AN AXIOMATIZATION FOR THE UNIVERSAL THEORY OF THE HEISENBERG GROUP

. The Heisenberg group, here denoted H , is the group of all 3 × 3 upper unitri-angular matrices with entries in the ring Z of integers. A.G. Myasnikov posed the question of whether or not the universal theory of H , in the language of H , is axiomatized, when the models are restricted to H -groups, by the quasi-identities true in H together with the assertion that the centralizers of noncentral elements be abelian. Based on earlier published partial results we here give a complete proof of a slightly stronger result.


Introduction
A (multiplicatively written) group G is commutative transitive, briefly CT, provided the relation of commutativity is transitive on G\{1}; equivalently, provided the centralizer of every element g ̸ = 1 is abelian.
Noncyclic free groups are universally equivalent, even elementarily equivalent.Myasnikov and Remeslennikov [MR] proved that their universal theory is axiomatized by the quasi-identities they satisfy together with commutative transitivity.Fixing a noncyclic free group F , they proved the analogous result in the language of F when the models are restricted to F -groups.
Let A be a countably infinite set well-ordered as {a 1 , a 2 , ..., a n , ...} = {a n+1 : n < ω} where ω is the first limit ordinal which we take as the set of nonnegative integers provided with its natural order.Let F ω (N 2 ) be the group free in the variety of all 2-nilpotent groups on the generators A. For each integer n ≥ 2 let F n (N 2 ) be the subgroup of F ω (N 2 ) generated (necessarily freely) by the initial segment {a 1 , a 2 , ..., a n } of A. The Heisenberg group is the group H of all 3 × 3 upper unitriangular matrices with entries in the ring Z of integers.It is free in N 2 on the generators (See [4]).We take the liberty of identifying F 2 (N 2 ) with H. Now F ω (N 2 ) is discriminated by the family of retractions F ω (N 2 ) → H.That means that given finitely many elements f 1 , ..., f k ∈ F ω (N 2 )\{1} there is a retraction F ω (N 2 ) → H which doesn't annihilate any of them.(H discriminates N 2 in the sense of Hanna Neumann [8].)From this it follows that F n (N 2 ), n ≥ 2, are universally equivalent; moreover, since the discrimination is done by retractions, they are universally equivalent in the language of H. (See [3]).
Let CT (1) or noncentral commutative transitivity, briefly NZCT, be the property that the relation of commutativity be transitive on G\Z(G) where Z(G) is the center of G. Equivalently, NZCT asserts that the centralizers of noncentral elements are abelian.A special case of a question posed by A.G. Myasnikov is whether or not the universal theory of noncyclic free 2-nilpotent groups is axiomatized by the quasi-identities they satisfy together with NZCT and whether or not that theory in the language of H is so axiomatized when the models are restricted to H-groups.
In this paper, in the case of the language of H, we answer that question in the positive.In fact we prove a slightly stronger result.The remainder of this paper contains four additional sections.In Section 2 we fix definitions and notation.In Section 3 we prove the main result.In Section 4 we ponder but do not settle the question in the language without parameters from H. Finally in Section 5 we suggest problems for future research.
Before closing the introduction we note that the variety A 2 of metabelian groups is discriminated by its rank 2 free group.From this it follows that the noncyclic free metabelian groups are universally equivalent.It is worth mentioning in passing that Remeslenikov and Stohr proved in [9] that the universal theory of the noncyclic free metabelian groups is axiomatized by the quasi-identities they satisfy together with commutative transitivity.

Pedantic Preliminaries
Let L 0 be the first order language with equality containing a binary operation symbol • , a unary operational symbol −1 and a constant symbol 1.If G is a (multiplicatively written) group L 0 [G] is obtained from L 0 by adjoining names g for the elements g ∈ G\{1} as new constant symbols.We find it convenient to commit the "abuses" of identifying g with g for all g ∈ G and replacing • with juxtaposition.Moreover, we find it convenient to write an inequation ∼ (s = t) as s ̸ = t.
An identity, in L 0 [G] (Note: ) is a universal sentence of the form ∀x (s(x) = t(x)) where x is a tuple of variables and s(x) and t(x) are terms of L 0 [G].Examples of identities are the group axioms, namely: where x is a tuple of variables and the s i ( x), t i ( x), s( x) and t( x) are terms of Every identity is equivalent to a quasi-identity since ∀x (s(x In particular, the group axioms are equivalent to quasi-identities. If G is a group and we let Q 0 (G) be the set of all quasi-identities of L 0 true in G and Q(G) be the set of all quasi-identities of L 0 [G] true in G.We view the group axioms as contained in Q 0 (G) ⊆ Q(G).We let T h 0 ∀ (G) be the set of all universal sentences of L 0 true in G and T h ∀ (G) be the set of all universal sentences of L 0 [G] true in G.Note that quantifier free sentences are viewed as special cases of universal sentences.In particular, the diagram of G, briefly diag(G), consisting of the atomic and negated atomic sentences of A G-group Γ is a model of the group axioms and diag(G).That is equivalent to the group Γ containing a distinguished copy of G as a subgroup.A G-polynomial is a group word on the elements of G and variables.(If you like, an element of the free product G * ⟨x 1 , ..., x n ; ⟩ for some n.)Note that, modulo the group axioms, every identity of L 0 (L 0 [G] ) is equivalent to one of the form ∀x (w(x) = 1) where w(x) is a group word (Gpolynomial) and every quasi-identity of L 0 (L 0 [G] ) is equivalent to one of the form where the u i (x) and w( x) are group words (G-polynomials).
In this paper by "ring" we shall always mean commutative ring with multiplicative identity 1 ̸ = 0. Subrings are required to contain 1 and homomorphisms are required to preserve 1.A ring R is residually-Z provided, given r ∈ R\{0}, there is a homomorphism R → Z which does not annihilate r.This forces R to have characteristic zero and we identify the minimum subring of R with Z. Hence, we view R as separated by retractions R → Z.A ring is locally residually-Z provided every finitely generated subring is residually-Z .Being locally residually-Z is equivalent to being a model of the quasi-identities of ring theory true in Z (See [4]).
It was proven in [4] that every model of ) Here an H-embedding is an embedding which is the identity on H. (The meanings of H-subgroup and H-homomorphism in the category of H-groups are readily apparent.) Let G be a group and g ∈ G.We let C G (g) be the centralizer of g in G. NZCT is the following universal sentence of L 0 : A group G satisfies NZCT if and only if C G (g) is abelian for all g ∈ G\Z(G).The following quasi-identity of L 0 [H] holds in H.
We shall prove in the next section that   where the p i ( x) and q j ( x) are

H-polynomials.
By writing the matrix of an existential sentence in disjunctive normal form one sees that every existential sentence of L 0 [H] is equivalent (modulo the group axioms) to a disjunction of primitive sentences and so holds in an H-group if and only if at least one disjunct does.
Assume momentarily that there is a universal sentence Then the finitely generated H-subgroup G 0 = ⟨a 1 , a 2 , g 1 , ...g k ⟩ of G also satisfies the above primitive sentence and, since universal sentences of L 0 [H] are preserved in H-subgroups, G 0 is a model of Q(H) ∪ diag(H) ∪ {τ }.Hence, if a counterexample exists, then so would a finitely generated counterexample exist.So it suffices to prove the result for finitely generated models.We shall find it convenient to prove, more generally, that the result holds for models G such that the quotient G = G/Z(G) is finitely generated.

The Lame Property and the Universal Theory of H
It was shown in [5] using a characterization due to Mal'cev [7] In other words, for each commutator [g 2 , g 1 ], each of the systems

Mimicking the construction of an existentially closed extension (See e.g. Hodges [H])
we may embed a model G of Q(H) ∪ diag(H) ∪ {τ } into U T 3 (R), If we could preserve τ at each step then, since universal sentences are preserved in direct unions, U T 3 (R) would also satisfy τ .If U T 3 (R) satisfies τ , then R is an integral domain.To see this let (r, s) ∈ R 2 and suppose rs = 0. Let satisfies τ , r = 0 or s = 0 and R indeed is an integral domain.
A residually-Z ring R is ω-residually-Z provided it is discriminated by the family of retractions R → Z .That is, given finitely many nonzero elements of R, there is a retraction R → Z which does not annihilate any of them.That is equivalent to R being an integral domain.Suppose first that R is an integral domain.Let r 1 , ..., r n be finitely many nonzero elements of R. Then the product r = r 1 • • • r n ̸ = 0 and there is a retraction ρ : R → Z which does not annihilate r so cannot annihilate and of r 1 , ..., r n .Conversely, if R is ωresidually-Z and r and s are nonzero elements of R, then there is a retraction ρ : R → Z such that ρ(r) ̸ = 0 and ρ(s) ̸ = 0; so, ρ(rs) = ρ(r)ρ(s) ̸ = 0 and hence rs ̸ = 0 and R must be an integral domain.From this it follows that a locally residually-Z ring R is locally Since universal sentences are preserved in direct unions and G = lim we have that G is a model of T h ∀ (H).The bottom line is that we would be finished if we could construct an overgroup U T 3 (R) in such a way that τ is preserved at each step of the construction.
Let us forget momentarily about this particular R and consider a possible property that a representation G ≤ H U T 3 (R) might satisfy.
where R is locally residually-Z.We say the representation satisfies the Lame Property provided, for where R locally residually-Z.The representation satisfies the Lame Property if and only if it satisfies the conjunction of the following two conditions: (1.)For all y =   1 y 12 y 13 0 1 0 0 0 1 is not a zero divisor in R. The contradiction shows that the representation satisfies (1.).
We next note that if the representation G ≤ H U T 3 (R) satisfies the Lame Property, then G satisfies τ .
For suppose [a 2 , y] = 1 so y =   1 y 12 y 13 0 1 0 0 0 1 ) is contradicted while if x 23 ̸ = 0, then y 12 = 0 otherwise (2.) is contradicted.It follows that either [y, a 1 ] = 1 or [a 2 , x] = 1 so τ holds in G.For a fixed representation G ≤ H U T 3 (R) satisfying the Lame Property is a sufficient condition for τ to hold in G; however, it is not a necessary condition for G to satisfy τ .(None the less we shall subsequently see that having at least one representation satisfying the Lame Property is necessary and sufficient.) The result (proven in [5]) that every 3-generator model of Q(H) ∪ diag(H) is already a model of the T h ∀ (G) provides a treasure trove of counterexamples.
Let G be a model of Q(H) ∪ diag(H) and let R be a locally residually-Z ring.We say that R is appropriate for G provided (1) G ≤ H U T 3 (R) and (2) R is generated by the entries of the elements of G. Now R = Z×Z = Ze 1 + Ze 2 where e 1 = (1, 0) and e 2 = (0, 1) is residually-Z .Consider the 3-generator subgroup G ≤ H U T 3 (R) generated by Let θ be an indeterminate over Z .Then the polynomial ring Z[θ] is residually-Z and we could have just as well embedded this G into U T 3 ( Z[θ] ) as the subgroup generated by Since θ is an entry of an element of G, Z[θ] is also appropriate for G. Since Z[θ] is an integral domain, the representation does satisfy the Lame Property.
Anticipating an application to be used later in this paper, suppose G 0 is a model of Q(H) ∪ diag(H) and let a i be a free generator of H, i ∈ {1, 2}.Suppose G is obtained from G 0 by extending C G 0 (a i ).That is

Using a big powers argument, we get a discriminating family of retractions
It follows that G is universally equivalent to G 0 .Now suppose we fix a model G 0 of Q(H) ∪ diag(H) ∪ {τ } which admits a representation G 0 ≤ H U T 3 (R) satisfying the Lame Property.Let (g 1 , g 2 ) ∈ G 2 0 and suppose the system Similarly, if the system Getting back to G 1 , suppose the system has a solution.So, if every model G of Q(H) ∪ diag(H) ∪ {τ } has at least one representation satisfying the Lame Property, then G embeds in U T 3 (R) where R is an integral domain and we are finished.To prove that every model of Q(H) ∪ diag(H) ∪ {τ } admits a representation satisfying the Lame Property, we need some results from model theory.
Let M be the model class operator.We first paraphrase a result from Bell and Slomson [1].Proposition 3.3.Let L be a first order language with equality.Let K be the class of all Lstructures and let X ⊆ K. Then there is a set S of sentences of L such that X = M(S) if and only if X is closed under isomorphism and ultraproducts and K\X is closed under ultraprowers.
Remark 3.4.The proof in [1] needed the Generalized Continuum Hypothesis.In view of Shelah's [10] improvement of Keisler's ultrapower theorem that hypothesis may be omitted.
The next result may be found in Hodges [H].Proposition 3.5.Let L be a first order language with equality.Let T be a set of sentences of L. Let T ∀ be the set of all universal sentences of L which are logical consequences of T .Then M(T ∀ ) consists of all L-substructures of models of T .1:9 Using Proposition 1, a striaghtforward but tedious verification reveals that the class of all models of Q(H) ∪ diag(H) ∪ {τ } which admit a representation satisfying the Lame Property is first order.Moreover, since this class is closed under H-subgroups, it has (as an application of Proposition 2) a set Φ of universal axioms in We would be finished if we could prove equality.It will suffice to establish the result for finitely generated models.We shall prove the result more generally for models G of Q(H) ∪ diag(H) ∪ {τ } such that the quotient G = G/Z(G) is finitely generated.
Proof of Theorem 3.6.Let G be a model of Q(H) ∪ diag(H) ∪ {τ }.We may assume without loss of generality that the quotient Thus, rank(C Since the C-rank of G is greater than 1 at least one of must have rank at least 2. We may assume rank Let T be a transversal for N in G so that every element g is uniquely expresses in the form t(g)υ(g) where t(g) ∈ T and υ(g) ∈ N .Assume further that, for all (p, q, z) ∈ Z 2 × Z( G), t(a p 1 a q 2 z) = a p 1 a q 2 and that t(x) = 1 for all x ∈ N .Let G 0 be the subgroup of G generated by the coset representatives of N in G together with Z( G) .Since b m−1 is killed off the C-rank of G 0 is less than that of G and so, by inductive hypothesis, G 0 is a model of Φ.
Examining the general form of a matrix in G and setting that guy equal to the identity matrix, we see that, modulo the law [x 1 , x 2 , x 3 ] = 1, the only relations in G are consequences of the relations in G 0 and the relations [C G 0 (a 2 ), b m−1 ] = 1.That is, G is the free rank 1 extension of C G 0 (a 2 ) relative to N 2 and hence G 0 and G are universally equivalent.Then G is a model of Φ since G 0 is.Since G ≤ G and universal sentences are preserved in subgroups, G is a model of Φ.That completes the induction and proves the theorem.

The Theory in the Base Language
We wish to ponder whether or not Q 0 (H) ∪ {N ZCT } axiomatizes T h 0 ∀ (H).To that end let G 0 be a model of Q 0 (H) ∪ {N ZCT }.We may assume G 0 is finitely generated.Suppose that G 0 is abelian.Then, since models of Q 0 (H) are torsion free, G 0 is free abelian of finite rank r.Choose a positive integer n such that It follows that G 0 embeds in G .So every universal sentence of L 0 true in G must also be true in G 0 .But G is universally equivalent to H. Therefore G 0 is a model of T h 0 ∀ (H).So it now suffices to assume G 0 is a finitely generated nonabelian model of Q 0 (H) ∪ {N ZCT }.A consequence of a result of Grätzer and Lasker [6] is that the quasivariety generated by H consists of all groups isomorphic to a subgroup of a direct product of a family of ultrapowers of H. View H as U T 3 (Z) and taking corresponding direct product of ultrapowers of Z we get a ring R such that G 0 embeds in U T 3 (R).Since quasi-identities are preserved in direct products and ultrapowers, R is a model of the quasi-identities true in Z.That is R is locally residually-Z.Further we may take R to be generated by the entries of a fixed finite set of generators for G 0 .Thus, R may be taken finitely generated.Therefore R is residually-Z and so separated by the family of retractions R → Z.Let G be the subgroup ⟨G 0 , H⟩ of U T 3 (R).The retractions R → Z induce group retractions G → H and these separate G.It follows that G is a model of Q(H) ∪ diag(H).Let us keep this G in mind as we move on.
Let us say a group is (G 0 , H)-group if it contains a distinguished copy of each of G 0 and H.The meanings of (G 0 , H)-subgroup and (G 0 , H)-homomorphism are readily apparent.A (G 0 , H)-ideal is the kernel of a (G 0 , H)-homomorphism.Equivalently, a (G 0 , H)-ideal in a (G 0 , H)-group G is a subgroup K normal in G such that K ∩ G 0 = {1} = K ∩ H.
We further observe that Z(G 0 ) coinciding with G 0 ∩ Z(U H (G 0 )) is a necessary condition for Question 4.1 to have a positive answer.Recall we are taking G 0 nonabelian.Let g 1 and g 2 be noncommuting elements of G 0 and g ∈ Z(G 0 )\(G 0 ∩ Z(U H (G 0 ))), then since g 1 and g 2 each commute with g / ∈ Z(U H (G 0 )) , NZCT would be violated.).We let S c be the set of all universal sentences of L 0 true in F n (N c ).Since F n (N c ) ≤ F r (N c ) for all 0 ≤ n ≤ r and universal sentences are preserved in subgroups, S c is actually the set of all universal sentences of L 0 true in every free c-nilpotent group.
For each integer n ≥ 0 we define CT (n) to be the following universal sentence of L 0 .

Since 1 −
e 1 = e 2 , R = Z × Z is generated by the entries of the elements of G. Since e 1 e 2 = 0, e 1 is a zero divisor in R. Now b ∈ C G (a 1 )\Z(G) and b 23 = e 1 is a zero divisor in R. Hence, the representation violates the Lame Property.Every 3-generator model of Q(H) ∪ diag(H) is already a model of T h ∀ (H).So this G = ⟨a 1 , a 2 , b⟩ satisfies τ .Now this G is obtained from H by extending C H (a 1 ) introducing a new parameter.(It is a rank 1 centralizer extension relative to the category N 2 of 2-nilpotent groups.)

 1 
Collecting and simplifying we see a typical element of G 1 has the form uY n [Y, w] where n ∈ Z and (u, w) ∈ G 2 0 .The matrix representing this element has the form ∈ C G 1 (a 2 ) and B = ∈ C G 1 (a 1 ).Assume further that [C, B] so (c 12 + nz 13 )b 23 = 0. Now, c 12 + nz 13 ̸ = 0 and b 23 ̸ = 0.Moreover, B has the form u[Y, w] with u ∈ C G 0 (a 1 ) looking like  .Since b 23 is a zero divisor in R that contradicts that the representation G 0 satisfies the Lame Property.Hence, either c 12 + nz 13 = 0 or b 23 = 0 so either[C, a 1 ] = 1 or [a 2 , B] = 1 and G 1 satisfies τ .


1 ) + rank(C 2 ) ≥ 2. Define the C-rank of G to be rank(C 1 ) + rank(C 2 ) − 1.The proof will proceed by induction on the C-rank.Suppose first that the C-rank of G is 1.That forces rank(C 1 ) = 1 = rank(C 2 ) and C i = ⟨a i ⟩ = ⟨a i Z(G)⟩, i = 1, 2. Let G ≤ H U T 3 (R) be any representation where R is locally residually-Z .It follows from the above that every element of C G (a 2 )\Z(G) looks like where m ∈ Z\{0} and every element of C G (a 1 )\Z(G) looks like where n ∈ Z\{0}.Now, for each k ∈ Z\{0}, the quasi-identity ∀x ((kx = 0) → (x = 0)) holds in Z .Hence, these quasi-identities hold in R and consequently every representation of G satisfies the Lame Property.The initial step of the induction has been established.Now suppose G has C-rank n > 1 and the result has been established for models with C-rank k with 1 ≤ k < n.Now let G ≤ H U T 3 (R) be any representation of G where R is locally residually-Z .Let us extend G to G by adjoining the elements as r varies over R. Since τ depends on the (1, 2) and (2, 3) entries only and since G has the same C-rank as G, we may replace G with G.This causes no harm since universal sentences are preserved in subgroups; so G will be a model of Φ whenever G is.

Question 4 . 2 .
Is Z(G 0 ) = G 0 ∩ Z(U H (G 0 ))? 5. Questions Let c ≥ 2 be an integer.Let r = max{2, c − 1}.Let s be any integer such that s ≥ r.Let G = F s (N c ).Then F ω (N c ) is discriminated by the family of retractions F ω (N c ) → G.It follows that the F n (N c ) have the same universal theory relative to L 0 for all n ≥ r and that the F n (N c ) have the same universal theory relative to L 0 [G] for all n ≥ s (See [GS 1]