Interpreting a field in its Heisenberg group
Rachael Alvir, Wesley Calvert, Grant Goodman, Valentina Harizanov, Julia Knight, Andrey Morozov, Russell Miller, Alexandra Soskova, Rose Weisshaar
aa r X i v : . [ m a t h . L O ] J un Interpreting a field in its Heisenberg group
R. Alvir, W. Calvert, G. Goodman, V. Harizanov, J. Knight,A. Morozov, R. Miller, A. Soskova, and R. Weisshaar ∗ June 23, 2020
Abstract
We improve on and generalize a 1960 result of Maltsev. For a field F , we denote by H ( F ) the Heisenberg group with entries in F . Maltsev[7] showed that there is a copy of F defined in H ( F ), using existentialformulas with an arbitrary non-commuting pair ( u, v ) as parameters. Weshow that F is interpreted in H ( F ) using computable Σ formulas withno parameters. We give two proofs. The first is an existence proof, relyingon a result of Harrison-Trainor, Melnikov, R. Miller, and Montalb´an [3].This proof allows the possibility that the elements of F are representedby tuples in H ( F ) of no fixed arity. The second proof is direct, givingexplicit finitary existential formulas that define the interpretation, withelements of F represented by triples in H ( F ). Looking at what was used toarrive at this parameter-free interpretation of F in H ( F ), we give generalconditions sufficient to eliminate parameters from interpretations. The Heisenberg group of a field F is the upper-triangular subgroup of GL ( F )in which all matrices have 1’s along the diagonal and 0’s below it. Maltsevshowed that there are existential formulas with parameters, which, for everyfield F , define F in its Heisenberg group H ( F ). In this article we will produceexistential formulas without parameters, which, for every field F , interpret F in H ( F ). Observing what is used to obtain this result, we will then formulatea general result on removing parameters from an interpretation.Languages are assumed to be computable, and structures are assumed tohave universe a subset of ω . For a given structure A , the atomic diagram D ( A )may be identified, via G¨odel numbering, with a subset of ω . We then identify ∗ The first, second, fourth, fifth, sixth, seventh and ninth authors are grateful for supportfrom NSF grant DMS itself with the characteristic function of D ( A ). Classes of structures havea fixed language, and are closed under isomorphism. The following notion, of“Turing computable embedding,” is from [1], based on the earlier notion of“Borel embedding” from [2]. Definition 1.1.
For classes
K, K ′ , we say that K is Turing computably em-bedded in K ′ , and we write K ≤ tc K ′ , if there is a Turing operator Θ : K → K ′ such that for all A , B ∈ K , A ∼ = B iff Θ( A ) ∼ = Θ( B ) . Medvedev reducibility is used to compare “problems,” where a problem is asubset of ω ω . The problems that concern us have the form “build a copy of A .” Definition 1.2.
1. For structures A and B , we say that A is Medvedevreducible to B , and we write A ≤ s B , if there is a Turing operator Φ thattakes copies of B to copies of A .2. For classes K and K ′ , where K ≤ tc K ′ via Θ , we say that the structures in K are uniformly Medvedev reducible to their Θ -images in K ′ , A ∈ K is uniformly Medvedev reducible to Θ( A ) if there is a single Turing operator Φ such that for all A ∈ K , A ≤ s Θ( A ) via Φ . Often, when we have a Turing computable embedding Θ : K → K ′ suchthat A ∈ K is uniformly Medvedev reducible to Θ( A ) as in (2) above, it isbecause there are simple formulas that define, for all A ∈ K , an interpretationof A in Θ( A ). Montalb´an defined a very general kind of interpretation of A in B that yields a uniform Medvedev reduction of A to B . In this definition,the tuples from B that represent elements of A may have arbitrary arity. Theinterpretation is defined by formulas that have no specific arity. Here, the arityof a formula is the number of its free variables. As usual, we often write B bothfor the structure and its domain. Definition 1.3 (Generalized computable Σ -definition) . Let R ⊆ B <ω , and let ϕ n (¯ x n ) n ∈ ω be a computable sequence of computable Σ formulas, where ϕ n (¯ x n ) has arity n . If for each n , ϕ n (¯ x n ) defines R ∩ B n , then we say that W n ϕ n (¯ x n ) is a generalized computable Σ definition of R . Thus a generalized computable Σ formula allows consideration of tuples ofall finite arities. Such a formula is technically not in L ω ω , as it uses infinitelymany free variables; however, it is a computable disjunction, over all n ∈ ω , of L ω ω formulas ϕ n with free variables x , . . . , x n . Generalized computable Σ formulas are exactly what is required in the following definition. Definition 1.4 (Montalb´an) . For a relational structure A = ( A, ( R i ) i ∈ I ) and astructure B , we say A is effectively interpreted in B if there exist a set D ⊆ B <ω and relations ∼ and R ∗ i on D such that1. ( D, ( R ∗ i ) i ∈ I ) / ∼ ∼ = A ,2. D , ± ∼ , and ± R ∗ i are defined by a computable sequence of generalizedcomputable Σ formulas with no parameters. ± ∼ denotes both ∼ and , and the sequence of formulas includesformulas for both ∼ and . Similarly, ± R ∗ i denotes both the relation R ∗ i andits negation, and the sequence of formulas includes one Σ formula for each ofthem. (One might call such a pair of formulas a ∆ definition of R ∗ i .) In theRussian tradition, a structure that is effectively interpreted in B is said to beΣ -definable in B .The following definition was first presented as [9, Defn. 3.1]. Definition 1.5. A computable functor from B to A is a pair of Turing operators Φ , Ψ such that:1. Φ takes copies of B to copies of A ,2. Ψ takes each triple ( B , f, B ) such that B i ∼ = B for i = 1 , and B ∼ = f B to a function g such that Φ( B ) ∼ = g Φ( B ) . Moreover, Ψ preserves identityand composition. Harrison-Trainor, Melnikov, Miller, and Montalb´an [3] proved the following.
Theorem 1.1.
For a pair of structures A and B , the following are equivalent:1. A is effectively interpreted in B ,2. there is a computable functor from B to A . Definition 1.6.
Suppose K ≤ tc K ′ via Θ . (1) We say that the structuresin K are uniformly effectively interpreted in their Θ -images if there is a fixedcollection of computable Σ formulas (without parameters) such that, for all A ∈ K , define an interpretation of A in Θ( A ) .(2) We say that Φ and Ψ form a uniform computable functor from thestructures Θ( A ) to A if these Turing operators serve for all A ∈ K . There is a uniform version of Theorem 1.1.
Theorem 1.2.
For classes
K, K ′ with K ≤ tc K ′ via Θ , the following are equiv-alent:1. there are computable Σ formulas (without parameters) which, for all A ∈ K , effectively interpret A in Θ( A ) ,2. there are uniform Turing operators Φ , Ψ that, for all A ∈ K , form acomputable functor from Θ( A ) to A . Note : In the proof of Theorem 1.1, it is important that D consist of tuplesof arbitrary arity. The next result is an immediate corollary of that theorem:take the computable functor always to have Φ B = A , ignoring its oracle andsimply computing A , while Ψ always computes the identity function. However,the direct proof that we give here exemplifies the use of arbitrarily long tuples. Proposition 1.3. If A is computable, then it is effectively interpreted in allstructures B . roof. Let D = B <ω . Let ¯ b ∼ ¯ c if ¯ b, ¯ c are tuples of the same length. Forsimplicity, suppose A = ( ω, R ), where R is binary. If A | = R ( m, n ), let R ∗ (¯ b, ¯ c )for all ¯ b of length m and ¯ c of length n . Then ( D, R ∗ ) / ∼ ∼ = A .It is natural to ask whether, when A ≤ s B , there must be an effectiveinterpretation of A in B . It is also natural to ask whether, when A is effectivelyinterpreted in ( B , ¯ b ) with parameters ¯ b , it must be effectively interpreted in B without parameters. Kalimullin [6] gave examples providing negative answersto both questions.Maltsev defined a Turing computable embedding of fields in 2-step nilpotentgroups. The embedding takes each field F to its Heisenberg group H ( F ). Toshow that the embedding preserves isomorphism, Maltsev gave uniform existen-tial formulas defining a copy of F in H ( F ). The definitions involved a pair ofparameters, whose orbit is defined by an existential (in fact, quantifier-free) for-mula. In Section 2, we recall Maltsev’s definitions. In Section 3, we describe auniform computable functor that, for all F , takes copies of H ( F ), with their iso-morphisms, to copies of F , with corresponding isomorphisms. By Theorem 1.2,it follows that there is a uniform effective interpretation of F in H ( F ) with noparameters. In Section 4, we give explicit finitary existential formulas that de-fine such an interpretation. In Section 5, we note that although F is effectivelyinterpretable in H ( F ) and H ( F ) is effectively interpretable in F , we do not,in general, have effective bi-interpretability. In Section 6, we generalize whatwe did in passing from Maltsev’s definition, with parameters, to the uniformeffective interpretation, with no parameters. F in H ( F ) In this section, we recall Maltsev’s embedding of fields in 2-step nilpotent groups,and his formulas that define a copy of the field in the group. Recall that for afield F , the Heisenberg group H ( F ) is the set of matrices of the form h ( a, b, c ) = a c b with entries in F . Note that h (0 , ,
0) is the identity matrix. We are interestedin non-commuting pairs in H ( F ). One such pair is ( h (1 , , , h (0 , , u = h ( u , u , u ) and v = h ( v , v , v ), let∆ ( u,v ) = (cid:12)(cid:12)(cid:12)(cid:12) u v u v (cid:12)(cid:12)(cid:12)(cid:12) . For a group G , we write Z ( G ) for the center. For group elements x, y , the commutator is [ x, y ] = x − y − xy . The following technical lemma provides muchof the information we need to show that F is defined, with parameters, in H ( F ).4 emma 2.1.
1. (a) For u and v , the commutator, [ u, v ] , is h (0 , , ∆ ( u,v ) ) , and(b) [ u, v ] = 1 iff ∆ ( u,v ) = 0 .2. Let u = h ( u , u , u ) , and let v = h ( v , v , v ) . If (cid:20) u u (cid:21) = (cid:20) (cid:21) , then u ∈ Z ( H ( F )) . If (cid:20) u u (cid:21) = (cid:20) (cid:21) , then [ u, v ] = 1 iff there exists α suchthat (cid:20) v v (cid:21) = α · (cid:20) u u (cid:21) .3. Z ( H ( F )) consists of the elements of the form h (0 , , c ) .4. If [ u, v ] = 1 , then x ∈ Z ( H ( F )) iff [ x, u ] = [ x, v ] = 1 .Proof. For Part 1, (a) is proved by direct computation, and (b) follows from(a). Parts 2 and 3 are easy consequences of Part 1. We prove Part 4. Suppose[ u, v ] = 1. If x ∈ Z ( H ( F )), then it commutes with both u and v . We must showthat if x commutes with both u and v , then x ∈ Z ( H ( F )). Let u = h ( u , u , u ), v = h ( v , v , v ), and x = h ( x , x , x ). By Part 2, since [ x, u ] = 1, there exists α such that (cid:20) x x (cid:21) = α (cid:20) u u (cid:21) . Similarly, since [ x, v ] = 1, there exists β suchthat (cid:20) x x (cid:21) = β (cid:20) v v (cid:21) . Since the vectors (cid:20) u u (cid:21) and (cid:20) v v (cid:21) , are linearlyindependent, this implies that α = β = 0. It follows that x = x = 0, so x ∈ Z ( H ).The next lemma tells us how, for any non-commuting pair u, v in the group( H ( F ) , ∗ ), we can define operations + and · , and an isomorphism f from F to( Z ( H ( F )) , + , · ). Lemma 2.2.
Let u = h ( u , u , u ) and v = h ( v , v , v ) be a non-commutingpair. Assume that α, β, γ ∈ F . Let x = h (0 , , α · ∆ ( u,v ) ) , y = h (0 , , β · ∆ ( u,v ) ) ,and z = h (0 , , γ · ∆ ( u,v ) ) . Then1. α + β = γ iff x ∗ y = z , where ∗ is the matrix multiplication.2. α · β = γ iff there exist x ′ and y ′ such that [ x ′ , u ] = [ y ′ , v ] = 1 , [ u, y ′ ] = y , [ x ′ , v ] = x , and z = [ x ′ , y ′ ] .Proof. For Part 1, matrix multiplication yields the fact that h (0 , , a ) ∗ h (0 , , b ) = h (0 , , a + b ) . Then α + β = γ iff x ∗ y = h (0 , , α · ∆ ( u,v ) ) ∗ h (0 , , β · ∆ ( u,v ) ) = h (0 , , γ · ∆ ( u,v ) ) = z . α · β = γ . We take x ′ = h ( α · u , α · u , y ′ = h ( β · v , β · v , ( x ′ ,u ) = 0, so [ x ′ , u ] = h (0 , ,
0) = 1.Similarly, [ y ′ , v ] = 1. Also, ∆ ( x ′ ,v ) = α · ∆ ( u,v ) , so [ x ′ , v ] = h (0 , , α · ∆ ( u,v ) ) = x .Similarly, ∆ ( u,y ′ ) = β · ∆ ( u,v ) , so [ u, y ′ ] = h (0 , , β · ∆ ( u,v ) ) = y . Finally,∆ ( x ′ ,y ′ ) = α · β · ∆ ( u,v ) = γ · ∆ ( u,v ) , so [ x ′ , y ′ ] = h (0 , , γ · ∆ ( u,v ) ) = z .Now, suppose we have x ′ and y ′ such that [ x ′ , u ] = [ y ′ , v ] = 1, [ u, y ′ ] = y ,[ x ′ , v ] = x , and [ x ′ , y ′ ] = z . Say that x ′ = h ( x ′ , x ′ , x ′ ) and y ′ = h ( y ′ , y ′ , y ′ ).Since [ x ′ , v ] = x , ∆ ( x ′ ,v ) = α · ∆ ( u,v ) , so (cid:20) x ′ x ′ (cid:21) = α (cid:20) u u (cid:21) . Since [ u, y ′ ] = y ,∆ ( u,y ′ ) = β · ∆ ( u,v ) , so (cid:20) y ′ y ′ (cid:21) = β (cid:20) v v (cid:21) . Combining these facts, we see that∆ ( x ′ ,y ′ ) = (cid:12)(cid:12)(cid:12)(cid:12) x ′ y ′ x ′ y ′ (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) α · u β · v α · u β · v (cid:12)(cid:12)(cid:12)(cid:12) = α · β · ∆ ( u,v ) . Since [ x ′ , y ′ ] = z ,∆ ( x ′ ,y ′ ) = γ · ∆ ( u,v ) . Since u and v do not commute, ∆ ( u,v ) = 0. Therefore, α · β = γ .The main result of the section follows directly from Lemmas 2.1 and 2.2. Theorem 2.3 (Maltsev, Morozov) . For an arbitrary non-commuting pair ( u, v ) in H ( F ) , we get F ( u,v ) = ( Z ( H ( F )) , ⊕ , ⊗ ( u,v ) ) where1. x ∈ Z ( H ( F )) iff [ x, u ] = [ x, v ] = 1 ,2. ⊕ is the group operation from H ( F ) ,3. ⊗ ( u,v ) is the set of triples ( x, y, z ) such that there exist x ′ , y ′ with [ x ′ , u ] = [ y ′ , v ] = 1 , [ x ′ , v ] = x , [ u, y ′ ] = y , and [ x ′ , y ′ ] = z ,4. the function g ( u,v ) taking α ∈ F to h (0 , , α · ∆ ( u,v ) ) ∈ H ( F ) is an isomor-phism between F and F ( u,v ) . Note : From Part 4, it is clear that h (0 , , ∆ ( u,v ) ) is the multiplicative identityin F ( u,v ) —we may write 1 ( u,v ) for this element. Proposition 2.4.
There is a uniform Medvedev reduction Φ of F to H ( F ) .Proof. Given G ∼ = H ( F ), we search for a non-commuting pair ( u, v ) in G , andthen use Maltsev’s definitions to get a copy of F computable from G .It turns out that the Medvedev reduction Φ is half of a computable functor.In the next section, we explain how to get the other half. In the previous section, we saw that for any field F and any non-commutingpair ( u, v ) in H ( F ), there is an isomorphic copy F ( u,v ) of F defined in H ( F ) byfinitary existential formulas with parameters ( u, v ). The defining formulas arethe same for all F . Hence, there is a uniform Turing operator Φ that, for all fields6 , takes copies of H ( F ) to copies of F . In this section, we describe a companionoperator Ψ so that Φ and Ψ together form a uniform computable functor. Forany field F , and any triple ( G , p, G ) such that G and G are copies of H ( F )and p is an isomorphism from G onto G , the function Ψ( G , p, G ) must be anisomorphism from Φ( G ) onto Φ( G ), and, moreover, the isomorphisms givenby Ψ must preserve identity and composition. We saw in the previous sectionthat for any field F , and any non-commuting pair ( u, v ) in H ( F ), the function g ( u,v ) taking α to h (0 , , α · ∆ ( u,v ) ) is an isomorphism from F onto F ( u,v ) . Weuse this g ( u,v ) below. Lemma 3.1.
For any F and any non-commuting pairs ( u, v ) , ( u ′ , v ′ ) in H ( F ) ,there is a natural isomorphism f ( u,v ) , ( u ′ ,v ′ ) from F ( u,v ) onto F ( u ′ ,v ′ ) . Moreover,the family of isomorphisms f ( u,v ) , ( u ′ ,v ′ ) is functorial; i.e.,1. for any non-commuting pair ( u, v ) , the function f ( u,v ) , ( u,v ) is the identity,2. for any three non-commuting pairs ( u, v ) , ( u ′ , v ′ ) , and ( u ′′ , v ′′ ) , f ( u,v ) , ( u ′′ ,v ′′ ) = f ( u ′ ,v ′ ) , ( u ′′ ,v ′′ ) ◦ f ( u,v ) , ( u ′ ,v ′ ) . Proof.
We let f ( u,v ) , ( u ′ ,v ′ ) = g ( u ′ ,v ′ ) ◦ g − u,v ) . This is an isomorphism from F ( u,v ) onto F ( u ′ ,v ′ ) . It is clear that f ( u,v ) , ( u,v ) is the identity. Consider non-commuting pairs ( u, v ), ( u ′ , v ′ ), and ( u ′′ , v ′′ ). We must show that f ( u ′ ,v ′ ) , ( u ′′ ,v ′′ ) ◦ f ( u,v ) , ( u ′ ,v ′ ) = f ( u,v ) , ( u ′′ ,v ′′ ) . We have: f ( u ′ ,v ′ ) , ( u ′′ ,v ′′ ) ◦ f ( u,v ) , ( u ′ ,v ′ ) = g ( u ′′ ,v ′′ ) ◦ g − u ′ ,v ′ ) ◦ g ( u ′ ,v ′ ) ◦ g − u,v ) == g ( u ′′ ,v ′′ ) ◦ g − u,v ) == f ( u,v ) , ( u ′′ ,v ′′ ) . The next lemma says that there is a uniform existential definition of thefamily of isomorphisms f ( u,v ) , ( u ′ ,v ′ ) . Lemma 3.2.
There is a finitary existential formula ψ ( u, v, u ′ , v ′ , x, y ) that,for any two non-commuting pairs ( u, v ) and ( u ′ , v ′ ) , defines the isomorphism f ( u,v ) , ( u ′ ,v ′ ) taking x ∈ F ( u,v ) to y ∈ F ( u ′ ,v ′ ) .Proof. Since the operation ⊗ ( u,v ) and 1 ( u ′ ,v ′ ) are definable by ∃ –formulas withparameters u, v and u ′ , v ′ respectively, it suffices to prove the equivalence f ( u,v ) , ( u ′ ,v ′ ) ( x ) = y ⇔ x ⊗ ( u,v ) ( u ′ ,v ′ ) = y. First assume that f ( u,v ) , ( u ′ ,v ′ ) ( x ) = y , i.e., y = g ( u ′ ,v ′ ) ◦ g − u,v ) ( x ). Let α = g − u,v ) ( x ), i.e., x = h (0 , , α · ∆ ( u,v ) ). It follows that y = h (cid:0) , , α · ∆ ( u ′ ,v ′ ) (cid:1) .7hen x ⊗ ( u,v ) ( u ′ v ′ ) = h (cid:0) , , α · ∆ ( u,v ) (cid:1) ⊗ ( u,v ) h (cid:0) , , ∆ ( u ′ ,v ′ ) (cid:1) == h (cid:0) , , α · ∆ ( u,v ) (cid:1) ⊗ ( u,v ) h (cid:18) , , ∆ ( u ′ ,v ′ ) ∆ ( u,v ) · ∆ ( u,v ) (cid:19) == h (cid:18) , , α · ∆ ( u ′ ,v ′ ) ∆ ( u,v ) · ∆ ( u,v ) (cid:19) == h (cid:0) , , α · ∆ ( u ′ ,v ′ ) (cid:1) = y. Assume now that x ⊗ ( u,v ) ( u ′ ,v ′ ) = y and let x = h (cid:0) , , α · ∆ ( u,v ) (cid:1) . Then y = x ⊗ ( u,v ) ( u ′ ,v ′ ) = h (cid:0) , , α · ∆ ( u,v ) (cid:1) ⊗ ( u,v ) h (cid:0) , , ∆ ( u ′ ,v ′ ) (cid:1) == h (cid:0) , , α · ∆ ( u ′ ,v ′ ) (cid:1) = g ( u ′ ,v ′ ) ◦ g − u,v ) ( x ) = f ( u,v ) , ( u ′ ,v ′ ) ( x ) . We will use Lemmas 3.1 and 3.2 to prove the following.
Proposition 3.3.
There is a uniform computable functor that, for all fields F ,takes H ( F ) to F .Proof. Let Φ be the uniform Medvedev reduction of F to H ( F ). Take copies G , G of H ( F ) and take p such that G ∼ = p G . We describe q = Ψ( G , p, G ) asfollows. Let ( u, v ) be the first non-commuting pair in G , and let ( u ′ , v ′ ) be thefirst non-commuting pair in G . Now, p takes ( u, v ) to a non-commuting pair( p ( u ) , p ( v )), and p maps F ( u,v ) isomorphically onto F ( p ( u ) ,p ( v )) . The function f ( p ( u ) ,p ( v )) , ( u ′ ,v ′ ) is an isomorphism from F ( p ( u ) ,p ( v )) onto F ( u ′ ,v ′ ) . We get anisomorphism q from F ( u,v ) onto F ( u ′ ,v ′ ) by composing p with f ( p ( u ) ,p ( v )) , ( u ′ ,v ′ ) .For x ∈ F ( u,v ) , we let q ( x ) = f ( p ( u ) ,p ( v )) , ( u ′ ,v ′ ) ( p ( x )). Since f ( p ( u ) ,p ( v )) , ( u ′ ,v ′ ) isdefined by an existential formula, with parameters p ( u ) , p ( v ) , u ′ , v ′ , we can applya uniform effective procedure to compute q from ( G , p, G ).If G = G and p is the identity, then ( u, v ) = ( u ′ , v ′ ), and by Lemma3.1, f ( u,v ) , ( u ′ ,v ′ ) is the identity. Consider G , G , G , all copies of G , with func-tions p , p such that G ∼ = p G and G ∼ = p G . Then p = p ◦ p is anisomorphism from G onto G . Let q = Ψ( G , p , G ), q = Ψ( G , p , G ), and q = Ψ( G , p , G ). We must show that q = q ◦ q . The idea is to transfer ev-erything to G and use Lemma 3.1. Let r be the result of transferring q downto G — r = f ( p ( u ) ,p ( v )) , ( p ( u ) ,p ( v )) . We have q ( x ) = y iff r ( p ( x )) = p ( y ).Let r be the result of transferring q down to G — r = f ( p ( u ) ,p ( v )) , ( u,v ) . Wehave q ( y ) = z iff r ( p ( y )) = z . We let r be the result of transferring q downto G — r = f ( p ( u ) ,p ( v )) , ( u,v ) . We have q ( x ) = z iff r ( p ( x )) = z . By Lemma3.1, r = r ◦ r . If q ( x ) = y and q ( y ) = z , then r ( p ( x )) = p ( y ), and r ( p ( y )) = z . Then r ( p ( x )) = z , so q ( x ) = z , as required. Corollary 3.4.
There is a uniform effective interpretation of F in H ( F ) .Proof. Apply the result from [3]. 8he result from [3] gives a uniform interpretation of F in H ( F ), valid forall countable fields F , using computable Σ formulas with no parameters. Thetuples from H ( F ) that represent elements of F may have arbitrary arity. In thenext section, we will do better.We note here that the uniform interpretation of F in H ( F ) given in thissection allows one to transfer the computable-model-theoretic properties of anygraph G to a 2-step-nilpotent group, without introducing any constants. Thisis not a new result: in [8], Mekler gave a related coding of graphs into 2-step-nilpotent groups, which, in concert with the completeness of graphs for suchproperties (see [5]), appears to yield the same fact, although Mekler’s codinghad different goals than completeness. Then, in [5], Hirschfeldt, Khoussainov,Shore, and Slinko used Maltsev’s interpretation of an integral domain in itsHeisenberg group with two parameters, along with the completeness of integraldomains, to re-establish it. More recently, [9] demonstrated the completenessof fields, by coding graphs into fields, From that result, along with Corollary3.4 and the usual definition of H ( F ) as a matrix group given by a set of triplesfrom F , we achieve a coding of graphs into 2-step-nilpotent groups, differentfrom Mekler’s coding, with no constants required. Our goal in this section is to give explicit existential formulas defining a uniformeffective interpretation of a field in its Heisenberg group. We discovered theformulas for this interpretation by examining the infinitary formulas used in theinterpretation in Corollary 3.4 and trimming them down to their essence, whichturned out to be finitary.
Theorem 4.1.
There are finitary existential formulas that, uniformly for everyfield F , define an effective interpretation of F in H ( F ) , with elements of F represented by triples of elements from H ( F ) . We offer intuition before giving the formal proof. The domain D of theinterpretation will consist of those triples ( u, v, x ) from H ( F ) with uv = vu and x in the center: for each single ( u, v ), we apply Maltsev’s definitions, with u , v as parameters, to get F ( u,v ) ∼ = F . We view the triples arranged as follows: F ( u,v ) F ( u ′ ,v ′ ) F ( u ′′ ,v ′′ ) · · · ( u, v, x )( u, v, x )( u, v, x )( u, v, x )... ( u ′ , v ′ , x )( u ′ , v ′ , x )( u ′ , v ′ , x )( u ′ , v ′ , x )... ( u ′′ , v ′′ , x )( u ′′ , v ′′ , x )( u ′′ , v ′′ , x )( u ′′ , v ′′ , x )...9ere each column can be seen as F ( u,v ) for some non-commuting pair ( u, v ).Now the system of isomorphisms from Lemma 3.1 will allow us to identifyeach element in one column with a single element from each other column, andmodding out by this identification will yield a single copy of F . Proof.
Let H be a group isomorphic to H ( F ). Recalling the natural isomor-phisms f ( u,v ) , ( u ′ ,v ′ ) defined in Lemma 3.1 for non-commuting pairs ( u, v ) and( u ′ , v ′ ), we define D ⊆ H , a binary relation ∼ on D , and ternary relations ⊕ , ⊙ (which are binary operations) on D , as follows.1. D is the set of triples ( u, v, x ) such that uv = vu and xu = ux and xv = vx . (Notice that, no matter which non-commuting pair ( u, v ) ischosen, the set of corresponding elements x is precisely the center Z ( H ),by Theorem 2.3.)2. ( u, v, x ) ∼ ( u ′ , v ′ , x ′ ) holds if and only if the isomorphism f ( u,v ) , ( u ′ ,v ′ ) from F ( u,v ) to F ( u ′ ,v ′ ) maps x to x ′ .3. ⊕ (( u, v, x ) , ( u ′ , v ′ , y ′ ) , ( u ′′ , v ′′ , z ′′ )) holds if there exist y, z ∈ H such that( u, v, y ) ∼ ( u ′ , v ′ , y ′ ) and ( u, v, z ) ∼ ( u ′′ , v ′′ , z ′′ ), and F ( u,v ) | = x + y = z .4. ⊙ (( u, v, x ) , ( u ′ , v ′ , y ′ ) , ( u ′′ , v ′′ , z ′′ )) holds if there exist y, z ∈ H such that( u, v, y ) ∼ ( u ′ , v ′ , y ′ ) and ( u, v, z ) ∼ ( u ′′ , v ′′ , z ′′ ), and F ( u,v ) | = x · y = z .Lemma 3.2 yielded a finitary existential formula defining the relation ( u, v, x ) ∼ ( u ′ , v ′ , x ′ ). Moreover, the field addition and multiplication were defined in F ( u,v ) by finitary existential formulas using u and v , which were parametersthere but here are elements of the triples in D . Finally, we must consider thenegations of the relations. First, ( u, v, x ) ( u ′ , v ′ , x ′ ) if and only if some y ′ commuting with u ′ and v ′ satisfies ( u, v, x ) ∼ ( u ′ , v ′ , y ′ ) and y ′ = x ′ – that is,just if f ( u,v ) , ( u ′ ,v ′ ) maps x to some element different from x ′ . Likewise, since + isa binary operation in F ( u,v ) , the negation of ⊕ (( u, v, x ) , ( u ′ , v ′ , y ′ ) , ( u ′′ , v ′′ , z ′′ ))is defined by saying that some w ′′ = z ′′ is the sum: ∃ w ′′ ([ w ′′ , u ′′ ] = 1 = [ w ′′ , v ′′ ] & w ′′ = z ′′ & ⊕ (( u, v, x ) , ( u ′ , v ′ , y ′ ) , ( u ′′ , v ′′ , w ′′ ))) , which is also existential, and similarly for the negation of ⊙ . Therefore, all ofthese sets have finitary existential definitions in the language of groups, with noparameters, as do the negations of ∼ , ⊕ , and ⊙ . (In fact, the complement of D is Σ as well.)The functoriality of the system of isomorphisms f ( u,v ) , ( u ′ ,v ′ ) (across all pairsof pairs of noncommuting elements) ensures that ∼ will be an equivalence rela-tion. Lemma 3.1 showed that f ( u,v ) , ( u,v ) is always the identity, giving reflexivity.Transitivity follows from the functorial property in that same lemma: f ( u,v ) , ( u ′′ ,v ′′ ) = f ( u ′ ,v ′ ) , ( u ′′ ,v ′′ ) ◦ f ( u,v ) , ( u ′ ,v ′ ) , and with ( u ′′ , v ′′ ) = ( u, v ), this property also yields the symmetry of ∼ .10he definitions of ⊕ and ⊙ essentially say to convert all three triples into ∼ -equivalent triples with the same initial coordinates u and v , and then tocheck whether the final coordinates satisfy Maltsev’s definitions of + and · in the field F ( u,v ) . Understood this way, they clearly respect the equivalence ∼ . Finally, by fixing any single noncommuting pair ( u, v ), we see that the set { ( u, v, x ) : x ∈ Z ( H ) } contains one element from each ∼ -class and, under ⊕ and ⊙ , is isomorphic to the field F ( u,v ) defined by Maltsev, which in turn isisomorphic to the original field F .It should be noted that, although this interpretation of F in H ( F ) wasdeveloped using computable functors on countable fields F , it is valid even when F is uncountable (or finite). A full proof requires checking that the system ofisomorphisms f ( u,v ) , ( u ′ ,v ′ ) remains functorial and existentially definable even inthe uncountable case, but this is straightforward.In Theorem 4.1, to eliminate parameters from Maltsev’s definition of F in H ( F ), we gave an interpretation of F in H ( F ), rather than another definition.(Recall that a definition is an interpretation in which the equivalence relationon the domain is simply equality.) We now demonstrate the impossibility ofstrengthening the theorem to give a parameter-free definition of F in H ( F ). Lemma 4.2.
In the Heisenberg group H ( F ) of a field F , the only singletonorbit is that of the identity element.Proof. Suppose that the element h ( a, b, c ) ∈ H ( F ) is fixed by all automorphisms.Its images under conjugation by h (1 , ,
0) and by h (0 , ,
0) are h ( a, b, c + b ) and h ( a, b, c − a ), so a = b = 0. But the automorphism of H ( F ) mapping h ( x, y, z )to h ( y, x, xy − z ), which interchanges h (1 , ,
0) with h (0 , , h (0 , , c ) to h (0 , , − c ), hence shows that c = 0 as well. Corollary 4.3.
There is no parameter-free definition of any field F in itsHeisenberg group H ( F ) by finitary formulas.Proof. Suppose that there were such a definition, and let D ⊆ H ( F ) n be itsdomain. By Lemma 4.2, the only ( x , . . . , x n ) ∈ ( H ( F )) n that is fixed byall automorphisms of H ( F ) is the tuple with all x i = e , the identity elementof H ( F ). So, for every ~x ∈ D except this identity tuple, there would be anautomorphism of H ( F ) mapping ~x to some ~y = ~x in D . With equality of n -tuples as the equivalence relation on D , this would yield an automorphism ofthe field F (viewed as D under the definable addition and multiplication) thatdoes not fix ~x . However, the prime subfield of F contains at least two elements,each of which must be fixed by every automorphism of F . If B is interpreted in A , we write B A for the copy of B given by the interpretationof B in A . The structures A and B are effectively bi-interpretable if there areuniformly relatively computable isomorphisms f from A onto A B A and g from11 onto B A B . In general, the isomorphism f would map each element of A to anequivalence class of equivalence classes of tuples in A . We would represent f bya relation R f that holds for a, ¯ a , . . . , ¯ a r if f maps a to the equivalence class ofthe tuple of equivalence classes of the ¯ a i ’s. Similarly, the isomorphism g wouldbe represented by a relation R g that holds for b, ¯ b , . . . , ¯ b r if g maps b to theequivalence class of the tuple of equivalence classes of the ¯ b i ’s. Saying that f and g are uniformly relatively computable is equivalent to saying that the relations R f , R g , have generalized computable Σ definitions without parameters.For a field F and its Heisenberg group H ( F ), when we define H ( F ) in F ,the elements of H ( F ) are represented by triples from F , and we have finitaryformulas, quantifier-free or existential, that define the group operation (as arelation). When we interpret F in H ( F ), the elements of F are representedby triples from H ( F ), and we have finitary existential formulas that define thefield operations and their negations (as ternary relations). Thus, in F H ( F ) F (thecopy of F interpreted in the copy of H ( F ) that is defined in F ), the elementsare equivalence classes of triples of triples. In H ( F ) F H ( F ) (the copy of H ( F )defined in the copy of F that is interpreted in H ( F )), the elements are triplesof equivalence classes of triples. So, an isomorphism f from F to F H ( F ) F isrepresented by a 10-ary relation R f on F , and an isomorphism g from H ( F ) to H ( F ) F H ( F ) —it is represented by a 10-ary relation R g on H ( F ).For a Turing computable embedding Θ of K in K ′ we have uniform effectivebi-interpretability if there are (generalized) computable Σ formulas with noparameters that, for all A ∈ K and B = Θ( A ), define isomorphisms from A to A B A and from B to B A B . After a talk by the fifth author, Montalb´an asked thefollowing very natural question. Question 5.1.
Do we have uniform effective bi-interpretability of F and H ( F ) ? The answer to this question is negative. In particular, Q and H ( Q ) arenot effectively bi-interpretable. One way to see this is to note that Q is rigid,while H ( Q ) is not—in particular, for any non-commuting pair, u, v ∈ H ( Q ),there is a group automorphism that takes ( u, v ) to ( v, u ). The negative answerto Question 5.1 then follows from [10, Lemma VI.26(4)], which states that if A and B are effectively bi-interpretable, then their automorphism groups areisomorphic.Morozov’s result shows which half of effective bi-interpretability causes thedifficulties. Proposition 5.1 (Morozov) . There is a finitary existential formula that, forall F , defines in F a specific isomorphism k from F to F H ( F ) F .Proof. In F , we have the copy of H ( F ), consisting of triples ( a, b, c ) (repre-senting h ( a, b, c )), for a, b, c ∈ F . The group operation, derived from matrixmultiplication, is ( a, b, c ) ∗ ( a ′ , b ′ , c ′ ) = ( a + a ′ , b + b ′ , c + c ′ + ab ′ ). The definitionsof the universe and the operation are quantifier-free, with no parameters. Wehave seen how to interpret F in H ( F ) using finitary existential formulas with12o parameters. There is a natural isomorphism k from F onto F H ( F ) F obtainedas follows. In H ( F ), let u = h (1 , ,
0) and v = h (0 , , ( u,v ) = 1.We have an isomorphism mapping F to F ( u,v ) that takes α to h (0 , , α ). We let k ( α ) be the ∼ -class of ( u, v, h (0 , , α )). The isomorphism k is defined in F byan existential formula. The complement of k is defined by saying that k ( α ) hassome other value.The other half of what we would need for uniform effective bi-interpretabilityis sometimes impossible, as remarked above in the case F = Q . We do not knowof any examples where F and H ( F ) are effectively bi-interpretable: the obstaclefor Q might hold in all cases. Problem 5.1.
For which fields F , if any, are the automorphism groups of F and H ( F ) isomorphic? Even if there are fields F such that Aut( F ) ∼ = Aut( H ( F )), we suspect that F and H ( F ) are not effectively bi-interpretable, simply because it is difficult tosee how one might give a computable Σ formula in the language of groups thatdefines a specific isomorphism from H ( F ) to H ( F ) F H ( F ) . Our first general definition and proposition follow closely the example of a fieldand its Heisenberg group.
Definition 6.1.
Let A be a structure for a computable relational language.Assume that its basic relations are R i , where R i is k i -ary. We say that A is effectively defined in B with parameters ¯ b if there exist D (¯ b ) ⊆ B <ω , and ± R i (¯ b ) ⊆ D (¯ b ) k i , defined by a uniformly computable sequence of generalizedcomputable Σ formulas with parameters ¯ b . Proposition 6.1.
Suppose A is effectively defined in B with parameters ¯ b . For ¯ c in the orbit of ¯ b , let A ¯ c be the copy of A defined by the same formulas, butwith parameters ¯ c replacing ¯ b . Then the following conditions together suffice togive an effective interpretation of A in B without parameters:1. The orbit of ¯ b is defined by a computable Σ formula ϕ (¯ u ) ;2. There is a generalized computable Σ formula ψ (¯ u, ¯ v, ¯ x, ¯ y ) such that forall ¯ c, ¯ d in the orbit of ¯ b , the formula ψ (¯ c, ¯ d, ¯ x, ¯ y ) defines an isomorphism f ¯ c, ¯ d from A ¯ c onto A ¯ d ; and3. The family of isomorphisms f ¯ c, ¯ d preserves identity and composition.Proof. We write D (¯ b ), ± R i (¯ b ) for the set and relations that give a copy of A and for the defining formulas (with parameters ¯ b ). We obtain a parameter-freeinterpretation of A in B as follows: 13. Let D consist of the tuples (¯ c, ¯ x ) such that ¯ c is in the orbit of ¯ b and ¯ x isin D (¯ c ). This is defined by a generalized computable Σ formula.2. Let ∼ be the set of pairs ((¯ c, ¯ x ) , ( ¯ d, ¯ y )) in D such that f ¯ c, ¯ d (¯ x ) = ¯ y . Thisis defined by a generalized computable Σ formula. For pairs (¯ c, ¯ x ), ( ¯ d, ¯ y )from D , it follows that (¯ c, ¯ x ) ( ¯ d, ¯ y ) if and only if( ∃ ¯ y ′ )(( ¯ d, ¯ y ′ ) ∈ D & f ¯ c, ¯ d (¯ x ) = ¯ y ′ & ¯ y ′ = ¯ y ) . Hence the negation of ∼ is also defined by a generalized computable Σ formula.3. We let R ∗ i be the set of k i -tuples ((¯ b , ¯ x ) , . . . , (¯ b k i , ¯ x k i )) in D k i such thatfor the tuple (¯ y , . . . , ¯ y k i ) with f ¯ b j , ¯ b (¯ x j ) = ¯ y j , we have (¯ y , . . . , ¯ y k i ) ∈ R i (¯ b ). This is defined by a generalized computable Σ formula. Thecomplementary relation ¬ R ∗ i is the set of tuples ((¯ b , ¯ x ) , . . . , (¯ b k i , ¯ x k i ))such that for ¯ y , . . . , ¯ y k i as above, (¯ y , . . . , ¯ y k i ) ∈ ¬ R i (¯ b ). This is alsodefined by a generalized computable Σ formula.The verification is identical to that of Theorem 4.1. Corollary 6.2.
In the situation of Proposition 6.1, if D (¯ b ) is contained in B n for some single n ∈ ω , then the ψ in item (2) and the formulas in Definition 6.1will simply be computable Σ formulas (as opposed to generalized computable Σ formulas) and the interpretation of A in B without parameters will also be bycomputable (as opposed to generalized) Σ formulas. The reader will have noticed that we only produced an interpretation of A in B , even though we originally had a definition (with parameters) of A in B .Corollary 4.3 shows that in general this is the best that can be done. On theother hand, we may extend Proposition 6.1 and remove parameters even in thecase where A is interpreted (as opposed to being defined) with parameters in B . Definition 6.2 (Effective Interpretation with Parameters) . We say that A ,with basic relations R i , k i -ary, is effectively interpreted with parameters ¯ b in B if there exist D ⊆ B <ω , ≡⊆ D , and R ∗ i ⊆ D k i such that1. ( D, ( R ∗ i ) i ) / ≡ ∼ = A ,2. D , ± ≡ , and ± R ∗ i are defined by a computable sequence of generalizedcomputable Σ formulas, with a fixed finite tuple of parameters ¯ b . Again, in the case where D ⊆ B n for some fixed n , the formulas definingthe effective interpretation are computable Σ formulas of the usual kind, withparameters ¯ b . Proposition 6.3.
Suppose that A (with basic relations R i , k i -ary) has an effec-tive interpretation in B with parameters ¯ b . For ¯ c in the orbit of ¯ b , let A ¯ c be thecopy of A obtained by replacing the parameters ¯ b by ¯ c in the defining formulas,with domain D ¯ c / ≡ ¯ c containing ≡ ¯ c -classes [¯ a ] ≡ ¯ c . Then the following conditionssuffice for an effective interpretation of A in B (without parameters): . The orbit of ¯ b is defined by a computable Σ formula ϕ (¯ x ) ;2. There is a relation F ⊆ B <ω , with a generalized computable Σ -definition,such that for every ¯ c and ¯ d in the orbit of ¯ b , the set of pairs (¯ x, ¯ y ) ∈ D ¯ c × D ¯ d with (¯ c, ¯ d, ¯ x, ¯ y ) ∈ F is invariant under ≡ ¯ c on ¯ x and under ≡ ¯ d on ¯ y , anddefines an isomorphism f ¯ c, ¯ d from A ¯ c onto A ¯ d ; and3. The family of isomorphisms f ¯ c, ¯ d preserves identity and composition.Proof. Let the new domain D consist of those tuples (¯ c, ¯ x ) with ¯ c in the orbitof ¯ b and ¯ x in D ¯ c . This is defined by a generalized computable Σ formula.Let the equivalence relation ∼ on D be the set of pairs ((¯ c, ¯ x ) , ( ¯ d, ¯ y )) ∈ D such that f ¯ c, ¯ d ([¯ x ] ≡ ¯ c ) = [¯ y ] ≡ ¯ d . This is defined by a generalized computable Σ formula. For (¯ c, ¯ x ), ( ¯ d, ¯ y ) ∈ D , we have (¯ c, ¯ x ) ( ¯ d, ¯ y ) if and only if( ∃ ¯ y ′ ∈ D ¯ d ) ( f ¯ c, ¯ d ([¯ x ] ≡ ¯ c ) = [¯ y ′ ] ≡ ¯ d & ¯ y ¯ d ¯ y ′ ) . Hence is also defined by a generalized computable Σ formula.Let R ∗ i be the set of k i -tuples ((¯ b , ¯ x ) , . . . , (¯ b k i , ¯ x k i )) in D k i such that for thetuple (¯ y , . . . , ¯ y k i ) with f ¯ b j , ¯ b (¯ x j ) = ¯ y j , we have (¯ y , . . . , ¯ y k i ) ∈ R i (¯ b ). This isdefined by a generalized computable Σ -formula. The complementary relation ¬ R ∗ i is the set of tuples ((¯ b , ¯ x ) , . . . , (¯ b k i , ¯ x k i )) such that for ¯ y , . . . , ¯ y k i as above,(¯ y , . . . , ¯ y k i ) ∈ ¬ R i (¯ b ). This too is defined by a generalized computable Σ formula. Finally, as in the proofs of Theorem 4.1 and Proposition 6.1, it is clearthat this yields an interpretation of A in B without parameters.A relation R ⊆ B <ω may have a definition that is generalized computable Σ α for a computable ordinal α , or generalized X -computable Σ α for an X -computable ordinal α , or generalized L ω ω , or generalized Σ α for a countableordinal α . The definition has the form W n ϕ n (¯ x n ), where the sequence of dis-juncts (each in L ω ω , but of different arities n ) is computable, or X -computable,or just countable. We note that each generalized L ω ω formula is generalized X -computable Σ α for an appropriately chosen X and α , and each generalizedΣ α -formula is generalized X -computable Σ α for an appropriately chosen X .As computable structure theorists, we have focused here on effective inter-pretations. Nevertheless, we wish to point out that our results apply not onlyto effective interpretations, but to all interpretations using generalized L ω ω for-mulas. The following theorem generalizes Proposition 6.3 and considers everyvariation we can imagine. Theorem 6.4.
Let A be a relational structure with basic relations R i that are k i -ary. Suppose there is an interpretation of A in B by generalized L ω ω formulas,with parameters ¯ b from B . For ¯ c in the orbit of ¯ b , let A ¯ c be the copy of A obtained by the interpretation with parameters ¯ c replacing ¯ b . Assume that thereis a generalized L ω ω -definable relation F defining, for each ¯ c and ¯ d in the orbitof ¯ b , an isomorphism f ¯ c, ¯ d : A ¯ c → A ¯ d as in Proposition 6.3, and that this familyis closed under composition, with the identity map as f ¯ c, ¯ c for all ¯ c .Then there is an interpretation of A in B by L ω ω formulas without param-eters. Moreover, the new interpretation satisfies all of the following. For each countable ordinal α , if the interpretation in ( B , ¯ b ) defines D , ≡ ,and each R i using Σ α formulas from L ω ω , and F and the orbit of ¯ b in B are both defined by Σ α formulas, then the parameter-free interpretationalso uses Σ α formulas to define these sets. • For each countable ordinal α , if the interpretation in ( B , ¯ b ) defines eachof D , ± ≡ , and ± R i using Σ α formulas, and F and the orbit of ¯ b in B are both defined by Σ α formulas, then the parameter-free interpretationalso uses Σ α formulas to define its domain, its equivalence relation ∼ , thecomplement , and its relations ± R i . (Defining and ¬ R i this way isrequired by the usual notion of effective Σ α interpretation.) • Let X ⊆ ω . If the interpretation in ( B , ¯ b ) used X -computable formulas,and F and the orbit of ¯ b in B are both defined by X -computable formulas,then the parameter-free interpretation also uses X -computable formulas.Of course, for every countable set of L ω ω formulas, there is an X thatcomputes them all. If the signature of A is infinite, and the formulas forthe interpretation of A in ( B , ¯ b ) are computable uniformly in X , then soare the formulas for the parameter-free interpretation of A in B .(With X = ∅ , X -computable formulas are simply computable formulas.) • If the interpretation in ( B , ¯ b ) had domain contained in B n for a single n , so that the defining formulas for this interpretation and for F in B are all in L ω ω (as opposed to generalized L ω ω ), then the parameter-freeinterpretation also uses (non-generalized) L ω ω formulas, and its domainis contained in B n + | ¯ b | . • If the interpretation in ( B , ¯ b ) used finitary formulas, and F and the orbitof ¯ b in B are both defined by finitary formulas, then the parameter-freeinterpretation also uses finitary formulas.Proof. We obtain the parameter-free interpretation just as in the proof of Propo-sition 6.3. Notice that, by a result of Scott in [11], the orbit of ¯ b must be defin-able by some L ω ω formula. Checking the specific claims is simply a matter ofwriting out the new formulas using the old ones. References [1] W. Calvert, D. F. Cummins, J.F. Knight, & S. Miller (Quinn), “Com-paring classes of finite structures,”
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