Turing Degrees and Automorphism Groups of Substructure Lattices
aa r X i v : . [ m a t h . L O ] N ov TURING DEGREES AND AUTOMORPHISM GROUPS OFSUBSTRUCTURE LATTICES
RUMEN DIMITROV, VALENTINA HARIZANOV, AND ANDREY MOROZOV
Abstract.
The study of automorphisms of computable and other structuresconnects computability theory with classical group theory. Among the non-computable countable structures, computably enumerable structures are oneof the most important objects of investigation in computable model theory.In this paper, we focus on the lattice structure of computably enumerablesubstructures of a given canonical computable structure. In particular, for aTuring degree d , we investigate the groups of d -computable automorphisms ofthe lattice of d -computably enumerable vector spaces, of the interval Booleanalgebra B η of the ordered set of rationals, and of the lattice of d -computablyenumerable subalgebras of B η . For these groups we show that Turing reducibil-ity can be used to substitute the group-theoretic embedding. We also provethat the Turing degree of the isomorphism types for these groups is the secondTuring jump of d , d ′′ . Automorphisms of effective structures
Computable model theory investigates algorithmic content (effectiveness) ofnotions, objects, and constructions in classical mathematics. In algebra this in-vestigation goes back to van der Waerden who in his
Modern Algebra introducedan explicitly given field as one the elements of which are uniquely representedby distinguishable symbols with which we can perform the field operations algo-rithmically. The formalization of an explicitly given field led to the notion of acomputable structure, one of the main objects of study in computable model the-ory. A structure is computable if its domain is computable and its relations andfunctions are uniformly computable. Further generalization and relativization ofnotion of a computable structure led to computably enumerable (abbreviated byc.e.) structures, as well as d -computable and d -c.e. computable structures fora given Turing degree d . In computability theory, Turing degrees are the mostimportant measure of relative difficulty of undecidable problems. All decidable MSC-class: 03D45 (Primary) 03C57, 08A35 (Secondary).The authors acknowledge partial support of the binational research grant DMS-1101123 fromthe National Science Foundation. The second author acknowledge support from the SimonsFoundation Collaboration Grant and from CCFF and Dean’s Research Chair awards of theGeorge Washington University.Preliminary version of this paper appeared in the CiE conference proceedings [1]. problems have Turing degree . There are uncountably many Turing degrees andthey are partially ordered by Turing reducibility, forming an upper semi-lattice.Order relations are pervasive in mathematics. One of the most importantsuch relations is the embedding of mathematical structures. The structures wefocus on are the automophism groups of various lattices of algebraic structuresthat are substructures of a large canonical computable structure. The studyof automorphisms of computable or computably enumerable structures connectscomputability theory and classical group theory. The set of all automorphisms ofa computable structure forms a group under composition. We are interested inmatching the embeddability of natural subgroups of this group with the Turingdegree ordering. It is also natural to ask questions about the complexity ofthe automorphism groups and its isomorphic copies since isomorphisms do notnecessarily preserve computability-theoretic properties.Our computability-theoretic notation is standard and as in [20, 19, 5, 6]. By D we denote the set of all Turing degrees, and by d = deg( D ) the Turing degreeof a set D . Hence = deg( ∅ ) . Turing jump operator is the main tool to obtainhigher Turing degrees. For a set D, the jump D ′ is the halting set relative to D. Turing established that d ′ = deg( D ′ ) > d . All computably enumerable sets haveTuring degrees ≤ ′ . By d ′′ we denote ( d ′ ) ′ , and so on. We recall the followingdefinition from computability theory. Definition 1.
A nonempty set of Turing degrees, I ⊆ D , is called a Turing ideal if: (1) ( ∀ a ∈ I )( ∀ b )[ b ≤ a ⇒ b ∈ I ] , and (2) ( ∀ a, b ∈ I )[ a ∨ b ∈ I ] . Notation 2.
Let I be a Turing ideal. Let M be a computable structure. Then: (1) Aut I ( M ) is the set of all d -computable automorphisms of M for any d ∈ I ; (2) If I = { s : s ≤ d } , then Aut I ( M ) is also denoted by Aut d ( M ) . When the structure M is ω with equality, then its automorphism group Aut ( M )is usually denoted by Sym ( ω ), the symmetric group of ω. Hence we have
Sym d ( ω ) = { f ∈ Sym ( ω ) : deg( f ) ≤ d } .(See [14, 6, 12, 13, 15, 16, 7] for previous computability-theoretic results about Aut ( M ).)The Turing degree spectrum of a countable structure A is DgSp ( A ) = { deg( B ) : B ∼ = A} , where deg( B ) is the Turing degree of the atomic diagram of B . Knight [10]proved that the degree spectrum of any structure is either a singleton or is upwardclosed. Only the degree spectrum of a so-called automorphically trivial structureis a singleton, and if the language is finite, that degree must be (see [9]).Automorphically trivial structures include all finite structures, and also somespecial infinite structures, such as the complete graph on countably many vertices. URING DEGREES AND AUTOMORPHISM GROUPS OF SUBSTRUCTURE LATTICES 3
Jockusch and Richter (see [17]) defined the
Turing degree of the isomorphism type of a structure, if it exists, to be the least degree in its Turing degree spectrum.Richter [17, 18] was first to systematically study such degrees. For these andmore recent results about these degrees see [6]. In this paper, we are especiallyinterested in the following result by Morozov.
Theorem 3. ( [12] ) The degree of the isomorphism type of the group Sym d ( ω ) is d ′′ . We extend this and other computability-theoretic results by Morozov to com-putable algebraic structures, in particular, vector spaces and certain Booleanalgebras. Preliminary version of this paper appeared in the CiE conference pro-ceedings [1].In the remainder of this section, we establish some general results about lat-tices of subspaces of a computable vector space and subalgebras of the intervalBoolean algebra of the ordered set of rationals. There are two main results in thepaper. In Section 2, we establish exact correspondence between embeddabilityof automorphism groups of substructures and the order relation of the corre-sponding Turing degrees (see Theorem 9). In Section 3, we compute the Turingdegrees of the isomorphism types of the corresponding automorphism groups (seeTheorem 15).Let V ∞ be a canonical computable ℵ -dimensional vector space over a com-putable field F, which has a computable basis. We can think of a presentation of V ∞ in which the vectors in V ∞ are the (codes of) finitely non-zero ω -sequences ofelements of F . By L we denote the lattice of all subspaces of V ∞ . For a Turingdegree d , by L d ( V ∞ ) we denote the following sublattice of L : L d ( V ∞ ) = { V ∈ L : V is d- computably enumerable } . Note that in the literature L ( V ∞ ) is usually denoted by L ( V ∞ ). About c.e.vector spaces see [11, 4, 2]. Computable vector spaces and their subspaces havebeen also studied in the context of reverse mathematics (see [3]).Guichard [8] established that there are countably many automorphisms of L ( V ∞ ) by showing that every computable automorphism is generated by a1 − V ∞ . Recall that a map µ : V ∞ → V ∞ is called a semilinear transformation of V ∞ if there is an automor-phism σ of F such that µ ( αu + βv ) = σ ( α ) µ ( u ) + σ ( β ) µ ( v )for every u, v ∈ V ∞ and every α, β ∈ F. By GSL d we denote the group of 1 − h µ, σ i such that deg ( µ ) ≤ d and deg ( σ ) ≤ d .We will also consider the structure B η , which is the interval Boolean algebraover the countable dense linear order without endpoints. Let Q be a fixed stan-dard copy of the rationals and let η be its order type. The elements of B η are RUMEN DIMITROV, VALENTINA HARIZANOV, AND ANDREY MOROZOV the finite unions and intersections of left-closed right-open intervals . For a Turingideal I , let L I ( B η ) be the lattice of all subalgebras of B η which are computablyenumerable (abbreviated c.e.) in some d ∈ I . Notation 4.
Note that L D ( B η ) is also denoted by L at ( B η ) , where D is the set ofall Turing degrees. Definition 5.
Note that L d ( B η ) is the lattice of all subalgebras of B η which arec.e. in d .Note that L ( B η ) is also denoted by L ( B η ) . In [8], Guichard proved that every element of
Aut ( L ( V ∞ )) is generated by anelement of GSL . This result can be relativized to an arbitrary Turing degree d . Theorem 6. ( [8] ) Every Φ ∈ Aut ( L d ( V ∞ )) is generated by some h µ, σ i ∈ GSL d . Moreover, if Φ is also generated by some other h µ , σ i ∈ GSL d , then there is γ ∈ F such that ( ∀ v ∈ V ∞ ) [ µ ( v ) = γµ ( v )] . It follows from this result that every Φ ∈ Aut ( L I ( V ∞ )) is generated by some h µ, σ i ∈ GSL I . Every automorphism of L I ( V ∞ ) is defined on the one-dimensional subspacesof V ∞ and can be uniquely extended to an automorphism of the entire lattice L .Hence, we can identify the automorphisms of L I ( V ∞ ) with their extensions toautomorphism of L . We will prove that every automorphism of L is generatedby a d -computable semilinear transformation and its restriction to L d ( V ∞ ) is anautomorphism of L d ( V ∞ ) . Proposition 7. (a)
Aut ( L I ( V ∞ )) = S d ∈ I Aut ( L d ( V ∞ )) (b) Aut ( L ) = S d ∈D Aut ( L d ( V ∞ )) Proof.
Part (b) follows immediately from (a).To prove (a) note first that S d ∈ I Aut ( L d ( V ∞ )) ⊆ Aut ( L I ( V ∞ )) by the discussionabove.Now, suppose Φ ∈ Aut ( L I ( V ∞ )). Let α , α , α , ... be a fixed computable enu-meration of the elements of the field F. Assume that v , v , v , ... is a computableenumeration of a computable basis of V ∞ . Following Guichard’s idea in [8], wedefine the following computable subspaces of V ∞ : V = span ( { v , v , v , ... } ) ,V = span ( { v , v , v , ... } ) ,V = span ( { v + v , v + v , v + v , ... } ) ,V = span ( { v + v , v + v , v + v , ... } ) ,V = span ( { v + α v , v + α v , v + α v , ... } ) . URING DEGREES AND AUTOMORPHISM GROUPS OF SUBSTRUCTURE LATTICES 5
Suppose that Φ( V i ) = W i ∈ L I ( V ∞ ) for i = 1 , . . . , d ∈ I such that W i ∈L d ( V ∞ ). Using Guichard’s method, we can prove that there is a d -computablesemilinear transformation that induces an automorphism Ψ of L d ( V ∞ ) and issuch that Ψ = Φ | L d ( V ∞ ) . Hence Φ as an automorphism of L I ( V ∞ ) is the uniqueextension of Ψ.We can also prove that every automorphism of Aut ( L d ( B η )) is generated by a d -computable automorphism of B η , and that every Φ ∈ Aut ( L I ( B η )) is generatedby a d -computable automorphism of B η for some d ∈ I. Proposition 8.
Aut ( L I ( B η )) ∼ = Aut I ( B η ) Proof.
Suppose ϕ : Aut I ( B η ) → Aut ( L I ( B η )) is an onto homomorphism thattakes every member σ ∈ Aut I ( B η ) to an automorphism ϕ ( σ ) of L I ( B η ) inducedby σ . We will show that Ker ( ϕ ) = id. Assume that σ is a nontrivial element of Ker ( ϕ ). It is easy to show that there exists an a = 0 such that σ ( a ) = a and σ ( a ) ∩ a = 0. Then, considering the image of the Boolean algebra { , a, a, } generated by a, we obtain ϕ ( σ )( { , a, a, } ) = { σ (0 } , σ ( a ) , σ ( a ) , σ (1) } = { , σ ( a ) , σ ( a ) , } 6 = { , a, a, } . Hence ϕ ( σ ) = id . Therefore, Ker ( ϕ ) is trivial and ϕ is an isomorphism.2. Turing reducibility and group embeddings for vector spacesand Boolean algebras
Morozov [13] showed that the correspondence a → Sym a ( ω ) can be used tosubstitute Turing reducibility with group-theoretic embedding. More precisely,he established that Sym a ( ω ) ֒ → Sym b ( ω ) ⇔ a ≤ b for every pair a , b of Turing degrees. It follows from this result that Sym a ( ω ) ∼ = Sym b ( ω ) ⇔ a = b . Here, we establish analogous results for vector spaces and Boolean algebras. Inthe proof of the next, main, theorem we will use the standard notation: [ x, y ] = x − y − xy and x y = y − xy. Theorem 9.
For any pair of Turing ideals
I, J we have:(a)
Aut ( L I ( V ∞ )) ֒ → Aut ( L J ( V ∞ )) ⇔ I ⊆ J (b) Aut I ( B η ) ֒ → Aut J ( B η ) ⇔ I ⊆ J (c) Aut ( L I ( B η )) ֒ → Aut ( L J ( B η )) ⇔ I ⊆ J Proof.
We will prove (a) and (b) only. Statement (c) follows easily from (b)and Proposition 8.(a) Assume I ⊆ J. Then it is straightforward to show that
Aut ( L I ( V ∞ )) ֒ → Aut ( L J ( V ∞ )) . RUMEN DIMITROV, VALENTINA HARIZANOV, AND ANDREY MOROZOV
Now, assume that
Aut ( L I ( V ∞ )) ֒ → Aut ( L J ( V ∞ )) . We will prove that I ⊆ J. Let d ∈ I . As usual, let { e , e , . . . } be a fixed computable basis of V ∞ . For h µ i , σ i i ∈ GSL d , we define h µ , σ i ∼ h µ , σ i iff:(1) σ = σ , and(2) there is α ∈ F such that α = 0 and ( ∀ v ∈ V ∞ ) [ µ ( v ) = αµ ( v )].Note that Aut ( L d ( V ∞ )) ∼ = GSL d / ∼ . We can define a group embedding δ : Sym d ( ω ) ֒ → GSL d / ∼ as follows. For any f ∈ Sym d ( ω ), we let δ ( f ) be the ∼ -equivalence class of a linear transformation h e f , id i such that(1) e f ( e i ) = e f ( i ) . Note that if δ ( f ) = δ ( f ), then e f = c e f for some c ∈ F , and thus( ∀ i ∈ ω ) [ e f ( i ) = e f ( e i ) = c e f ( e i ) = ce f ( i ) ] . Since the vectors e i , i ∈ ω , are independent, we must have( ∀ i ∈ ω ) [ f ( i ) = f ( i )] . Therefore, there exists a map K : Sym d ( ω ) ֒ → GSL J / ∼ such that if f ∈ Sym d ( ω ) , then K ( f ) is a b -computable (for some b ∈ J ) lineartransformation of V ∞ modulo scalar multiplication.We claim that if a set A is c.e. in d , then A is c.e. in some c ∈ J. Fix A ⊆ ω such that A is c.e. in d , and let h : ω → ω be a d -computable enumeration of A , that is, rng ( h ) = A. Fix a partition of the natural numbers into uniformlycomputable infinite sets R i for i ∈ Z with enumerations R i = { c i < c i < · · · } . Let the permutations g , g , w, b ∈ Sym d ( ω ) be defined as follows: w ( c ji ) = c ji +1 for each i ∈ Z and j ∈ ω , g = Q j ∈ ω ( c j , c j +10 ), g = Q j ∈ ω ( c j +10 , c j +20 ), and b = Q n,t ∈ ω ∧ h ( t )= n ( c tn , c t +1 n ).We will also use the following abbreviation: w n = w · · · w | {z } . n times Then we have n / ∈ A ⇔ (cid:0) [ g , b w n ] = 1 ∧ [ g , b w n ] = 1 (cid:1) .This is because g and b w n commute iff n is not enumerated into A at an oddstage t , and, similarly, g and b w n commute iff n is not enumerated into A at aneven stage t. Let e g = K ( g ), e g = K ( g ), e w = K ( w ), and e b = K ( b ) . Each e g , e g , e w and e b is computable in some (possibly different degrees in J ) . Since J is anideal there is a fixed c ∈ J that computes all of them. Then URING DEGREES AND AUTOMORPHISM GROUPS OF SUBSTRUCTURE LATTICES 7 n / ∈ A ⇔ (cid:16) [ K ( g ) , K ( b ) ( K ( w )) n ] = 1 ∧ [ K ( g ) , K ( b ) ( K ( w )) n ] = 1 (cid:17) ⇔ (cid:16) [ e g , e b e w n ] / ∼ = 1 ∧ [ e g , e b e w n ] / ∼ = 1 (cid:17) .We will now show that [ e g , e b e w n ] ≁ c ∈ J . Let τ n = def [ e g , e b e w n ]. Then τ n ≁ ⇔ τ n ( e ) and e are linearly independent, or( ∃ m ∈ ω ) ( ∃ α = 0) [ τ n ( e ) = αe ∧ τ n ( e m ) = αe m ] . Let A have Turing degree d . Then A and A are both c.e. in d , and, therefore, A is computable in c . Hence d ≤ c and so d ∈ J .(b) We will now prove that Aut I ( B η ) ֒ → Aut J ( B η ) ⇔ I ⊆ J. The proof is a corollary of the fact that
Sym I ( ω ) ֒ → Sym J ( ω ) ⇔ I ⊆ J. We first define an embedding H : Sym ( ω ) ֒ → Aut ( B η ) . For any p ∈ Sym ( ω ) , define e p : Q → Q as follows: e p ( x ) = (cid:26) x if x is negative, p ([ x ]) + { x } if x is non-negative , where [ x ], and { x } are the integer and fractional parts of x, respectively . Then for any p ∈ Sym ( ω ), let H ( p ) be an automorphism of B η defined asfollows. If I ∈ B η , then H ( p )( I ) = e p ( I ) . It is important to note that H ( p ) is uniformly computable in p. Now, suppose
Aut I ( B η ) ֒ → Aut J ( B η ). Then Sym I ( ω ) ֒ → H Aut I ( B η ) ֒ → Aut J ( B η ) ֒ → ID Sym J ( ω ) , and so Sym I ( ω ) ֒ → Sym J ( ω ) . Thus, by Morozov’s result, I ⊆ J. Corollary 10.
For any pair of Turing degrees a , b we have Aut ( L a ( V ∞ )) ֒ → Aut ( L b ( V ∞ )) ⇔ a ≤ b ; Aut a ( B η ) ֒ → Aut b ( B η ) ⇔ a ≤ b . RUMEN DIMITROV, VALENTINA HARIZANOV, AND ANDREY MOROZOV Turing degrees of the isomorphism types of automorphismgroups
In this section, we will determine the Turing degree spectra of both
GSL d and Aut d ( B η ). More precisely, we will show that each of them is the upper cone withthe least element d ′′ . For the statement of the main theorem we use terminologyand notation from the following definition.
Definition 11.
A permutation p on a set M will be called: ( i ) 1 inf inf on M if it is a product of infinitely many -cycles and infinitelymany -cycles; ( ii ) 1 inf fin on M if it is a product of infinitely many -cycles and finitely many -cycles. The main results about the degree spectra of
GSL and Aut ( B η ) will use thefollowing embeddability theorem, which is interesting on its own. Theorem 12.
Let G be an X -computable group, and let H : Sym ( ω ) ֒ → G bean embedding (of any complexity). Suppose that for every inf inf permutation p ∈ Sym ( ω ) , the image H ( p ) is not a conjugate of the image of any inf fin permutation in Sym ( ω ) .Then ′′ ≤ deg( X ) . Proof.
Let A be a Π -complete set and let R ( x, t ) be a computable predicatesuch that n ∈ A ⇔ ( ∃ ∞ t ) R ( n, t ) . We will prove that A ≤ T X. Fix a partition of ω into uniformly computableinfinite sets S i,j for i ∈ Z and j ∈ { , } with enumerations S i,j = { c i,j < c i,j < · · · } . The sets S i, and S i, will be referred to as the left and the right parts ofthe i -th column S i = S i, ∪ S i, . This reference will be useful in the definitions ofcertain maps below. We can graphically present this partition as follows: · · · .. c − , c − , c − , .. c − , c − , c − , .. c , c , c , .. c , c , c , .. c , c , c , .. c , c , c , S − , S − , | {z } S , S , | {z } S , S , | {z } column S − column S column S · · · We will now define the following maps.(i) w ( c ki +1 ,j ) = def c ki,j for each i ∈ Z , k ∈ ω and j = 1 , . Clearly, the map w is such that w ( S i +1 , ) = S i, and w ( S i +1 , ) = S i, . It mapsthe left (right) part of the ( i + 1)-st column to the left (right) part of the i -thcolumn for each i . URING DEGREES AND AUTOMORPHISM GROUPS OF SUBSTRUCTURE LATTICES 9 (ii) p = def Q k ∈ ω ( c k , , c k , )It is easy to see that the map p switches the left and right parts of the 0-thcolumn (i.e., p ( S , ) = S , and p ( S , ) = S , ), and is identity on all otherelements of ω. (iii) p n = def p w n = w − n p w n Note that the map p n switches the left and right parts of the n -th column (i.e., p n ( S n, ) = S n, and p n ( S n, ) = S n, ), and is identity on all other elements of ω .(iv) z ( k ) = def k = 0 , k = 2 ,k − k = 2 t ≥ ,k + 2 if k = 2 t + 1 . Note that the map z is a permutation of ω , which contains only one infinitecycle and (0).(v) τ = def (0 , k ∈ Z , we have τ z k = (cid:26) (0 , k ) if k ≥ , (0 , | k | + 1) if k ≤ , so(2) ( ∀ n, m ∈ ω ) ( ∃ n , m ∈ Z ) [ (cid:0) τ z n (cid:1) τ zm = ( n, m )] . Note that property (2) guarantees that any 1 inf fin permutation on ω can berepresented as a finite product of the permutations τ and z. (vi) We will now construct a permutation b on ω with the following properties: b ↾ S n, = id ↾ S n, b ↾ S n, is (cid:26) inf inf on S n, if n ∈ A, inf fin on S n, if n / ∈ A .We will define b in stages. At each stage s we will have E s = def dom ( b s ) = rng ( b s ) . ConstructionStage b ↾ S i = def id for i ≤ − , and E = S i ≤− S i . Stage s + 1 = h n, t i .Case 1. If R ( n, t ), then find the least elements p, q, r ∈ S n, such that p, q, r / ∈ E s . Let b s +1 = b s · ( p, q ) and assume that b s +1 ( r ) = r. Thus, we have E s +1 = E s ∪ { p, q, r } and b s +1 ↾ E s = b s .Case 2. If ¬ R ( n, t ), then find the least elements p, q, r ∈ S n, such that p, q, r / ∈ E s . Let b s +1 ↾ E s = b s and b s +1 ( p ) = p, b s +1 ( q ) = q, b s +1 ( r ) = r. Then E s +1 = E s ∪ { p, q, r } . End of construction.
By construction, dom ( b ) = rng ( b ) = ω. It follows that if n ∈ A, then ( ∃ ∞ t ) R ( n, t ) , so Case 1 applies infinitely oftenfor this n , and hence the map b switches infinitely many pairs in the right partof the n -th column. Therefore, b ↾ S n, is 1 inf inf and b ↾ S n, = id .If n / ∈ A, then ( ∃ < ∞ t ) R ( n, t ), so Case 1 applies finitely often for this n , andhence the map b switches only finitely many pairs in the right part of the n -thcolumn. Therefore, b ↾ S n, is 1 inf fin and b ↾ S n, = id .In both cases, the map b p n reverses the action of b on the left part and theright part of the n -th column S n , while for k = n , we have b p n ↾ S k = b ↾ S k .Then b · b p n is inf inf on S n if n ∈ A, inf fin on S n if n / ∈ A,id on S k if n = k. Therefore, b · b p n is (cid:26) inf inf on ω if n ∈ A, inf fin on ω if n / ∈ A. Finally, note that on ω, every computable 1 inf inf permutation is the conjugateof a fixed computable 1 inf inf permutation and some other computable permuta-tion. Therefore, assume that f is a fixed computable 1 inf inf permutation suchthat for every 1 inf inf permutation q ∈ Sym ( ω ):( ∃ h ∈ Sym ( ω )) [ q = f h ] . Hence for every n , we have(3) n ∈ A ⇔ b · b p n is a 1 inf inf permutation on ω ⇔ ( ∃ h ∈ Sym ( ω )) [ b · b p n = f h ] ⇔ ( ∃ u ∈ H ( Sym ( ω ))) [ H ( b ) · H ( b ) H ( p n ) = H ( f ) u ] , and,(4) n / ∈ A ⇔ b · b p n is a 1 inf fin permutation on ω ⇔ b · b p n = Q ( i,j ) ∈ F (cid:16) τ z i (cid:17) τ zj ⇔ H ( b ) · H ( b ) H ( p n ) = Q ( i,j ) ∈ F (cid:16) H ( τ ) H ( z ) i (cid:17) H ( τ ) H ( z ) j . The set F in the last equality in (4) denotes some finite set of pairwise disjointcycles. For the map H : Sym ( ω ) ֒ → G, note that H ( p n ) = H ( w ) − n · H ( p ) · H ( w ) n . We claim that the last equivalence in (3) can be strengthened so that we have:(5) n ∈ A ⇔ ( ∃ u ∈ G ) [ H ( b ) · H ( b ) H ( p n ) = H ( f ) u ] . To prove ( ⇒ ) in 5), assume that n ∈ A. Then( ∃ u ∈ H ( Sym ( ω ))) [ H ( b ) · H ( b ) H ( p n ) = H ( f ) u ] , hence( ∃ u ∈ G ) [ H ( b ) · H ( b ) H ( p n ) = H ( f ) u ] . URING DEGREES AND AUTOMORPHISM GROUPS OF SUBSTRUCTURE LATTICES 11
We will prove ( ⇐ =) by contradiction. Assume that for some fixed u ∈ G we have H ( b ) · H ( b ) H ( p n ) = H ( f ) u , but n / ∈ A. Then, by (4), we have the following:(i) b · b p n is a 1 inf fin permutation on ω, (ii) H ( b ) · H ( b ) H ( p n ) is the image of the 1 inf fin permutation b · b p n , while(iii) H ( f ) is the image of the 1 inf inf permutation f. This contradicts our assumption that the image under H of the 1 inf fin permu-tation b · b p n cannot be the conjugate of the image of any 1 inf inf permutation,including f. We will now show that A is computable in the group G , and hence A ≤ T X. For a given n ∈ ω, simultaneosly search for a finite set F of pairwise disjointcycles and an u ∈ G such that either the last equality in (4) holds: H ( b ) · H ( b ) H ( p n ) = Y ( i,j ) ∈ F (cid:16) H ( τ ) H ( z ) i (cid:17) H ( τ ) H ( z ) j , or the following equality from (5) holds: H ( b ) · H ( b ) H ( p n ) = H ( f ) u . If the former search succeeds, then n / ∈ A, while if the latter search succeeds,then n ∈ A. We will now prove our main results about the degree spectra of automorphismgroups.
Theorem 13.
The degree of the isomorphisms type of the group
GSL is ′′ . Proof.
Let V = { v , v , . . . } be a computable basis of V ∞ . Define H : Sym ( ω ) ֒ → GSL so that for any p ∈ Sym ( ω ) the image H ( p ) = h L, id i is a semilinear map suchthat L ( v i ) = v p ( i ) for every i ∈ ω. We claim that under H , the image of a 1 inf inf permutation from Sym ( ω ) cannotbe a conjugate of the image of a 1 inf fin permutation from Sym ( ω ). To establishthis fact, suppose that h f, id i , h f , id i ∈ GSL are the images of some 1 inf inf and1 inf fin computable permutations on ω , respectively. Suppose that h f, id i and h f , id i are conjugates, and let h h, σ i ∈ GSL be such that h f, id i h h,σ i = h f , id i . Note that the map h : V ∞ → V ∞ is 1 − V ∞ . To simplify the notation,we will refer to the semilinear maps h f, id i , h f , id i , and h h, σ i simply as f, f ,and h, respectively. We can view f ↾ V and f ↾ V as 1 inf inf and 1 inf fin permutations on V, respectively. We will prove that f satisfies the property:(6) ( ∃ W ⊂ fin V ∞ ) ( ∀ v ∈ V ∞ ) [( v − f ( v )) ∈ W ] . where, W ⊂ fin V ∞ stands for W being a finite-dimensional subspace of V ∞ . Toprove (6), assume that B = { x , . . . , x k , y , . . . , y k } ⊆ V is such that f ↾ V = Q i ≤ k ( x i , y i ) . Note that for every v ∈ V ∞ , there are v ∈ span ( V − B ) and v ∈ span ( B ) such that v = v + v . Then f ( v ) = f ( v ) + f ( v ) = v + f ( v ) , and so v − f ( v ) = v + v − v − f ( v ) = v − f ( v ) ∈ span ( B )since f ( v ) ∈ span ( B ) . Therefore, W = span ( B ) is a finite-dimensional subspaceof V ∞ for which property ( D ) holds.We will now prove that f h does not satisfy property (6), which will contradictthe assumption that f h = f . Thus, assume that W is a finite-dimensionalsubspace of V ∞ such that(7) ( ∀ x ∈ V ∞ ) [ (cid:0) x − f h ( x ) (cid:1) ∈ W ] . Let W = h ( W ) and note that W is finite-dimensional . Let B be a finite subsetof the basis V such that ( ∀ x ∈ W ) [ supp V ( x ) ⊆ B ] , where supp V ( x ) denotes the support of x with respect to the basis V. We will now find u ∈ V ∞ such that u − f ( u ) / ∈ W . Since f ↾ V is a 1 inf inf permutation on V, there are infinitely many pairs ( u, v ) ∈ V × V such that(8) u = v, f ( u ) = v and f ( v ) = u. Since B is finite, we can also find u , v ∈ V − B , which have property (8).Then:(i) u − f ( u ) = u − v = 0 , and(ii) u − f ( u ) = ( u − v ) / ∈ span ( B ) because B ∪ { u , v } ⊆ V .Since W ⊆ span ( B ), we have u − f ( u ) / ∈ W . Therefore, ( h − ( u ) − h − ( f ( u ))) / ∈ h − ( W ),and so ( h − ( u ) − h − f hh − ( u )) / ∈ W. If we let x = h − ( u ), we obtain x − f h ( x ) / ∈ W, which contradicts (7).Thus, we constructed an embedding H : Sym ( ω ) ֒ → GSL such that the im-ages of any 1 inf inf and 1 inf fin permutations from Sym ( ω ) cannot be conjugatesin GSL . By Theorem 12, we conclude that ∅ ′′ is computable in any copy of URING DEGREES AND AUTOMORPHISM GROUPS OF SUBSTRUCTURE LATTICES 13
GSL . Clearly, we can construct a specific copy of
GSL , which is computablein ∅ ′′ . Therefore, the degree of the isomorphisms type of
GSL is ′′ . We will now establish a similar result for the Boolean algebra B η . Recall thatwe identify the elements of B η with finite unions and intersections of left-closedright-open intervals of the linear order of the set Q of rationals. Theorem 14.
The degree of the isomorphisms type of the group
Aut ( B η ) is ′′ . Proof.
The map H : Sym ( ω ) ֒ → Aut ( B η ) was defined in the proof of part(b) of Theorem 9 in section 1. Since H ( p ) is uniformly computable in p , where p ∈ Sym ( ω ) , H takes elements of Sym ( ω ) to elements of Aut ( B η ) . In theremainder of the proof we will be concerned with the restriction of H to Sym ( ω ) . We claim that under H , the image of a 1 inf inf permutation from Sym ( ω ) cannotbe a conjugate of the image of a 1 inf fin permutation from Sym ( ω ).Let Φ( x, ϕ ) denote the statement( ∀ y ≤ x ) [ y = 0 ⇒ ( ∃ z ≤ y ) ( ϕ ( z ) = z )] , and let Ψ( ϕ ) denote the statement( ∃ z ) [ z = sup { x : Φ( x, ϕ ) } ] . The statement Φ is in the language of Boolean algebras expanded with a functionsymbol ϕ with an intended interpretation being an element of Aut ( B η ) . Notethat sup can be defined in the language of Boolean algebras.Recall that in the proof of part (b) of Theorem 9, for p ∈ Sym ( ω ) , we defined e p : Q → Q as follows: e p ( x ) = (cid:26) x if x is negative, p ([ x ]) + { x } if x is non-negative . If ϕ = H ( p ) , where p is a 1 inf fin permutation on ω , then the set A = { x ∈ Q : e p ( x ) = x } is the union of finitely many intervals of the type [ n, n + 1) for anatural number n . If u is the join of these intervals in B η , then u is the largestelement that satisfies Φ( u, ϕ ). In this case Ψ( ϕ ).If ϕ = H ( p ) , where p is a 1 inf inf permutation on ω , then the set A = { x ∈ Q : e p ( x ) = x } is the union of intervals of the type [ n, n + 1) for a natural number n .In this case A is also the union of infinitely such intervals. The formula Φ( u, ϕ )is satisfied exactly by those u that are the unions of finitely many intervals thatconsist entirely of elements of A . Clearly, { u : Φ( u, ϕ ) } has no supremum in thiscase and so q Ψ( ϕ ) . Thus, we have established that:(1) Ψ( ϕ ) holds when ϕ is the image, under H, of an 1 inf fin permutation,(2) q Ψ( ϕ ) holds when ϕ is the image, under H, of an 1 inf inf permutation.We will now prove that for any formula Θ in the language of Boolean algebras,expanded with function symbols for the elements of Aut ( B η ) and quantifiers ranging over the elements of a Boolean algebra, we have the following for everyfunction f ∈ Aut ( B η ): Θ( −→ x , −→ θ ) ⇔ Θ( f ( −→ x ) , −→ θ f ) , where −→ θ = θ , . . . , θ n and −→ θ f = θ f , . . . , θ fn are new function symbols (with theusual intended interpretation). In our notation Θ( −→ x , −→ θ ), −→ x are the free variablesand −→ θ are the new function symbols used in Θ.We will proceed by induction on the complexity of the formula Φ.If Θ( −→ x , −→ θ ) is quantifier-free, we can consider the following atomic formulas: x = y, x = 0 , x = 1 , θ ( x ) = y, x ∧ y = z, x ∨ y = z, x = y . We will presentonly the case when Θ is the formula θ ( x ) = y. Let f ∈ Aut ( B η ) . Then we have: θ ( x ) = y ⇔ f θ ( x ) = f ( y ) ⇔ f θf − f ( x ) = f ( y ) ⇔ θ f ( f ( x )) = f ( y ) . Let Θ be the formula ∃ z Θ ( z, −→ x , −→ θ ) and let f ∈ Aut ( B η ) . We have that ∃ z Θ ( z, −→ x , −→ θ ) ⇔ Θ ( c, −→ x , −→ θ ) for some c ∈ B η . Then, using theinductive hypothesis,Θ ( c, −→ x , −→ θ ) ⇔ Θ ( f ( c ) , f ( −→ x ) , −→ θ f ) ⇔ ∃ z Θ ( z, f ( −→ x ) , −→ θ f ) . This completes the induction.By the previous considerations, we have( ∀ f ∈ Aut ( B η )) [Ψ( ϕ ) ⇔ Ψ( f − ϕf )] . It follows from (1) and (2) above that, under H, the image of any 1 inf inf per-mutation from Sym ( ω ) cannot be the conjugate of the image of any 1 inf fin permutation from Sym ( ω ) . By Theorem 12, we conclude that ∅ ′′ is computable in any copy of Aut ( B η ) . Clearly, we can construct a specific copy of
Aut ( B η ), which is computable in ∅ ′′ . Therefore, the degree of the isomorphisms type of
Aut ( B η ) is ′′ . Note that the results of the previous theorems in this section can be easilyrelativized to any Turing degree d . Theorem 15.
The degree of the isomorphisms type of each of the groups
GSL d and Aut d ( B η ) is d ′′ . References [1] R. Dimitrov, V.Harizanov, and A. Morozov, Automorphism groups of substructure latticesof vector spaces in computable algebra, in: 12 th Conference on Computability in Europe , Lecture Notes in Computer Science
Annals of Pure and Applied Logic
URING DEGREES AND AUTOMORPHISM GROUPS OF SUBSTRUCTURE LATTICES 15 [3] R. Downey, D.R. Hirschfeldt, A.M. Kach, S. Lempp, J.R. Mileti, and A. Montalb´an,
Journal of Algebra
314 (2007), pp. 888–894.[4] R.G. Downey and J.B. Remmel, Computable algebras and closure systems: coding prop-erties, in: Yu.L. Ershov, S.S. Goncharov, A. Nerode, and J.B. Remmel, eds.,
Handbook ofRecursive Mathematics , vol. 2, Studies in Logic and the Foundations of Mathematics 139,pp. 997–1039. North-Holland, Amsterdam, (1998).[5] Yu.L. Ershov and S.S. Goncharov,
Constructive Models , Siberian School of Algebra andLogic, (English translation). Kluwer Academic/Plenum Publishers, (2000).[6] E. Fokina, V. Harizanov, and A. Melnikov, Computable model theory,
Turing’s Legacy:Developments from Turing Ideas in Logic , R. Downey, editor, pp. 124–194. CambridgeUniversity Press/ASL (2014).[7] S. Goncharov, V. Harizanov, J. Knight, A. Morozov, and A. Romina, On automorphictuples of elements in computable models,
Siberian Mathematical Journal
46, (Englishtranslation), 405–412 (2005).[8] D.R. Guichard, Automorphisms of substructure lattices in recursive algebra,
Annals ofPure and Applied Logic
25, 47–58 (1983).[9] V. Harizanov and R. Miller, Spectra of structures and relations,
Journal of Symbolic Logic
72, 324–348 (2007).[10] J.F. Knight, Degrees coded in jumps of orderings,
Journal of Symbolic Logic
51, 1034–1042(1986).[11] G. Metakides and A. Nerode, Recursively enumerable vector spaces , Annals of Pure andApplied Logic
11, 147–171 (1977).[12] A.S. Morozov, Permutations and implicit definability,
Algebra and Logic
27 (English trans-lation), 12–24 (1988).[13] A.S. Morozov, Turing reducibility as algebraic embeddability,
Siberian Mathematical Jour-nal
38 (English translation), 312–313 (1997).[14] A.S. Morozov, Groups of computable automorphisms, in: Yu.L. Ershov, S.S. Goncharov,A. Nerode, and J.B. Remmel, eds.,
Handbook of Recursive Mathematics , vol. 1, Studies inLogic and the Foundations of Mathematics 139, pp. 311–345. North-Holland, Amsterdam,(1998).[15] A.S. Morozov, On theories of classes of groups of recursive permutations,
Trudy InstitutaMatematiki (Novosibirsk) 12 (1989), Mat. Logika i Algoritm. Probl., pp. 91–104 (Russian).(English translation in:
Siberian Advances in Mathematics
1, 138–153 (1991).[16] A.S. Morozov, Computable groups of automorphisms of models,
Algebra and Logic
Degrees of Unsolvability of Models , Ph.D. dissertation, University of Illinoisat Urbana-Champaign, (1977).[18] L.J. Richter, Degrees of structures,
Journal of Symbolic Logic
46, 723–731 (1981).[19] H. Rogers, Jr.,
Theory of Recursive Functions and Effective Computability . McGraw-Hill,New York, (1967).[20] R.I. Soare,
Recursively Enumerable Sets and Degrees.
Springer-Verlag, Berlin, (1987).
Department of Mathematics, Western Illinois University, Macomb, IL 61455,USA
E-mail address : [email protected] Department of Mathematics, George Washington University, Washington, DC20052, USA
E-mail address : [email protected] Sobolev Institute of Mathematics, Novosibirsk, 630090, Russia
E-mail address ::