On bi-embeddable categoricity of algebraic structures
aa r X i v : . [ m a t h . L O ] M a y ON BI-EMBEDDABLE CATEGORICITY OF ALGEBRAICSTRUCTURES
NIKOLAY BAZHENOV, DINO ROSSEGGER, AND MAXIM ZUBKOV
Abstract.
In several classes of countable structures it is known that every hy-perarithmetic structure has a computable presentation up to bi-embeddability.In this article we investigate the complexity of embeddings between bi-embed-dable structures in two such classes, the classes of linear orders and Booleanalgebras. We show that if L is a computable linear order of Hausdorff rank n , then for every bi-embeddable copy of it there is an embedding computablein 2 n − L be a computable linear order of Hausdorff rank n ≥
1, then (2 n − does not compute embeddings between it and all its com-putable bi-embeddable copies. We obtain that for Boolean algebras which arenot superatomic, there is no hyperarithmetic degree computing embeddingsbetween all its computable bi-embeddable copies. On the other hand, if acomputable Boolean algebra is superatomic, then there is a least computableordinal α such that ( α ) computes embeddings between all its computable bi-embeddable copies. The main technique used in this proof is a new variationof Ash and Knight’s pairs of structures theorem. Introduction
It is well-known that arbitrary isomorphic algebraic structures A and B possessthe same algebraic properties. In contrast to this fact, the algorithmic properties of A and B can be strikingly different. One of the first examples of this phenomenonwas witnessed by Mal’tsev [Mal62]: While the standard recursive copy A of Q ω (i.e.the divisible torsion-free abelian group of countably infinite rank) has an algorithmfor linear dependence, Mal’tsev built a copy B of Q ω with no such algorithm.These kinds of algorithmic discrepancies motivated a plethora of questions, shap-ing modern computable structure theory. One of these questions can be (informally)stated as follows: What is the simplest possible presentation of a given structure A ?Or more concretely, given some countable structure A , which is “hard to compute”,is it possible to find a copy B ∼ = A such that B is “computationally more simple”than A itself?Quite remarkably, one of the pioneering results of computable structure the-ory deals with the questions above: In 1955, Spector [Spe55] showed that everyhyperarithmetic well-order is isomorphic to a computable one.In turn, this classic result raises a new question — How hard is it to computeisomorphisms between well-orders? Ash [Ash86] answered this question by showingthat if α is such that ω δ + n ≤ α < ω δ + n +1 , where δ is either 0 or a limit ordinal Mathematics Subject Classification.
Key words and phrases. computable structures, linear ordering, boolean algebra, computablecategoricity,bi-embeddability.Bazhenov and Zubkov are supported by the RSF grant (project No. 18-11-00028). and n ∈ ω , then for any two orderings A and B of order type α , there is a ∆ A ⊕ B δ +2 n computable isomorphism, but there are orderings ˆ A and ˆ B of this order type suchthat for β < δ + 2 n , ∆ A ⊕ B β does not compute any isomorphisms. Recently, Alvirand Rossegger [AR] generalized this result to arbitrary scattered linear orderings bygiving precise bounds on the complexity of Scott sentences in this class. Note thatin their result one loses effectiveness, however, obtaining bounds on the complexityof the isomorphisms in the Borel hierarchy.By using the fact that any countable superatomic Boolean algebra is isomorphicto the interval algebra of an ordinal Int ( α ), one can obtain similar results to thoseof Ash for superatomic Boolean algebras [Ash87, AK00]. In general, there is alarge body of literature dealing with the algorithmic complexity of isomorphisms:We refer the reader to the surveys [Ash98, FHM14] and the monograph [AK00] forthe results on Turing degrees of isomorphisms. For the subrecursive complexity ofisomorphisms, the reader is referred to the surveys [CR98, Mel17, BDKM19].One weakening of the notion of isomorphism is bi-embeddability or, as it is some-times called in the literature, equimorphism : Definition 1.
Two structures are bi-embeddable if either is isomorphic to a sub-structure of the other.
Given two structures A and B we write A ֒ → B to denote that A is embeddablein B , i.e., that there is a 1-1 structure preserving map from A to B . We also write f : A ֒ → B to say that f : A → B is an embedding of A in B . If A ֒ → B and B ֒ → A , i.e., if A and B are bi-embeddable, then we write A ≈ B .Montalb´an [Mon05] was able to obtain an analogue to Spector’s result for alllinear orders with respect to bi-embeddability. Theorem 1 (Montalb´an) . Every hyperarithmetic linear order is bi-embeddable witha computable one.
Greenberg and Montalb´an subsequently obtained the same result for Booleanalgebras, abelian groups, and compact metric spaces [GM08]. For the class ofequivalence structures one can even obtain that every equivalence structure is bi-embeddable with a computable one [FRSM19]. These remarkable results raisethe question of how hard it is to compute embeddings between bi-embeddablestructures.In this article we obtain partial answers to these questions by calculating thecomplexity of embeddings between bi-embeddable countable Boolean algebras andlinear orders of finite Hausdorff rank.To do this we study the notions of relative bi-embeddable categoricity and degreeof bi-embeddable categoricity. These notions are analogues of well studied notionsfor isomorphism, see [AKMS89, Chi90, FKM10, CFS13, FHM14].
Definition 2.
A countable (not necessarily computable) structure A is relatively∆ α bi-embeddably categorical if for any bi-embeddable copy B , A and B are bi-embeddable by ∆ A ⊕ B α embeddings. A computable structure is relatively computablybi-embeddably categorical if α = 1 . Definition 3.
Let A be a computable structure. If d is the least Turing degreecomputing embeddings between any two computable structures bi-embeddable with A , then d is the degree of bi-embeddable categoricity of A . N BI-EMBEDDABLE CATEGORICITY OF ALGEBRAIC STRUCTURES 3
Computable bi-embeddable categoricity and degrees of bi-embeddable categoric-ity were studied in a more general context in [BFRS]. Notice that a structure maynot have a degree of bi-embeddable categoricity. As we will discuss later in this ar-ticle, even natural examples such as the order type of the rational numbers, η , andthe atomless Boolean algebra do not have a degree of bi-embeddable categoricity. Inrelated work, Bazhenov, Fokina, Rossegger, and San Mauro [BFRS19] studied thecomplexity of embeddings between equivalence structures and showed that everycomputable equivalence relation has either degree of bi-embeddable categoricity , ′ or ′′ .Our main results show that linear orders of finite Hausdorff rank n are rela-tively ∆ n +2 bi-embeddably categorical, but not relatively ∆ n +1 , and that allcomputable superatomic Boolean algebras have a degree of bi-embeddable cate-goricity depending on their Frech´et rank. We present the necessary preliminariesand our results about linear orders in Section 2. In Section 3 we present our resultsabout Boolean algebras. We give a short summary about α -systems and then provea variation of Ash and Knight’s [AK90] pairs of structure theorem which we willuse to calculate the degrees of bi-embeddable categoricity for superatomic Booleanalgebras. 2. Linear orders
Preliminaries.
A linear order L is given by a pair ( L, ≤ L ), where L is a setand ≤ L is a binary relation on L satisfying the usual axioms of linear orders. If L is infinite, then we assume that L = ω . Given L , we let < L be the induced strictordering, i.e. for all x, y ∈ L , we have x < L y if and only if x ≤ L y and y L x .We will also use interval notation: [ x, y ] L = { z : x ≤ L z ≤ L y } ; and we make useof the following additional relations on linear orders. • The successor relation S L given by S L ( x, y ) ⇔ x < L y ∧ ∀ z ( z ≤ L x ∨ y ≤ L z ) , • and the block relation F L ( x, y ) defined by F L ( x, y ) ⇔ ( [ x, y ] L is finite if x ≤ L y ;[ y, x ] L is finite if y ≤ L x. We will drop subscripts if the order is clear from context.It is not hard to see that the block relation is an equivalence relation on L whichagrees with ≤ L , and that it is definable by a computable Σ formula in L ω ,ω . Wecall its equivalence classes blocks and denote the block of x ∈ L as [ x ] L . As theblock relation agrees with the ordering, we can take the quotient structure andobtain the factor ordering or condensation . It is denoted by L /F L and defined asusual by [ x ] L ≤ L /F L [ y ] L ⇔ x ≤ L y .Taking condensations can be iterated finitely often in an obvious way — byfactoring through the block relation of the previous condensation. In order toobtain a notion of iterated condensation for all ordinals, we define the α -blockrelation for all ordinals α . Definition 4.
Given a linear order L , the α -block relation F α L on L is defined byinduction as follows. Let [ x ] α L denote the F α L -equivalence class of x ∈ L . Then for x, y ∈ L , (1) F L ( x, y ) ⇔ x = y , NIKOLAY BAZHENOV, DINO ROSSEGGER, AND MAXIM ZUBKOV (2) if α = β + 1 , then F α L ( x, y ) ⇔ F L /F β L ([ x ] β L , [ y ] β L ) , and (3) if α is a limit ordinal, then F α L ( x, y ) ⇔ ( ∃ β < α ) F L /F β L ( x, y ) . It is not hard to see that for a finite α , L /F α L agrees with taking condensationsiteratively α times, and that F α L is Σ α definable. To simplify notation, we set L ( α ) F = L /F α L . Definition 5.
The
Hausdorff rank of a linear order L is the least α such that L ( α ) F = L ( α +1) F = 1 . As usual, we will identify with ω the order type of the natural numbers, with ζ the order type of the integers, and with η the order type of the rationals. The uniquefinite order type with precisely n elements is denoted by n . We let stand for theempty ordering. Further, if L is a linear order, then L ∗ is its reverse ordering, i.e., x ≤ L ∗ y ⇔ y ≤ L x . Definition 6.
A linear order is scattered if it has no subordering of order type η . Definition 7.
The class VD of linear orderings is defined by (1) VD = { , } , (2) VD α = (cid:26) P i ∈ τ L i : L i ∈ S β<α VD β , τ ∈ { ω, ω ∗ , ζ } (cid:27) , and (3) VD = S α VD α .The V D -rank of a linear order L is the least α such that L ∈ VD α ; and the V D ∗ -rank of L is the least α such that L is a finite sum of linear orders in VD α . The following theorem due to Hausdorff is well known.
Theorem 2.
A countable linear order is scattered if and only if it has countableVD-rank. Furthermore, the
V D -rank of a scattered linear order is equal to itsHausdorff rank.
A linear order L is indecomposable if whenever L = A + B , we have that L can be embedded in either A or B . It is not hard to prove that a linear orderbi-embeddable with an indecomposable one is also indecomposable.A signed tree is a pair h T, s T i , where T is a well-founded subtree of ω <ω (i.e. adownwards closed subset of ω <ω with no infinite paths), and s T is a map, called sign function , from T to { + , −} . We will usually write T instead of h T, s T i . Givena signed tree T , let T σ be the subtree of T with root σ .We associate with every signed tree T a linear order lin ( T ) whose order type isdefined by recursion as follows. If T = {∅} and s T ( ∅ ) = +, then lin ( T ) = ω . If T = {∅} and s T ( ∅ ) = − , then lin ( T ) = ω ∗ . Now, assume that lin ( T σ ) has beendefined for σ ∈ T \ {∅} . If s T ( ∅ ) = +, then lin ( T ) ∼ = X i ∈ ω X j ≤ i lin ( T h j i ) . If s T ( ∅ ) = − , then lin ( T ) ∼ = X i ∈ ω ∗ X j ≤ i lin ( T h j i ) . Linear orders of order type lin ( T ) for some signed tree T are called h-indecom-posable linear orders . Montalb´an [Mon06] showed that every indecomposable linear N BI-EMBEDDABLE CATEGORICITY OF ALGEBRAIC STRUCTURES 5 order is bi-embeddable with an h-indecomposable linear order. Furthermore, for anindecomposable linear order L of finite Hausdorff rank, the rank of the signed treeassociated with its bi-embeddable h-indecomposable order is equal to the Hausdorffrank of L .2.2. Upper bounds.
In this section we give an upper bound on the complexityof embeddings between two bi-embeddable linear orders of finite Hausdorff rank.Notice that there is only one non-scattered order type up to bi-embeddability —the ordering of the rational numbers, η . It is not relatively ∆ α bi-embeddablycategorical for any computable ordinal α . To see this, fix a standard copy of η (inwhich one can compute an infinite decreasing sequence) and a copy of the Harrisonordering H = ω CK1 (1 + η ) without hyperarithmetic infinite decreasing sequences.Any embedding of η into H will compute an infinite decreasing sequence, so therecan be no hyperarithmetic embedding.Results of Montalb´an show that any hyperarithmetic linear order, and, moreover,any linear order of computable Hausdorff rank is bi-embeddable with a computableone. This section and the next one provide sharp bounds on the complexity ofembeddings between linear orders of finite Hausdorff rank. Lemma 1.
Fix n ∈ ω . There are Turing operators Φ and Ψ such that if A and B are of VD-rank n , a, b ∈ ω , ◦ ∈ { <, >, ≤ , ≥ , ( , [ } and m ≤ n (1) Φ ( A ⊕ B ) (2 m +1) ( a, b, m, ◦ ) = ( if [ ◦ a ] m ֒ → [ ◦ b ] m otherwise , (2) Ψ ( A ⊕ B ) (2 m ) ( a, b, m, ◦ , − ) = ( [ ◦ a ] m ֒ → [ ◦ b ] m if Φ ( A ⊕ B ) (2 m +1) ( a, b, m, ◦ ) = 1 ↑ otherwisewhere [ ◦ x ] m is the subordering on { y : y ◦ x & y ∈ [ x ] m } if ◦ ∈ { <, >, ≤ , ≥} and [( x ] m is the subordering on { y : y ∈ ( x , x ] } if x = h x , x i with x ∈ [ x ] m and [ > x ] if x [ x ] m . Similarly, [[ x ] m is the subordering on { y : y ∈ [ x , x ] } if x = h x , x i with x ∈ [ x ] m and [ > x ] if x [ x ] m .Proof. The proof is by induction on m . Fix a, b ∈ ω and, as for m = 0 the lemmatrivializes, assume that m = 1. Consider [ ◦ a ] and [ ◦ b ] . The two orders can be ofthe following types: finite, ω , ω ∗ or ζ . It is easy to see that they are bi-embeddableif and only if they are isomorphic. We can determine whether the two orderingsare isomorphic and subsequently define an embedding between them by checkingwhether they have a first and, or, last element, calculating their size in case theyare finite, and calculating their successor relation. All of this can be done withinthree jumps over the diagrams, for example the following formula is satisfied by A if [ > a ] has a greatest element:(1) ∃ y ∀ x ( x ∈ [ a ] → x ≤ y )As the relation x ∈ [ a ] is Σ , the formula is Π and thus, using ( A ⊕ B ) (3) as anoracle we can decide whether it is true or not. It is straightforward to define Φ byevaluating the formulas corresponding to the statements mentioned above so thatΦ ( A ⊕ B ) (3) ( a, b, , ◦ ) = 1 iff [ ◦ a ] ֒ → [ ◦ b ] .Defining an embedding is even easier and only takes two jumps over the diagramsin case that [ ◦ a ] ֒ → [ ◦ b ] . Assume ◦ equals > , the other cases are symmetric.As [ ◦ a ] must have a least element, the unique element c satisfying the following NIKOLAY BAZHENOV, DINO ROSSEGGER, AND MAXIM ZUBKOV formula: c ∈ [ a ] ∧ c > a ∧ ∀ y (( y ∈ [ a ] ∧ y > a ) → y ≥ c )This formula is a conjunction of a computable Σ formula and a computable Π formula and therefore within two jumps over A we can find the least element.Likewise, we can find the least element d in [ ◦ b ] . Now let s A , s B be the successorfunction on A and B respectively, then we define Ψ byΨ ( A ⊕ B ) ( a, b, , >, x ) = ( y, ∃ ks k A ( c ) = x ∧ s k B ( d ) = y ↑ , otherwise . It is not hard to see that Ψ defined like this is a well-defined computable operator.Assume that the theorem holds for all k < n . We can establish the order typeof [ ◦ a ] n /F n − and [ ◦ b ] n /F n − in similar fashion as in the base case by checkingwhether there are least and, or, greatest elements and the size of the orders in casethey are finite. For instance to check whether [ > a ] n has a greatest element weonly need to replace x ∈ [ a ] by x ∈ [ a ] n in Eq. (1). The resulting formula is Σ n +1 and thus in 2 n + 1 jumps over the diagrams we can evaluate it and the formulasdefining the other properties.This is however not enough to say that [ ◦ a ] n ֒ → [ ◦ b ] n . We will describe in detailthe case when ◦ is ≥ and [ ≥ a ] n /F n − ∼ = [ ≥ b ] n /F n − ∼ = ω . The other cases aresymmetric. We can pick the least natural number from each of the ( n − ≥ a ] n , respectively [ ≥ b ] n and write them in order, i.e., a a a a a a a a a a . . .b b b b b b b b b b . . . We have to find embeddings of the ( n − a i into [ ≥ b ] n preserving order.We can do this inductively as follows:We have that [ ≥ a ] n ֒ → [ ≥ b ] n only if there are m , a < · · · < a k = a < · · · < a m such that a i ∈ [ a ] n for all i , and b < · · · < b m ∈ [ b j ] n − for the least j such that[ ≤ a ] n − ֒ → [ ≤ b ] n − ∧ ( a , a ] ֒ → ( b , b ] ∧ · · · ∧ [ > a m ] ֒ → [ > b m ] n − , where all the half-open intervals are contained in an ( n − n − n − n − n − A ⊕ B ) (2( n − . If the statement is true, then we proceed with a in place of a andby restricting our search for elements b i to elements in blocks greater than [ b j ] n − .If the statement is false, then we stop and conclude that [ ≥ a ] n is not embeddableinto [ ≥ b ] n .Now it is easy to see that [ ≥ a ] n ֒ → [ ≥ b ] n if and only if the above statement holdsfor all a i , and to verify this uniformly we need another jump, i.e., ( A ⊕ B ) (2 n +1) .Using this as an oracle we can define Φ as required.We can define Ψ by induction in the same manner as we verified whether [ ≥ a ] n ֒ → [ ≥ b ] n . First obtain the elements a , . . . and b , . . . as above. Then given x determine the a i such that x ∈ [ a i ] n − . First find the least j satisfying the conditiondescribed above for a , then for a , and so on to obtain the interval of [ ≥ b ] n inwhich x embeds if such an interval exists. If it exists then use the hypothesis todefine the embedding, otherwise stay undefined. N BI-EMBEDDABLE CATEGORICITY OF ALGEBRAIC STRUCTURES 7
Given the arguments above this can be done from ( A ⊕ B ) (2( n − = ( A ⊕ B ) (2 n ) . (cid:3) Theorem 3.
Suppose that L is a scattered linear order of finite V D ∗ -rank n . Thenit is relatively ∆ n bi-embeddably categorical.Proof. First consider the case where L has V D -rank n . Every bi-embeddable copyof it must also have V D -rank n . Let A , B be bi-embeddable with L and fix a ∈ A . Then there is an embedding f : A ֒ → B sending a to f ( a ). In particular,[ < a ] n ֒ → [ < f ( a )] n and [ ≥ a ] n ֒ → [ ≥ f ( a )] n . We show how to construct a ∆ A ⊕ B n embedding of [ ≥ a ] n ֒ → [ ≥ f ( a )] n in case that [ ≥ a ] n /F n − ∼ = ω . It is easy toadapt the construction for the case when [ ≥ a ] n /F n − ∼ = m for some m ∈ ω .We proceed similarly as in the proof of Lemma 1. First we obtain a representationof [ ≥ a ] n /F n − ∼ = ω and [ ≥ f ( a )] n /F n − using the least elements in each block.Thus we obtain ordered sequences a , . . . and b , . . . . We then inductively define theembedding. The fact that [ ≥ a ] n ֒ → [ ≥ f ( a )] n implies that there are m , a < · · · a m ] n − ֒ → [ > b m ] n − , where all the half-open intervals are contained in an ( n − j such that for [ b j ] n − this condition holds. Now, obtaining the sequences a , . . . and b , . . . is ∆ A ⊕ B n (the sequence is described by finite conjunctions ofcomputable Π n − and Σ n − sentences). We then know that the condition issatisfied for a and using Φ from Lemma 1 finding m and the elements a i and b i is ∆ A ⊕ B n − = ∆ A ⊕ B n . Using Ψ from Lemma 1, we can then define the embedding a ֒ → [ b , b j ). Having defined an embedding of [ a ] n − + · · · + [ a i ] n − into [ b , b l )for some l , we can define the an embedding for [ a i +1 ] n − similarly to the case of a with the difference that we restrict our search to elements in blocks greater thanthe block of b l .This clearly yields a ∆ A ⊕ B n embedding of [ ≥ a ] into [ ≥ f ( a )]. We can definea similar embedding to embed [ < a ] into [ < f ( a )] and thus also obtain a ∆ A ⊕ B n embedding of A into B .Now, say L has V D ∗ rank n . Then every bi-embeddable copy of it must alsohave V D ∗ rank n and furthermore if A ≈ B ≈ L and A = A + · · · + A n whereeach A i is of V D rank less or equal than n , then we have B = B + · · · + B n where V D ( B i ) = V D ( A i ) and A i ≈ B i for all i ≤ n . Fix elements a i and b i from each ofthe A i , respectively B i . Then it is not hard to see that using the same strategy asin the case where L has V D rank n , we can define an embedding. The only thingwe need to change is that if x ∈ [ a i ] n , then we need to define the embedding for x by only considering elements in [ b i ] n . This is again ∆ n . (cid:3) Lower Bounds.
In this section we prove the following theorem.
Theorem 4.
Suppose that L is a computable scattered linear order of finite Haus-dorff rank n + 1 . Then there are computable linear orders G , B bi-embeddable with L such that there is no ∆ n +1 -embedding of G into B . Lemma 2.
Let L be a computable scattered indecomposable linear order of fi-nite Hausdorff rank n + 1 . Then there are computable linear orders G and B bi-embeddable with L such that there is no ∆ n +1 embedding of G into B . NIKOLAY BAZHENOV, DINO ROSSEGGER, AND MAXIM ZUBKOV
Proof. As L is indecomposable of rank n + 1, there is an h-indecomposable linearorder together with its signed tree T of rank n + 1. Given T of rank n + 1, let σ ∈ T of length n and let P ( σ ) be the tree { τ : τ (cid:22) σ } with sign function inherited from T . Assume that s T ( ∅ ) = +, the case where s T ( ∅ ) = − is analogous. Our ordering G is of order type P i ∈ ω (cid:16)P j ≤ i lin ( T h j i ) + lin ( P ( σ )) (cid:17) . Let G n be a standard copyof ω with the elements labelled by the nodes of the tree of height 1, i.e., we canwrite G n as G n = t + g + t , + t , + g + t , + t , + t , + g + . . . . Clearly we can take G such that there is a computable function ψ : G → G n taking x in the i th copy of lin ( T h j i ) in G to t j,i and x in the i th copy of lin ( P ( σ )) in G to g i . We now build B such that no embedding B ֒ → G has degree ∆ n +1 . We firstbuild a ∆ n +1 computable linear order B n such that B n ∼ = G n but that there is no∆ n +1 embedding of B n into G n . Towards that fix a listing ( ϕ e ) e ∈ ω of all partial∆ n +1 computable functions. We construct B n in stages. At stage 0, B n is G n withthe difference that every element g i is replaced by successive elements b i, , b i, , i.e.,we can write B n as B n = t + b , + b , + t , + t , + b , + b , + t , + . . . . We want to satisfy the requirements R e : ϕ e : B n ֒ → G n . At stage s , if for e < s , s is the first stage greater than e such that ϕ e,s ( b e, ) ↓ = x for some x and g k is least such that x ≥ g k , then put elements into ( b e, , b e, ) suchthat | ( b e, , b e, ) | > | [ t , x ] | . This ensures that ϕ e can not be an embedding. Thisfinishes the construction.Note that in B n all the intervals [ b e, , b e, ] are finite with uniformly computablefirst and last element. Notice furthermore that everything except those intervals inthe ordering B n is computable. Thus if we replaced those intervals with computableintervals we would get a computable ordering.We now construct for every i < n a ∆ i +1 computable ordering B i such that B = B ≈ G and that there are ∆ i +1 computable embeddings from B i into B i − .We use a well known result that says if a linear order L is ∆ , then ω · L and ω ∗ · L are computable, see [AK00, Theorem 9.11]. This theorem relativizes and theproof is constructive in the sense that it constructs a computable copy of { ω, ω ∗ }· L ,given ∆ order L . Its only caveat is that it is nonuniform. If L has a least element,this element has to be fixed non-uniformly. However, this poses no problem to ourconstruction, as we wish to jump-invert the intervals [ b i, , b i, ] and have uniformlycomputable least and greatest elements for those.So we take our ∆ n +1 computable linear order B n and replace every interval[ b i, , b i, ] by a copy of ω · [ b i, , b i, ] if σ (0) = + and by a copy of ω ∗ · [ b i, , b i, ]otherwise. Elements labelled t i,j are replaced by computable disjoint copies of T h i i . We obtain a linear order B n − with suborderings B n − i corresponding tothe jump inverted intervals [ b i, , b i, ]. If B n − i has a least or greatest element,then the procedure will also return us these elements. Notice that if for example[ b i, , b i, ] ∼ = n and σ (0) = +, then B n − i ∼ = ω · n . Furthermore, B n − is a computablesum of uniformly computable and uniformly ∆ n − linear orders and thus itself N BI-EMBEDDABLE CATEGORICITY OF ALGEBRAIC STRUCTURES 9 ∆ n − . We repeat this procedure, replacing B ji with ω · B ji or ω ∗ · B ji dependingon whether σ ( j ) = + or σ ( j ) = − . The resulting linear order B j is ∆ n +1 − n − j ) computable. Thus by induction we end up with a linear order B = B which iscomputable, and it is easy to see B ∼ = P i ∈ ω (cid:16)P j ≤ i lin ( T h j i ) + lin ( P ( σ )) · k i (cid:17) where k i = | [ b i, , b i, ] | . Clearly B ≈ G .Our construction of B i from B i +1 also provides us with ∆ i computable embed-dings ϕ i : B i +1 → B i . Assume that there is a ∆ n +1 embedding χ of B into G ,then as can be seen from Fig. 1 the composition of the embeddings ϕ i and ψ givesa ∆ n +1 embedding of B n into G n , a contradiction. ω ∼ = B n B n − B B ω ∼ = G n G ϕ n − ϕ χψ Figure 1.
Morphisms between B and G (cid:3) Jullien [Jul] showed that every countable scattered linear order can be decom-posed into finitely many indecomposable linear orders and that there exists a min-imal decomposition which is unique up to bi-embeddability, see also [Mon06].Notice that if L has Hausdorff rank α and a minimal decomposition into inde-composable linear orders L + · · · + L n , then at least one of the L i must have rank α . Exploiting this properties, we can prove Theorem 4. Proof of Theorem 4.
Let L have Hausdorff rank α and L + · · · + L n be a minimaldecomposition, where L i is of rank α . We take G to be the linear order lin ( T ) + · · · + G i + · · · + lin ( T n ) where lin ( T k ) is a computable h-indecomposable linear orderbi-embeddable with L k and G i is the linear order G from Lemma 2 for L i . Fromnow on we will write G as the sum G + G i + G .For B we use the linear order G + B i + G where G and G are as for G and B i is the linear order B from Lemma 2 for L i . We claim that no embedding of B into G can be ∆ n +1 .To prove this we first construct a computable labelling of G by taking our labelling ψ from Lemma 2 and extending it to a labelling of G by labelling all elements in G and G with two new labels h , h . We also can canonically extend the embeddings ϕ i constructed in the proof of Lemma 2 to embeddings of G + B i +1 + G into G + B i + G .Let χ be an embedding of B into G and let b ϕi, be images of b i, in B n under ϕ n ◦ · · · ◦ ϕ . Then, it is not hard to see, that if χ sends infinitely many elements b ϕi, to G , then G + B i is bi-embeddable with G , contradicting the minimality ofthe decomposition. On the other hand if there is one b ϕi, that goes to G , thencofinitely many elements b ϕi, must go to G . As B i is bi-embeddable with anyof its end segments, this implies that B i + G is bi-embeddable with G — againcontradicting the minimality of the decomposition. Now, if χ was ∆ n +1 , then ϕ n ◦ · · · ◦ ϕ ◦ χ ◦ ψ would embed an end segment of B n into N G and we can get a ∆ n +1 embedding by shifting the embedding to theright by a finite number m . (cid:3) Boolean algebras
The reader is referred to, e.g., monographs [Gon97, Kop89] for the backgroundon countable Boolean algebras. We treat Boolean algebras as structures in the lan-guage {∪ , ∩ , ( ) ; 0 , } . Any Boolean algebra B admits a natural partial ordering: x ≤ B y iff x ∪ y = y . We always assume that 0 B = 1 B .Suppose that L is a linear order with least element. Then the corresponding interval Boolean algebra Int ( L ) is defined as follows: • the domain of Int ( L ) is the smallest set containing all finite unions of L -intervals:[ a , b ) ∪ [ a , b ) ∪ · · · ∪ [ a n , b n ) or [ a , b ) ∪ [ a , b ) ∪ · · · ∪ [ a n , ∞ )where a < L b < L a < L b < L · · · < L a n < L b n ; • the functions of Int ( L ) are the usual set-theoretic operations.For more details, see, e.g., Section 15 of [Kop89].Let B be a Boolean algebra. An element a ∈ B is an atom if a is a minimalnon-zero element in B . The algebra B is atomic if for every non-zero b ∈ B , thereis an atom a such that a ≤ B b . The algebra B is superatomic if all subalgebras of B are atomic.The following fact is well-known: A countable Boolean algebra B is superatomicif and only if there are a countable ordinal α and a non-zero natural number m such that B ∼ = Int ( ω α · m ) (see Theorem 1 of [Gon73] or p. 277 of [Kop89]).Furthermore, this superatomicity criterion admits a natural “effectivization”:Goncharov (Theorem 2 of [Gon73]) proved that a superatomic Boolean algebra B has a computable copy if and only if B ∼ = Int ( ω α · m ), where α is a computableordinal and 0 < m < ω .The main result of this section is the following Theorem 5.
Let α be a non-zero computable ordinal, and let m be a non-zero nat-ural number. The superatomic Boolean algebra Int ( ω α · m ) has degree of b.e. cate-goricity equal to h α i = ( (2 α − , if α < ω, (2 α ) , if α ≥ ω. Theorem 5 and Theorem 2.1 of [Baz13] together imply that for a computablesuperatomic Boolean algebra, its degree of b.e. categoricity is equal to its degreeof categoricity. Nevertheless, we note that the proofs of [Baz13] cannot be directlytransferred to the bi-embeddability setting: the key tool of this section is a newtechnique which employs limitwise monotonic functions (see Subsection 3.2).As a consequence of Theorem 5, we obtain a complete description of degrees ofb.e. categoricity for Boolean algebras:
Corollary 1.
Let B be a computable Boolean algebra. Then B satisfies one of thefollowing two conditions: (a) There is a computable ordinal α such that ( α ) is the degree of b.e. cate-goricity for B . N BI-EMBEDDABLE CATEGORICITY OF ALGEBRAIC STRUCTURES 11 (b) B is not hyperarithmetically b.e. categorical, and B does not have degree ofb.e. categoricity. The further discourse is arranged as follows. Subsection 3.1 contains neces-sary preliminaries on the technique of pairs of structures, developed by Ash andKnight [AK90, AK00]. Furthermore, we give a new theorem, which allows to encodesome specific families of limitwise monotonic functions into families of computablestructures (Theorem 7). Since the proof of Theorem 7 contains a lot of bulkytechnical details, this proof is delegated to the last subsection (Subsection 3.5).Subsection 3.2 discusses some auxiliary facts about limitwise monotonic func-tions: in particular, Proposition 1 shows how one can encode the oracle ( α ) , where α < ω CK , in a “sufficiently good” limitwise monotonic fashion. Subsection 3.3proves Theorem 5, and Subsection 3.4 obtains Corollary 1.3.1. Pairs of computable structures.
Let d be a Turing degree. A function F : ω → ω is d -limitwise monotonic if there is a d -computable function f : ω × ω → ω such that:(a) f ( x, s ) ≤ f ( x, s + 1) for all x and s ;(b) F ( x ) = lim s f ( x, s ) for all x .As per usual, we fix a path through Kleene’s O , and we identify computableordinals with their notations along this path.Let α be a non-zero computable ordinal. For the sake of convenience, we use thefollowing notation: h α i := ( ( α − , if α < ω, ( α ) , if α ≥ ω. For a language L , infinitary formulas of L are formulas of the logic L ω ,ω . For acountable ordinal α , infinitary Σ α and Π α formulas are defined in a standard way(see, e.g., [AK00, Chapter 6]).Suppose that A and B are L -structures, and α is a countable ordinal. We saythat A ≤ α B if every infinitary Π α sentence true in A is true in B . The relations ≤ α are called standard back-and-forth relations .Ash [Ash86] provided a complete description of standard back-and-forth relationsfor countable well-orders — here is a small excerpt from the description: Lemma 3 (Lemma 7 of [Ash86]; see also Lemma 15.10 of [AK00]) . Let β be acountable ordinal, and let k be a non-zero natural number. Then ω β · ( k + 1) ≤ β +1 ω β · k and ω β · k (cid:2) β +1 ω β · ( k + 1) . Let α be a non-zero computable ordinal. A family K = { A i : i ∈ ω } of L -structures is called α -friendly if the structures A i are uniformly computable, andthe relations B β = (cid:8) ( i, ¯ a, j, ¯ b ) : i, j ∈ ω, ¯ a is from A i , ¯ b is from A j , ( A i , ¯ a ) ≤ β ( A j , ¯ b ) (cid:9) are c.e., uniformly in β < α . Theorem 6 (Theorem 4.2 of [Baz17]) . Let α be a non-zero computable ordinal.Suppose that { A k : k ∈ ω } is an α -friendly family of L -structures such that A k +1 ≤ α A k for all k ∈ ω . Then for any h α i -limitwise monotonic function g ( x ) , there isa uniformly computable sequence of structures ( C n ) n ∈ ω such that for every n , wehave C n ∼ = A g ( n ) . Theorem 6 admits the following generalization:
Theorem 7.
Let α be a computable infinite ordinal. Suppose that ( α i ) i ∈ ω is acomputable sequence of non-zero ordinals such that α i < α for all i . Suppose that { A ik } i,k ∈ ω is an α -friendly family of L -structures such that A ik +1 ≤ α i A ik for all i and k . Consider a family of functions ( f i ) i ∈ ω such that each f i is h α i i -limitwisemonotonic, uniformly in i ∈ ω , i.e. there is an index e ∈ ω such that for every i ,the binary function Φ ∅ h αi i e ( i, · , · ) approximates f i in a limitwise monotonic fashion.Then there is a uniformly computable sequence of structures ( C in ) i,n ∈ ω such that C in ∼ = A if i ( n ) for all i and n . Recall that the proof of Theorem 7 is given in Subsection 3.5.3.2.
Limitwise monotonic functions.
In this subsection, we give two useful factsabout limitwise monotonicity. Recall that a partial function ξ ( x ) dominates apartial function ψ ( x ) if ( ∀ ∞ x )[ ψ ( x ) ↓ ⇒ ξ ( x ) ↓ > ψ ( x )] . Lemma 4 (follows from Theorem 4.5.4 of [Soa16]) . Let A be an oracle. If a totalfunction g ( x ) dominates every partial A -computable function, then A ′ ≤ T g ⊕ A .Proof. Fix an A -effective enumeration { A ′ [ s ] } s ∈ ω of the set A ′ , and consider thefollowing partial A -computable function: ψ ( x ) := ( the least s such that x ∈ A ′ [ s ] , if x ∈ A ′ , undefined , if x A ′ . Since the function g dominates ψ , there is a number e such that:( ∀ x ≥ e )( x ∈ A ′ ⇔ x ∈ A ′ [ g ( x )]) . Therefore, A ′ ≤ T g ⊕ A . (cid:3) Proposition 1.
Let α be a computable ordinal. There is a total function f [ α ] ( x ) with the following property: If a total function g dominates f [ α ] , then g ≥ T ( α ) .Furthermore: (a) If α = β + 1 , then f [ α ] is ( β ) -limitwise monotonic. (b) If α is a limit ordinal, then f [ α ] is ( α ) -computable.Proof. Our proof employs recursion on α .Without loss of generality, we may assume that for any oracle A , the partialcomputable function ϕ A satisfies the following: ϕ A ( x ) := ( the least s such that ϕ Ax,s ( x ) ↓ , if such s exists , undefined , otherwise . We fix a Turing operator Ψ with the following property: Let δ be a computablelimit ordinal, and let ( γ i ) i ∈ ω be the standard fundamental sequence for δ (thissequence is induced by the notation of δ along our path through O ). Then we have ∅ ( δ ) = Ψ X where X = {h i, x i : i ∈ ω, x ∈ ∅ ( γ i ) } . In order to obtain the desired functions, we simultaneously define f [ α ] and aTuring operator Φ( α ; · ) satisfying the following condition: For any α and any g , if g ( x ) > f [ α ] ( x ) for all numbers x , then we have ∅ ( α ) = Φ g ( α ; · ). N BI-EMBEDDABLE CATEGORICITY OF ALGEBRAIC STRUCTURES 13
Case α = 0 . We define f [0] ( x ) := 0 and Φ(0; · ) := 0. Case α = β + 1 . Assume that the desired objects f [ β ] and Φ( β ; · ) have beenalready defined. If β = γ + 1, then f [ β ] is ( γ ) -limitwise monotonic. Thus, wededuce that f [ β ] ≤ T ( β ) . We set f [ β +1] (2 x ) := f [ β ] ( x ) ,f [ β +1] (2 x + 1) := 1 + X i ≤ x { ϕ ∅ ( β ) i ( x ) : ϕ ∅ ( β ) i ( x ) ↓} . It is not hard to show that the function f [ β +1] is ( β ) -limitwise monotonic.Assume that a total function g dominates f [ β +1] . Then the function ˜ g ( x ) := g (2 x )dominates f [ β ] , and hence, we have g ≥ T ˜ g ≥ T ( β ) . On the other hand, thefunction b g ( x ) := g (2 x + 1) dominates every partial ( β ) -computable function ϕ ∅ ( β ) i .Therefore, by Lemma 4, b g ⊕ ( β ) computes ( β +1) . Thus, we obtain: g ≡ T g ⊕ ( β ) ≥ T b g ⊕ ( β ) ≥ T ( β +1) . Now we describe how to define Φ( β + 1; · ). For the sake of simplicity, we assumethat g ( x ) > f [ β +1] ( x ) for all x , and we discuss the definition of Φ g ( β + 1; · ) in thiscase. Since ˜ g ( y ) > f [ β ] ( y ) for all y , the function Φ ˜ g ( β ; · ) is equal to the characteristicfunction of ∅ ( β ) .Note that for any y , if the value ϕ ∅ ( β ) ( y ) is defined, then b g ( y ) > ϕ ∅ ( β ) ( y ). Ouragreement about the partial function ϕ A implies that for all y ∈ ω , we have y ∈ ∅ ( β +1) ⇔ ϕ ∅ ( β ) y, b g ( y ) ( y ) ↓ . Since ∅ ( β ) = Φ ˜ g ( β ; · ), it is now clear how one can obtain the desired equalityΦ g ( β + 1; · ) = ∅ ( β +1) . Case of limit α . Consider the standard fundamental sequence ( β i ) i ∈ ω for theordinal α . The function f [ α ] is defined as follows: f [ α ] ( h i, x i ) := f [ β i ] ( x ) . By recursion on α , it is not hard to show that f [ α ] is ( α ) -computable.Assume that a total function g dominates f [ α ] . Consider the functions g i ( x ) := g ( h i, x i ), i ∈ ω . Since g dominates f [ α ] , there is a number j such that( ∀ i ≥ j )( ∀ x )( g i ( x ) > f [ β i ] ( x )) . Hence, for every i ≥ j , we have ∅ ( β i ) = Φ g i ( β i ; · ). Therefore, we deduce ( α ) ≡ T {h i, y i : i ≥ j , y ∈ ∅ ( β i ) } ≤ T {h i, x, g i ( x ) i : i ≥ j , x ∈ ω } ≤ T g. The definition of the object Φ( α ; · ) can be easily recovered from the followingobservation: If g ( x ) > f [ α ] ( x ) for all x , then ∅ ( α ) = Ψ X where X = {h i, x i : i ∈ ω, x ∈ ∅ ( β i ) } , and ∅ ( β i ) = Φ g i ( β i ; · ) for all i ∈ ω. Proposition 1 is proved. (cid:3)
Proof of Theorem 5.
Let B := Int ( ω α · m ). Using a standard Cantor–Bendixson analysis (see, e.g., Section 17.2 of [Kop89]), one can show the following:If a superatomic Boolean algebra A is bi-embeddable with B , then B and A areisomorphic.For the interested reader, we note that in Corollary 4.34 of [GM08], the followingfact is proved: For countable ordinals β and γ , we have β ≤ γ if and only if Int ( ω β )is embeddable into Int ( ω γ ).Ash (Theorem 5 in [Ash87]) proved that the structure B is relatively ∆ α cate-gorical. Therefore, it is sufficient to provide two computable isomorphic copies A and C of B such that every isomorphic embedding h : A ֒ → C computes the degree h α i .Choose a computable structure C ∼ = B with the following property: Given anelement b ∈ C , one can effectively compute a pair θ ( b ) := ( β, ℓ ) such that β ≤ α , ℓ ∈ ω , and the relative algebra C ↾ b is isomorphic to Int ( ω β · ℓ ). Such a copy C wasconstructed, e.g., in Proposition 15.15 of [AK00]; see [Baz16] for a more detaileddiscussion. Case I.
Let α be a successor ordinal, i.e. α = β + 1.Choose a h β +1 i -limitwise monotonic function f ( x ) from Proposition 1 satisfy-ing the following: if a total function g ( x ) dominates f , then g ≥ T h β +2 i = h α i .By Lemma 3, we have ω β · ( k + 1) ≤ β +1 ω β · k for every non-zero k ∈ ω . Further-more, the family { ω β · k : k ≥ } is (2 β + 1)-friendly (see, e.g., Proposition 15.11in [AK00]). Therefore, one can apply Theorem 6 and produce a computable se-quence of linear orders ( L n ) n ∈ ω such that L n ∼ = ω β · (1 + f ( n )) . Consider computable Boolean algebras D j := X n ∈ ω Int ( L n ) , where 1 ≤ j ≤ m, A := X ≤ j ≤ m D j . For n ∈ ω , let e n be the element induced by Int ( L n ) inside D . It is not hard toshow that every D j is isomorphic to Int ( ω β +1 ), and A ∼ = Int ( ω β +1 · m ).Assume that h is an isomorphic embedding from A into C . Then for every n ∈ ω ,the relative algebra C ↾ h ( e n ) must be isomorphic to Int ( ω β · ℓ n ) for some finite ℓ n ≥ f ( n ). Thus, the function ξ ( n ) := ℓ n is computable in h , and ξ dominates f . Therefore, we deduce h ≥ T ξ ≥ T h α i . Case II.
Let α be a limit ordinal. For simplicity, we give the proof for the casewhen B ∼ = Int ( ω α ). A general case of Int ( ω α · m ) is treated in a similar way.Consider the standard fundamental sequence ( β i ) i ∈ ω for α . Without loss ofgenerality, we may assume that for every i ∈ ω , 0 < β i < β i +1 and β i = ( γ i , if β i < ω, γ i + 1 , if β i ≥ ω. Fix a total function f [ α ] from Proposition 1. Recall that for any i and x , wehave f [ α ] ( h i, x i ) = f [ β i ] ( x ), where the functions f [ β i ] are h γ i i -limitwise monotonic,uniformly in i ∈ ω . N BI-EMBEDDABLE CATEGORICITY OF ALGEBRAIC STRUCTURES 15
Note the following: If λ ≥ γ and κ ≥ γ , then ω λ ≡ γ ω κ (see, e.g., Lemma 15.9in [AK00]). Hence, for every i ∈ ω , we have: ω γ i ≥ γ i ω γ i +1 ≥ γ i ω γ i +2 ≥ γ i · · · ≥ γ i ω γ i + j ≥ γ i . . . . From now on, for the sake of convenience, we use the notation γ [ i ] := γ i .We employ Theorem 7 with the following parameters: • the sequence α i := 2 γ i ; • the α -friendly family consisting of A ik := ω γ [ i + k ] ; and • the functions f i := f [ β i ] .This gives us a computable sequence of linear orders: L h i,k i ∼ = ω γ [ i + f [ βi ] ( k )] . We define a Boolean algebra A := X n ∈ ω Int ( L n ) . It is not difficult to show that A is a computable isomorphic copy of Int ( ω α ). For n ∈ ω , let e ( n ) denote the element induced by Int ( L n ) inside A .Suppose that h is an isomorphic embedding from A into C . We define a totalfunction ξ — for numbers i and k , the value ξ ( h i, k i ) is computed as follows.(1) Find the ordinal δ < α such that the relative algebra C ↾ h ( e ( h i, k i )) isisomorphic to Int ( ω δ · ℓ ) for some ℓ ∈ ω . Clearly, δ ≥ γ [ i + f [ β i ] ( k )].(2) Define the value ξ ( h i, k i ) as the least number j such that γ [ j ] > δ .Note that ξ ( h i, k i ) > i + f [ β i ] ( k ) = i + f [ α ] ( h i, k i ). Thus, the function ξ dominates f [ α ] . Hence, by Proposition 1, we deduce that h ≥ T ξ ≥ T ( α ) . This concludes theproof of Theorem 5. (cid:3) Proof of Corollary 1. If B is superatomic, then by Theorem 5, B satisfiesthe first condition. Thus, we will assume that B is not superatomic, i.e. B isbi-embeddable with the atomless Boolean algebra Int (1 + η ).Let A be a computable copy of Int (1 + η ). Fix a computable linear order L isomorphic to ω CK · (1 + η ) such that L has no hyperarithmetic descending chains.Set C := Int ( L ). Clearly, both A and C are bi-embeddable with B .Towards a contradiction, assume that there is a hyperarithmetic embedding h from A into C . Choose a non-zero element a ∈ A such that h ( a ) can be representedin the following form:(2) [ u ( a ) , v ( a )) L ∪ [ u ( a ) , v ( a )) L ∪ · · · ∪ [ u k ( a ) ( a ) , v k ( a ) ( a )) L , where u ( a ) < L v ( a ) < L u ( a ) < L v ( a ) < L · · · < L u k ( a ) ( a ) < L v k ( a ) ( a ). Set b := a and w := v k ( a ) ( a ).Now suppose that the elements b , b , . . . , b n ∈ A and w > L w > L · · · > L w n are already defined. Choose arbitrary non-zero c and d from A with c ∩ d = 0 A and c ∪ d = b n . For the elements h ( c ) and h ( d ), consider their representations of theform (2). Clearly, we have either v k ( c ) ( c ) < L w n or v k ( d ) ( d ) < L w n . Choose w n +1 as the element from { v k ( c ) ( c ) , v k ( d ) ( d ) } which is strictly less than w n in L .Then the constructed sequence ( w n ) n ∈ ω is hyperarithmetic and strictly descend-ing, which contradicts the choice of L . Therefore, the structure B is not hyperar-itmetically b.e. categorical. By Theorem 3.1 of [BFRS], this implies that B has nodegree of b.e. categoricity. (cid:3) Proof of Theorem 7.
The proof employs the α -systems technique. For adetailed exposition of this method, the reader is referred to [AK00]. Here we giveonly the necessary definitions and results.For sets L and U , an alternating tree on L and U is a tree P consisting of non-empty finite sequences σ = l u l u . . . , where l i ∈ L and u j ∈ U . We assume thatevery σ ∈ P has a proper extension in P .An instruction function for P is a function q from the set of sequences in P ofodd length (i.e. those with last term in L ) to U , such that if q ( σ ) = u , then σu ∈ P .A run of ( P, q ) is a path π = l u l u . . . through P such that u n +1 = q ( l u l u . . . u n l n )for every n ∈ ω . An enumeration function on L is a function E from L to the setof all finite subsets of ω . If π = l u l u . . . is a path through P , then E ( π ) = S i ∈ ω E ( l i ).Suppose that L and U are c.e. sets, E is a partial computable enumerationfunction on L , and P is a c.e. alternating tree on L and U . We assume that all σ ∈ P start with the same symbol ˆ l ∈ L .Fix a non-zero computable ordinal α . Suppose that ≤ β , β < α , are binaryrelations on L such that ≤ β are c.e., uniformly in β < α .The structure ( L, U, ˆ l, P, E, ( ≤ β ) β<α )is an α -system if is satisfies the following conditions:(1) ≤ β is reflexive and transitive, for β < α ;(2) ( l ≤ γ l ′ ) ⇒ ( l ≤ β l ′ ), for β < γ < α ;(3) if l ≤ l ′ , then E ( l ) ⊆ E ( l ′ );(4) if σu ∈ P , where σ ends in l ∈ L ; and l ≤ β l ≤ β · · · ≤ β k − l k , for α > β > β > · · · > β k , then there exists l ∗ ∈ L such that σul ∗ ∈ P ,and l i ≤ β i l ∗ for all i ≤ k . Theorem 8 (Ash and Knight [AK00, Theorem 14.1]) . Let ( L, U, ˆ l, P, E, ( ≤ β ) β<α ) be an α -system. Then for any ∆ α instruction function q , there is a run π of ( P, q ) such that E ( π ) is c.e., while π itself is ∆ α . Moreover, from a ∆ α indexfor q , together with a computable sequence of c.e. and computable indices for thecomponents of the α -system, we can effectively determine a ∆ α index for π and ac.e. index for E ( π ) . Now we are ready to give the proof of Theorem 7. Essentially, the constructiondescribed below extends the proof of Theorem 18.9 from [AK00].Fix an index e such that for each i ∈ ω , the function Φ ∅ h αi i e ( i, · , · ) approximates f i in a limitwise monotonic way. Set: g ( i, x, s ) := Φ ∅ h αi i e ( i, x, s ) . Note that f i ( x ) = lim s g ( i, x, s ).Given two indices M, n ∈ ω , we define (in a uniform way) the following objects:(a) an α M -system S M = ( L M , U M , ˆ l M , P M , E M , ( ≤ β,M ) β<α M ) (the same forall n ∈ ω ), and N BI-EMBEDDABLE CATEGORICITY OF ALGEBRAIC STRUCTURES 17 (b) a ∆ α M instruction function q M,n , with ∆ α M index that can be computedeffectively from M and n .The desired structure C Mn is obtained from a run of ( P M , q M,n ). The uniformity ofTheorem 8 guarantees that the structures C Mn are uniformly computable.We focus on a given pair ( M, n ), hence, we will slightly abuse the notations, andomit the subscript M in the names of our objects.Assume that C is an infinite computable set of constants, for the universe of C Mn .We also assume that all the structures A Mk , k ∈ ω , have the same universe A . Let F be the set of finite partial 1-1 functions p from C to A . Let U = ω and let L consist of the pairs in ω × F , and one extra element ˆ l := ( − , ∅ ). If ( k, p ) ∈ ω × F ,then ( k, p ) represents ( A Mk , p ).The standard enumeration function E st is defined as follows. Assume that m ∈ ω , and ¯ x is the sequence of the first m variables. For an m -tuple ¯ a in an L -structure B , the set E st ( B , ¯ a ) consists of the basic formulas ψ (¯ x ), with G¨odel number lessthan m , such that B | = ψ (¯ a ).Following [AK00], we extend the standard enumeration function and the stan-dard back-and-forth relations on pairs ( A Mk , ¯ a ) to tuples ( k, p ) ∈ L . If l = ( k, p ),where k ∈ ω and p maps ¯ b to ¯ a , then E ( l ) = E st ( A Mk , ¯ a ). We set E (ˆ l ) = ∅ .Assume that β < α M . If l = ( k, p ) and l ′ = ( j, q ), where p maps ¯ b to ¯ a and q maps ¯ b ′ to ¯ a ′ , then l ≤ β l ′ if and only if ¯ b ⊆ ¯ b ′ and ( A Mk , ¯ a ) ≤ β ( A Mj , ¯ a ′ ). We write l ⊆ l ′ if k = j and p ⊆ q . For all l ∈ L , we let ˆ l ≤ β l and ˆ l ⊆ l .The tree P consists of the finite sequences σ = ˆ lu l u . . . such that u k ∈ U , l k ∈ L , and the following conditions hold:(1) l k has the form ( u k , p k );(2) dom ( p k ) and ran ( p k ) include the first k elements of the sets C and A ,respectively;(3) u = 0;(4) if u k +1 = u k , then u k +1 = u k + 1;(5) if u k = u k +1 , then l k ⊆ l k +1 , and in any case, l k ≤ l k +1 .We have described all ingredients of our α M -system. Note that the description isuniform in M ∈ ω . Lemma 5. ( L, U, ˆ l, P, E, ( ≤ β ) β<α M ) is an α M -system.Proof. It is easy to check that the sets
L, U, P are c.e. and the function E is partialcomputable. Since the family { A ik : i, k ∈ ω } is α -friendly, the relations ≤ β are c.e.uniformly in β < α M (note that this fact is true even for all β < α , but we will notuse this). It is not hard to verify the first three conditions from the definition of an α M -system.Assume that σu ∈ P , where σ ends in l , and l ≤ β l ≤ β · · · ≤ β k − l k , for α M > β > · · · > β k . We want to find l ∗ ∈ L such that σul ∗ ∈ P and l i ≤ β i l ∗ for all i ≤ k . Suppose that l i = ( v i , p i ). Using the properties of the standard back-and-forth relations (see [AK00, § p ∗ i ⊇ p i , i ≤ k , suchthat p ∗ k = p k and ( v i , p ∗ i ) ≤ β i ( v i − , p ∗ i − ), for 0 < i ≤ k . Note that l i ≤ β i ( v , p ∗ )for all i ≤ k .If u = v , then we build an extension q ⊇ p ∗ (to include the necessary elementsin the domain and range of our function) and obtain the desired l ∗ = ( v , q ). Now assume that u = v + 1. Since A Mv +1 ≤ α M A Mv , there is a function q such that( v , p ∗ ) ≤ β ( v + 1 , q ). Again, we find a suitable extension q ⊇ q and producethe desired l ∗ = ( v + 1 , q ). (cid:3) The instruction function q M,n is defined as follows. We set q M,n (ˆ l ) = 0. For anelement σ = ˆ lu l . . . u s l s in P , let q M,n ( σ ) = (cid:26) u s + 1 , if g ( M, n, s ) > u s ,u s , otherwise . Given numbers
M, n and a ∆ α M index for the function g ( M, · , · ), we can computea ∆ α M index for q M,n .We apply Theorem 8, and for each M and n , we obtain a run π M,n =ˆ lu M,n l M,n u M,n . . . of ( P M , q M,n ) such that E ( π M,n ) is c.e., uniformly in
M, n . Sup-pose that l M,ns = ( u ns , p ns ).Recall that g ( M, n, s ) ≤ g ( M, n, s + 1) and lim s g ( M, n, s ) = f M ( n ). Hence, itis easy to show that for all s , we have u ns ≤ g ( M, n, s ). Moreover, there is a stage s ∗ such that u ns = f M ( n ) for all s ≥ s ∗ . The mapping F − = S s ≥ s ∗ p ns is a 1-1function from C onto A Mf M ( n ) . The map F induces a structure C Mn on the universe C such that D ( C Mn ) = E ( π M,n ).So, we constructed a uniformly computable sequence { C Mn } M,n ∈ ω such that C Mn ∼ = A Mf M ( n ) for all M and n . Theorem 7 is proved. (cid:3) References [AK90] C. J. Ash and J. F. Knight. Pairs of recursive structures.
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Sobolev Institute of Mathematics, 4 Acad. Koptyug Ave. and Novosibirsk StateUniversity, 2 Pirogova St., Novosibirsk, 630090, Russia
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