VISIÓN COLECTIVA

Theorem 3.1. Let A be a finitely generated algebra of finite signature with congru-

ence θ of finite index. Then θ is finitely generated.

Furthermore, a generating set may be found of size bounded in terms of the size of the finite-generating set of A, the index of θ, the signature of A, and the arity of its operation symbols.

index. Let T be a transversal of the θ-classes. Let Γ0 be the set of all pairs of the formhb0, bi whereb0 =QA_(b

0, . . . , br−1) for some fundamental operation symbol Qin the signature of A (with whatever rank, r), {b0, . . . , br−1} ⊆ T, and b is the unique

θ-representative of b0 drawn from T. Let Γ1 := {ha, bi ∈ θ | a ∈ X, b ∈ T}. Let Γ := Γ0 ∪Γ1, and note that, by our hypotheses, Γ is finite. We shall estimate the size of Γ at the conclusion of the proof. We claim thatθ = CgAΓ. By construction, Γ⊆θ; so, we need only verify that θ⊆CgAΓ.

Take ha, bi ∈ θ. By the symmetry and transitivity of CgAΓ, we need only show the case that b ∈ T, using, of course, too, that T is a transversal for θ. Since A is generated by X, we can write a =tA_(a

0, . . . , am−1) for some term t of rank, say, m and {a0, . . . , am−1} ⊆X. Writea for ha0, . . . , am−1i.

We induct on the complexity of t. The basis step consists of two cases, corre- sponding to the two possibilities for t: It is either a variable or a constant. First, suppose that t is a variable. Then, a = ai ∈ X, for some i, and hence ha, bi ∈ Γ1. Now, suppose that t is a constant. Then a =cA, for some nullary operation symbol

cin the signature of A. This puts ha, bi ∈Γ0, by construction.

For the inductive step, suppose that t =Qt0. . . tq−1 for some fundamental oper- ation symbol Q of rank q and each ti a term. Adopting an inductive hypothesis, we

assume that for eachi < q,hti(a), bii ∈CgAΓ, wherebi is the uniqueθ-representative

of ti(a) in T. Since CgAΓ respects the operations of A, we get that

a=QA(tA₀(a), . . . , tA_q−1(a)) CgAΓ∩θ QA(b0, . . . , bq−1) Γ0 b.

Thus, by the transitivity of CgAΓ, we have that ha, bi ∈ CgAΓ. It follows that,

θ ⊆CgAΓ, which proves the first part of the theorem.

For the second part, let n be the cardinality of the generating set X of A. Letp

be the number of fundamental operation symbols given by the signature, and letRbe their maximum arity. Finally, lett:=|T|. Then, as we found thatθwas generated by

Γ = Γ0∪Γ1, we need only count each of these. It is not difficult to see that|Γ0| ≤p·tR and that |Γ1|=n. Thus, we note, for convenience that |Γ| ≤n+t·pR.

For a given algebra A and α, β ∈ConA, we define a congruence ∆α

β on β by

∆α_β := Cgβ{hha, ai,hb, bii |a α b}.

The following is a consequence of Proposition 7.1 from Freese and McKenzie 1987; however, the proof of this particular detail is omitted in their exposition, so we record it here for convenience.

Theorem 3.2. Let A be any algebra in a variety with a difference term, d. Let ζ be

the center of A, and let 1 denote its highest congruence. Then

|A|= ζ/∆ 1 ζ · |A/ζ|.

Proof. Let T be a transversal of the ζ-classes of A, and write r(a) for the unique

ζ-representative from T of a given a ∈ A. We use the following map π : A →

ζ/∆1

ζ×A/ζ: π(x) =hhx, r(x)i/∆ζ1, x/ζi,for a givenx∈A. Suppose thatπ(x) = π(y).

Then, evidently,x ζ y, and hence r(x) =r(y). But, then, byhx, r(x)i∆1

ζhy, r(y)iand

Lemma 4.11, we get that x=y. Thus, we see that π is one-to-one.

Now, take any u, v, z ∈ A with hu, vi ∈ ζ. We need to find an element x ∈ A

so that hx, r(x)i∆1_ζhu, vi and x ζ z. We shall use x = d(u, v, r(z)). Note that x =

d(u, v, r(z))ζ r(z)ζ z. It follows also that r(x) = r(z). Now, since ζ is abelian, we can applyd coordinate-wise, “vertically,” to the pairs

hu, vi∆1_ζhu, vi hv, vi∆1_ζ hu, ui hr(z), r(z)i∆1_ζ hu, ui,

The following is also noted by Freese and McKenzie, but it has a brief proof and so is recorded here for convenience.

Lemma 3.3. Let A be any algebra in a congruence modular variety. Then ζ_A/∆1A

ζA

is abelian.

Proof. Writeζ =ζ_A. Also, write ∆ for ∆1A

ζ . By Proposition 4.32, we need only show

that [1ζ,1ζ] ⊆ ∆. For i = 0,1, let ηi be the kernel of the ith projection map from ζ

ontoA. Note that, for any hx0, x1i,hx00, y00i ∈ζ, we have that

hx0, x1iη0hx0, x0i∆hx00, x

0

0iη0hx00, x

0

1i;

thus, 1ζ = ∆∨η0. Similarly, one can show that 1ζ = ∆∨η1. Of course, we also have that [η0, η1]⊆η0∩η1 = 0ζ. Thus, by Proposition A.31 and the additivity of the

commutator in congruence modular varieties (see Theorem A.42), we have that

[1ζ,1ζ] = [∆∨η0,∆∨η1]⊆∆,

as desired.

We shall also make use of the following fact, noted in Freese and McKenzie (1987) (see equation 10, p. 83).

Theorem 3.4. Let A be an abelian algebra in a variety with Mal’cev term p. Then

for any t, a term operation on A of rank, say, n,

A |=t(x1, . . . , xn)≈t(z, . . . , z) + n

X

j=1

t(j)(xj, z),

where each t(j)_{, j = 1, . . . , n, is a binary term operation defined by}

t(j)(u, v) :=p(t(v, . . . , v, u, v, . . . , v), t(v, . . . , v), v),

The above has a straightforward proof that applies only the following well-known result (see McKenzie, McNulty, and Taylor (1987), Theorem 4.155).

Theorem 3.5. Let A be an abelian algebra in a variety with Mal’cev term p. Let

z ∈A, and define operations x+y :=p(x, z, y) and −x := p(z, x, z). Let R = {r ∈

Pol1A|r(z) =z}. ThenhA,+,−, z, rir∈R is a module with underlying abelian group

hA,+,−, zi. Furthermore, for any natural number r and any s ∈ PolrA such that s(z, . . . , z) = z, we get the identity

s(x1, . . . , xr) = n

X

i=1

si(xi),

where each si ∈Pol1A is defined by si(x) =s(z, . . . , z, x, z, . . . , z), with x appearing

in the ith place. (Abelian algebras in a Mal’cev variety are thus called affine.)

Corollary 3.6. Let A be in V, a Mal’cev variety of abelian algebras. Suppose also

that A is n-generated, for some natural number n. Then |A| ≤ |FV(1)| · |FV(2)|n ≤ |FV(2)|n+1.

Proof. We can takeFV({z}),FV({y1, . . . , yn}), and, for eachi= 1, . . . , n,FV({z, yi})

to be subalgebras ofFV({z, y1, . . . , yn}). (We shall let each of the variablesz, y1, . . . , yn denote both an index and the projection function indexed by that index; see A.1.1 for details.) Note also that FV({z}) is isomorphic to FV(1), while, for eachi = 1, . . . , n,

FV({z, yi}) is isomorphic to FV(2). Let S be the subset of FV({z})×FV({z, y1})×

· · · ×FV({z, yn}) with elements of the form

ht(z, . . . , z), t(1)(y1, z), . . . , t(n)(yn, z)i,

where t is any term operation of rank n. By Theorem 3.4, the map from S into

F :=FV({y1, . . . , yn}) defined by ht(z, . . . , z), t(1)(y1, z), . . . , t(n)(yn, z)i 7→t(z, . . . , z) + n X j=1 t(j)(yj, z)

is onto. Thus, |FV(n)| ≤ |FV(1)| · |FV(2)|n. Since everyn-generated algebra inV is a

homomorphic image of F, the result follows.

Given any variety V and for each natural numberN, let V(N) _{denote the variety} based on the set of N-variable equations that hold across V.

Theorem 3.7. Let V be a Mal’cev variety of finite signature such that |FV(2)|=m,

a natural number, and such that each of its algebras is of nilpotence class k. Let p

be the number of fundamental operations provided in the signature and let R be the

maximum of their ranks. There is a function F : _N×_N → _N (with its definition

depending on m, p, R) so that for all N high enough, for all n-generated B ∈ V(N)

of nilpotence class c ≤ k, |B| ≤ F(n, c). In particular, for all high enough N, V(N)

is locally finite, and hence V itself is locally finite.

Proof. LetV be as described in the hypotheses. By Theorems A.21 and 4.48, we can

find a natural number N0 ≥2 such that for all N > N0, V(N) is Mal’cev and so that for all B∈ V(N)_{, B is nilpotent of class c.}

We shall define f recursively in the parameter c. For the basis step, note that if

c= 1, we have that B is abelian. Then, it is not too difficult to see that we can use Corollary 3.6 to find that |B| ≤ mn+1_{. Thus, we set F(n,1) =m}n+1_{. Now, suppose}

F(n, c0) has been defined for all c0 < c, for some c≤ k, and that F(n, c0) provides a bound on then-generated algebras of nilpotence class c0 inV(N)_{. Using Theorem 3.2,} we can write |B| = |ζB/∆ζ1BB| · |B/ζB|. Note that B/ζB is nilpotent of class c−1

(see Theorem 4.30 and Proposition 4.28) and is also, of course, n-generated. Thus, we have that the index of ζB is bounded by t :=F(n, c−1). Thus, as shown in the

proof of Theorem 3.1, as a congruence, ζB is generated by n +ptR elements. We claim further that, as a subalgebra, ζ_B is generated by 2n+ptR elements. Indeed, from Theorem A.25, for any Mal’cev algebra A and X ⊆ A × A, we have that

CgAX = SgA×AX ∪0A. In light of the fact that 0A is n-generated in A×A, the claim is apparent. By Lemma 3.3, ζ_B/∆1B ζB is abelian. As ζB/∆ 1B ζB is also (2n + pt R_)-generated,

we may apply Corollary 3.6 to learn that |ζB/∆1ζBB| ≤ m

2n+ptR₊₁

, and hence |B| ≤

m2n+ptR+1_{·t. Thus, we set}

F(n, c) = m2n+ptR+1·t,

with t=F(n, c−1).