditions there cannot be two equalities while two inequalities imply the third. We verify this fact for condition (a) (it can be similarly verified for conditions (b) and (c)).

Suppose that r_i = r_i+1, s_i = s_i+1. If 0 = r_i = r_i+1, we contradict the number of groups of the system. Let 06=ri=ri+1. Then

Q(H_risiti, H_ri+1si+1ti+1) =Q(H_{risimin(ti,ti+1)}, H_{00max(ti,ti+1)})

contradicts the minimality of the number of nonassociative loops of the system.

Also, the equalities

ri=ri+1, ti=ti+1 or si =si+1, ti =ti+1

cannot hold.

We now prove that the implication

ri < ri+1&si ≤si+1 →ti > ti+1. is true. The implications

r_i ≤r_i+1&s_i < s_i+1 →t_i > t_i+1, r_i< r_i+1 &t_i > t_i+1 →s_i ≤(<) s_i+1. can be proved similarly. Indeed, if ti+1≤ti then

Q(Hrisiti, Hri+1si+1ti+1) =Q(Hrisiti, H00ti+1)

contradicts the minimality of the number of nonassociative loops. The proof is complete.

Proposition 3. The quasivariety r1 . . . rn

s₁ . . . s_n

t₁ . . . t_n

is contained in the quasivariety

r₁⁰ . . . r⁰_n

s⁰₁ . . . s⁰_n

t⁰₁ . . . t⁰_n

if and only if max

i (r_i⁰, s⁰_i, t⁰_i) ≥ max

i (ri, si, ti) and for every ri 6= 0 there is a triple





 r⁰_j s⁰_j t⁰_j





 such thatr_i ≥r⁰_j >0, s_i≥s⁰_j and t_i ≥t⁰_j. Proof. Sufficiency. For everyiand j(i),

H_risiti ∈Q H_r⁰

js⁰_jt⁰_j, Z

p^{max(r0i,s0i,t0i)}

ifri6= 0.

Necessity. The inequality max(r_i⁰, s⁰_i, t⁰_i)≥max(r_i, s_i, t_i) is obvious, since the exponent of the first quasivariety is not greater than the exponent of the

second. Suppose that the second condition is not fulfilled, i.e., for some i≤n and every j ≤m either r_i < r⁰_j ors_i < s⁰_j, or t_i < t⁰_j. Then the quasiidentity ϕrisiti is false in Hrisiti and is true according to Proposition 1 in the loop H_r⁰

js⁰_jt⁰_j. The proof is complete.

Corrolary. If

r₁ . . . r_n

s₁ . . . s_n

t₁ . . . t_n

=

r⁰₁ . . . r_n⁰

s⁰₁ . . . s⁰_n

t⁰₁ . . . t⁰_n then the matrices









r1 . . . rn

s1 . . . sn

t1 . . . tn







 and









r₁⁰ . . . r⁰_n

s⁰₁ . . . s⁰_n

t⁰₁ . . . t⁰_n









do coincide.

Proof. Indeed, from the assumption we have maxi (ri, si, ti) = max

j (r_j⁰, s⁰_j, t⁰_j).

If r_i 6= 0, according to Proposition 3,

ri ≤r_j⁰, si ≤s⁰_j, ti ≤t⁰_j, r⁰_j ≤r_k, s⁰_j ≤s_k, t⁰_j ≤t_k for some indices j, k. Hence

r_i≤r_k, s_i≤s_k, t_i≤t_k,

which, according to Proposition 2, are true only when i=k.

Consequently,

ri =r_j⁰, si =s⁰_j, ti =t⁰_j, which is what had to be proved.

Proposition 4. Fork >1the quasiidentities of the loopF₃(N_2,3k)have the following forms:

(i)x^3k = 1;

(ii)

[x, y, z], u, v

= 1;

(iii) the associator quasiidentities of the loopF3(N2,3);

(iv)x³ =

n

Q

i=1

[x3i−2, x3i−1, x_3i]→x³ = 1,n= 1,2, . . .;

(v) x^3k−1 = 1→[x, y, z] = 1.

Proof. Let N be a quasivariety defined by the identities of (i), (ii) and quasiidentities of (iii), (iv), (v) and assume that N 6= Q(F₃(N_2,3k)). Let us investigate a finitely generated loop L ∈ N, L /∈ Q(F₃(N_2,3)). Assume that L=Fn/H. LetM =F_n⁰ ∩H. Then there is an elementu∈F_n⁰,u /∈M, which is not approximated by the loop F₃(N_2,3k). Consequently, the quasiidentity M = 1→u= 1 is true in the loop F₃(N_2,3) and is false inF_n/M. But this is not true, since in Fn/M the quasiidentities (iii) are true.

Corollary. The quasivarietyLgenerated by a finite commutative Mou- fang loop contains a continuous set of subquasivarieties if and only if in some 3-subloop ofL one of the quasiidentities

(a)x³ =

n

Q

i=1

[x3i−2, x3i−1, x3i]→x³ = 1,2, . . .,

(b)the associator quasiidentities of the loop F3(N2,3), is false.

Proposition 5.Fork≥2, the latticeLqQ(Z₃^k×F3(N2,3))is not modular.

Indeed, we easily convince ourselves that the quasivarieties Q(Z₃2 ×F₃(N_2,3)), Q(F₃(N_2,3)), Q(Z₃2), Q(Z₃)

form a nonmodular sublattice of five elements in the latticeLqQ(Z₃^k×F₃(N2,3)), k≥2.

REFERENCES

[1] M.I. Kargapolov and Yu.I. Merzlyakov, Fundamentals of Group Theory, 3nd Edition.

“Nauka”, Moskow, 1982. (Russian)

[2] V.I. Ursu,On the lattice of quasivarieties of commutative Moufang loops with nilpotency class≤2.I. Rev. Roumaine Math. Pures Appl.54(2009), 33–51.

[3] A.I. Mal’cev,Algebraic Systems. “Nauka”, Moskow, 1970. (Russian)

Received 1 March 2006 Romanian Academy

“Simion Stoilow” Institute of Mathematics P.O. Box 1-764, Bucharest, Romania&

Technical University Chi¸sin˘au, Republica Moldova [email protected], [email protected]