Bi-embeddability spectra and bases of spectra
aa r X i v : . [ m a t h . L O ] A p r BI-EMBEDDABILITY SPECTRA AND BASES OF SPECTRA
EKATERINA FOKINA, DINO ROSSEGGER, AND LUCA SAN MAURO
Abstract.
We study degree spectra of structures with respect to the bi-em-beddability relation. The bi-embeddability spectrum of a structure is the fam-ily of Turing degrees of its bi-embeddable copies. To facilitate our study weintroduce the notions of bi-embeddable triviality and basis of a spectrum.Using bi-embeddable triviality we show that several known families of de-grees are bi-embeddability spectra of structures. We then characterize the bi-embeddability spectra of linear orderings and study bases of bi-embeddabilityspectra of strongly locally finite graphs.
The study of degrees realized by structures is a central topic in computablestructure theory initiated by Richter [1] who was the first to study the degrees ofisomorphic copies of structures. Knight [2] studied degree spectra of structures, theset of degrees of isomorphic copies of a given structure. Since then the questionwhich sets of degrees can be realized by degree spectra has been widely studied, seefor instance [3]–[8].In recent years researchers studied degree spectra under equivalence relationsother than isomorphism. Fokina, Semukhin, and Turetsky [9] gave the followingdefinition.
Definition 1.
Given a structure A and an equivalence relation ∼ , the degree spec-trum of A under ∼ is DgSp ∼ (A) = { deg (B) ∶ B ∼ A} . Definitions analogous to this were given by Yu Liang [10] and Montalb´an [11].Under this notion the classical degree spectrum of a structure A is DgSp ≅ (A) .In [9] Fokina, Semukhin, and Turetsky investigated degree spectra under Σ n equiv-alence, DgSp ≡ n (A) . Two structures are Σ n equivalent, A ≡ n B , if every first orderΣ n sentence true of A is true of B and vice versa. Andrews and Miller [7] studiedspectra of theories, the family of degrees of models of a complete theory T . Interms of the above definition the theory spectrum of T is the spectrum of a model A of T under elementary equivalence, DgSp ≡ (A) . Recently, Rossegger [12] inves-tigated elementary bi-embeddability spectra of structures, DgSp ≊ (A) . Two struc-tures are elementary bi-embeddable if either is elementary embeddable in the other.Elementary bi-embeddability lies in between isomorphism and elementary equiv-alence in the sense that two isomorphic structures are elementary bi-embeddable Mathematics Subject Classification.
Key words and phrases. degree spectra, bi-embeddability spectra, bases.The authors were supported by the Austrian Science Fund FWF through project P 27527. and two elementary bi-embeddable structures are elementary equivalent but noneof these implications reverses. It furthermore lies in between isomorphism andbi-embeddability, the topic of the present study.Two structures A and B are bi-embeddable , written A ≈ B , if either is embeddablein the other. Bi-embeddability has been studied in both descriptive set theory andcomputable structure theory. Louveau and Rosendal [13] proved that the relationof bi-embeddability for countable graphs is complete among analytic equivalencerelations under Borel reducibility; this contrasts to the case of the isomorphismrelation which is far from complete on countable graphs. The effective theory be-haves quite differently: Fokina, Friedman, Harizanov, McCoy, and Mont´alban [14]proved that the isomorphism relation on several classes of computable structure(e.g., graphs, trees, and linear orderings) is complete among Σ equivalence rela-tions, while Friedman and Fokina [15] observed that the same does not hold forthe bi-embeddability relation on linear orderings (this follows from Montalban’sanalysis [16] of the bi-embeddable type of hyperarithmetic linear orderings, that wewill discuss in Section 3). Recently, Bazhenov, Fokina, Rossegger, and San Mauroinvestigated another computational aspect of the bi-embeddability relation. Theystudied the complexity of embeddings structures. To facilitate this study, they in-troduced computable bi-embeddable categoricity [17] and classified the degrees ofcomputable bi-embeddable categoricity for equivalence structures [18].The focus of this paper is the degree spectrum of A under bi-embeddability, orfor short bi-embeddability spectrum of A , DgSp ≈ (A) = { deg (B) ∶ B ≈ A} . Obtaining examples of sets of degrees which are, or are not, bi-embeddability spec-tra of structures is in general difficult, since the bi-embeddability relation does notseem to possess strong combinatorial properties one could use to construct such ex-amples. However, for many of the examples constructed for classical degree spectra,a thorough analysis of their construction shows that their isomorphism spectrumcoincides with their bi-embeddability spectrum. Either because the structure is b.e. trivial , i.e., its isomorphism type and bi-embeddability type coincide, or everybi-embeddable copy computes an isomorphic copy, in which case we say that thestructure is a basis for its bi-embeddability spectrum.Given a single structure A we say that A is a ∼ basis of B if A ∼ B and
DgSp ≅ (A) = DgSp ∼ (B) . Apart from the above observation another motivationto study bases of spectra arises from the comparison of degree spectra under dif-ferent equivalence relations. Given two equivalence relations ∼ , ∼ on structuresand a structure A , a common question is if there is a structure B such that DgSp ∼ (B) = DgSp ∼ (A) . In general this structure B might look very differentthan A from a structural point of view. Thus, given A it might be hard to find B .Therefore it is useful to restrict B to some specific class of structures. The notionof a basis captures this question nicely for the most restrictive class of structures I-EMBEDDABILITY SPECTRA AND BASES OF SPECTRA 3 one could want B to be in, the ∼ type of A . Note that, while our definition of abasis only captures the case where ∼ is isomorphism, it can be adapted to capturethe general case without much effort.In the present article we study the phenomenon of b.e. triviality and bi-embed-dability bases of structures. Thus, if we say that A is a basis of B we mean that A is abi-embeddability basis. In Section 1 we give some examples of b.e. trivial structuresand use these to obtain examples of well known families of degrees that are realizedas bi-embeddability spectra. In Section 2 we give a more general definition of abasis where we allow families of structures. This definition is motivated by thenotion of basis in topology and linear algebra. In Section 3 we give a completecharacterization of the bi-embeddability spectra of linear orderings and in Section 4we show that in a subclass of strongly locally finite graphs every structure has abasis consisting of a single structure. We close by stating a number of open questionswe consider interesting for future research.1. B.e. triviality
In this section we show for several families of degrees known to be isomorphismspectra that they are bi-embeddability spectra. For many families in the literaturethe isomorphism type of the structure realizing it coincides with its bi-embeddabilitytype. We call such structures b.e. trivial.
Definition 2.
A structure A is b.e. trivial if any bi-embeddable copy B of A isisomorphic to A .A stronger condition that implies b.e. triviality is that any endomorphism of astructure is an automorphism. To see that this is strictly stronger consider theinfinite complete graph. It is b.e. trivial but does have endomorphisms which arenot automorphisms.Apart from being b.e. trivial, many families of degrees can be realized by rigidstructures, i.e., structures which do not possess non-trivial automorphisms. How-ever, in general there is no connection between the number of automorphisms of astructure and b.e. triviality. Proposition 1.
There is a b.e. trivial structure that is not rigid and there is arigid structure that is not b.e. trivial.Proof.
For an example of a countable b.e. trivial structure that is not rigid considerthe complete infinite graph. It is b.e. trivial but not rigid. In fact it has continuummany automorphisms as every permutation of its universe is an automorphism.On the other hand, for an example of a rigid structure that is not b.e. trivialconsider a tree in the language of graphs, where the number of successors of a vertexis strictly monotonic in the canonic lexicographical ordering on the tree. This treeis rigid as any automorphism must map a vertex to a vertex with the same numberof children. It is however not b.e. trivial as it is bi-embeddable with two disjointcopies of itself. (cid:3)
EKATERINA FOKINA, DINO ROSSEGGER, AND LUCA SAN MAURO
The complete graph we used in the above proposition as an example of a b.e.trivial but not rigid structure is an example of an automorphically trivial structure.Recall that a structure is automorphically trivial if there is a finite subset of itsuniverse such that every permutation of its universe that fixes this subset pointwiseis an automorphism.
Proposition 2.
Automorphically trivial structures are b.e. trivial.Proof.
Let A be automorphically trivial and B ≈ A . Assume µ ∶ A → B and ν ∶ B → A are embeddings, and that S is a finite substructure of A such thatevery permutation of A fixing S pointwise is an automorphism. We have that B is isomorphic to a substructure of A by ν and thus every permutation that fixes ν (B) ∩ S is an automorphism of ν (B) . Let S be the pullback of ν (B) ∩ S along ν . Then S witnesses that B is automorphically trivial. We can inductively define S n + switching the roles of A , B and µ , ν when n is odd. Observe that for all n , S n + is isomorphic to a substructure of S n . Because S was finite we will find afixpoint, i.e., there is an n such that S n + ≅ S n . Let k be the first even numbersuch that S k + ≅ S k . Since we constructed S k + by pulling back S k along ν we havethat ν is an isomorphism between S k + and S k .We can now build an isomorphism f ∶ B → A . At stage 0 let f be ν ↾ S k + , theabove-mentioned isomorphism between S k + and S k . At stage s , if f ( s ) is alreadydefined or not in B proceed to the next stage. Otherwise take the least x ∈ A thatis not in the range of f and let f ( s ) = x . Then proceed to the next stage.Clearly in the limit f will be a bijection between B and A . To see that it is anisomorphism let T = dom ( f ) at some stage s . We have that ν ( T ) ∩ f ( T ) ⊇ S k andthus there is a permutation π of A fixing S k pointwise such that π ( f ( T )) = ν ( T ) .By automorphic triviality of A we have that f ( T ) ≅ π ( f ( T )) = ν ( T ) ≅ T. Thus, at every stage s , f is a partial isomorphism from A to B and therefore in thelimit an isomorphism. (cid:3) Knight [2] showed that if a structure is automorphically trivial, then its degreespectrum is a singleton, and that otherwise it is upwards closed. By the aboveproposition also the bi-embeddability spectrum of automorphically trivial struc-tures is a singleton. Clearly every bi-embeddability spectrum of a structure isthe union of the degree spectra of structures in its bi-embeddability type. Thus,Knight’s result also holds for bi-embeddability spectra.
Corollary 3. If A is automorphically trivial, then its bi-embeddability spectrum isa singleton. Otherwise it is upwards closed. We now look at examples of b.e. trivial structures that appear in the literature.The following definition appears in [19].
I-EMBEDDABILITY SPECTRA AND BASES OF SPECTRA 5
Definition 3.
Let X ⊆ ω and n ∈ ω . The graph G ({ n } ⊕ X ) is an ω chain with an n + m if m ∈ X and a 4 cycle attachedto m if m /∈ X . Proposition 4.
Let X ⊆ ω , F be a family of sets and G be the disjoint union of thegraphs G ({ n } ⊕ F ) for F ∈ F and n ∈ X . Then G is b.e. trivial.Proof. It is easy to see that for any set Y and n ∈ ω , G ({ n } ⊕ Y ) is b.e. trivial ascycles of length m only embed into cycles of length m .Now, say G is bi-embeddable with A , say f ∶ G → A and g ∶ A → G . Let G ({ n } ⊕ F ) be a component of G , then g ( f ( G ({ n } ⊕ F ))) must be in a component containinga substructure isomorphic to G ({ n } ⊕ F ) . By construction the only componentlike this is G ({ n } ⊕ F ) and as it is b.e. trivial we get that g is the inverse of f on G ({ n } ⊕ F ) . We have that for every n ∈ X and F ∈ F , G contains exactly onecomponent isomorphic to G ({ n } ⊕ F ) and no other components. Therefore, g isthe inverse of f , and thus, f is an isomorphism. (cid:3) Graphs of the form required in Proposition 4 were used in [19] to show thatthe class of non computable degrees and the class of hyperimmune degrees areisomorphism spectra. We now get the same result for bi-embeddability spectra.
Theorem 5. (1)
For every Turing degree a there is a graph G such that DgSp ≈ (G) ={ d ∶ d ≥ a } . (2) There is a graph G such that DgSp ≈ (G) = { d ∶ d > } . (3) There is a graph G such that DgSp ≈ (G) = { d ∶ d is hyperimmune } .Proof. For (1), given a set X ∈ a consider the graph using { } ⊕ X . It is not hardto see that G ({ } ⊕ X ) is b.e. trivial and DgSp ≈ ( G ({ } ⊕ X )) = { d ∶ d ≥ a } . Items(2) and (3) follow directly from Proposition 4 and the results in [19]. The proofsgiven there follow the ideas of Wehner’s proof that the noncomputable degrees arethe spectrum of a structure [4] but with some differences.We sketch the proof of (2). Wehner considered the family of finite sets F ={{ n } ⊕ F ∶ F finite ∧ F ≠ W n } . He showed that this family is X -computablyenumerable if and only if X is not computable and coded this family into a structure H such that H is X -computable if and only if the family is X -c.e. It is unclear howto produce a b.e. trivial structure such that F is c.e. in every of its bi-embeddablecopies. Indeed, one can show that the usual encoding using “bouquet graphs”(see [12]) has a computable bi-embeddable copy. However, if we consider the graph G obtained by taking the disjoint unions of the graphs G ({ n } ⊕ F ) for { n } ⊕ F ∈ F we obtain a b.e. trivial structure by Proposition 4. Csima and Kalimullin showedthat G is X -computable if and only if there is Y ≡ T X such that for all e ∈ ω , Y [ e ] is finite and Y [ e ] ≠ W e . They then showed that the degrees with this property areexactly the non-computable degrees. (cid:3) Here Y [ e ] denotes the e th column of Y , i.e., Y [ e ] = { y ∶ ⟨ e, y ⟩ ∈ Y } . EKATERINA FOKINA, DINO ROSSEGGER, AND LUCA SAN MAURO
There are also other spectra known to be bi-embeddability spectra. In [12]Rossegger observed that for all computable successor ordinals α and β , { d ∶ d ( α ) ≥ ( β ) } is the bi-embeddability spectrum of a structure; he constructed b.e. trivialstructures having such spectra. It is doubtful whether this result can be extended toinclude limit ordinals. Soskov [20] gave an example of an isomorphism spectrum ofa structure A such that DgSp ≅ (A) ⊆ { d ∶ d ≥ ( ω ) } and showed that no structurehas { d ∶ d ( ω ) ∈ DgSp ≅ (A)} as its isomorphism spectrum. Faizrahmanov, Kach,Kalimullin, and Montalb´an [21] recently showed that no structure realizes the family { d ∶ d ( ω ) ≥ a ( ω ) } for a ≥ ( ω ) as its isomorphism spectrum.Andrews and Miller [7] showed that the family { d ∶ d ( ω + ) ≥ ( ω ⋅ + ) } is not thetheory spectrum of a structure. Rossegger’s result therefore gives an example of abi-embeddability spectrum which can not be a theory spectrum.2. Basis
All examples of bi-embeddability spectra seen so far have been realized by exam-ples which are b.e. trivial, i.e., their bi-embeddability type and their isomorphismtype coincides. B.e. triviality is purely model theoretic. Since we are interestedin degree spectra, a more general property of a structure A is when we can find astructure B in A ’s bi-embeddability type such that DgSp ≈ (A) = DgSp ≅ (B) . In this case we say that B is a b.e. basis for A . We now give a general definition ofa basis. Definition 4.
Given a structure A and an equivalence relation ∼ we say that afamily B of structures is a ∼ basis for A if(1) ∀ B ∈ B B ∼ A ,(2) ∀ B , C ∈ B DgSp ≅ ( B ) /⊆ DgSp ≅ ( C ) ,(3) and DgSp ∼ (A) = ⋃ B∈ B DgSp ≅ (B) .Recall the notion of Muchnik reducibility; a set of reals P is Muchnik reducible to a set of reals Q , P ≤ w Q , if every real in Q computes a real in P . In termsof structures one usually says that A ≤ w B if every structure in the isomorphismtype of B computes a structure in the isomorphism type of A , which is equivalentto saying that DgSp ≅ (B) ⊆ DgSp ≅ (A) . Let A and B be families of structures.Muchnik reducibility extends naturally to such families. A ≤ w B ∶ ⇔ ⋃ B∈ B DgSp ≅ (B) ⊆ ⋃ A∈ A DgSp ≅ (A) Using this we get the following characterization of a ∼ basis. Proposition 6.
Let A be the family of structures bi-embeddable with A . The family B ⊆ A is a ∼ basis of A if and only if B is a minimum with respect to inclusionsuch that B ≤ w A . I-EMBEDDABILITY SPECTRA AND BASES OF SPECTRA 7
All b.e. trivial structures exhibited in Section 1 clearly have a singleton bi-embeddability basis, themselves. It is unclear whether there exist structures withcountable or even finite bi-embeddability basis greater than one. This question canbe viewed as the computability theoretic analogue to a conjecture by Thomass´e inmodel theory stating that the number of isomorphism types in the bi-embeddabilitytype of a relational countable structure is either 1, ℵ , or 2 ℵ , see [22] for more onthis conjecture.However, if we consider bases for other equivalence relations, the analogue ofthis question has been answered positively. Andrews and Miller [7] have shownthat there is a theory whose degree spectrum is the union of two cones, i.e., thereis a complete theory T such that for A ⊧ T , DgSp ≡ (A) = { d ≥ a } ∪ { d ≥ b } for twoincomparable Turing degrees a and b . Fokina, Semukhin, and Turetsky [9] showedthat the same holds for Σ n equivalence with n >
2. Hence, for Σ n equivalence andelementary equivalence there are structures with a basis of size 2.3. Linear orderings
Montalb´an [16] showed that all hyperarithmetic linear orderings are bi-embed-dable with a computable one, and thus their bi-embeddability spectrum containsall Turing degrees. The following is a relativization of his theorem [16, Theorem1.2].
Theorem 7.
Let X ⊆ ω . If a linear ordering is hyperarithmetic in X then it isbi-embeddable with an X -computable linear ordering. This theorem implies that every linear ordering has a singleton bi-embeddabilitybasis.The proof of the original theorem is involved and most of it is not computabil-ity theoretic. Its relativization, Theorem 7, can be obtained by relativizing thecomputability theoretic part.As a corollary we obtain a characterization of the bi-embeddability spectra oflinear orderings in terms of their Hausdorff rank. Before we state the corollary weintroduce the required notions.
Definition 5.
Let L = ( L, ≤ ) be a linear ordering. For x, y ∈ L let x ∼ y if x = y ,for α a countable limit ordinal x ∼ α y if x ∼ γ y for some γ < α and for α = β + x ∼ α y if the intervals [[ x ] ∼ β , [ y ] ∼ β ] or [[ x ] ∼ β , [ y ] ∼ β ] are finite.The Hausdorff rank of L , r ( L ) , is the least countable ordinal α such that L / ∼ α is finite.Hausdorff [23] showed that a linear ordering is scattered , i.e., it does not embeda copy of η , if and only if it has countable Hausdorff rank. Clearly, if L is notscattered then it is bi-embeddable with η , and thus has a computable bi-embeddablecopy. In [16] it was shown that a scattered linear ordering is bi-embeddable witha computable one if and only if it has computable Hausdorff rank. Given a set EKATERINA FOKINA, DINO ROSSEGGER, AND LUCA SAN MAURO X ⊆ ω we write ω X for the first non X -computable ordinal. We can now state arelativization of this theorem. Theorem 8.
Let X ⊆ ω . A scattered linear ordering L has an X -computable bi-embeddable copy if and only if r ( L ) < ω X . In other words, L has an X -computable copy if and only if it computes a copyof its Hausdorff rank, i.e., X ≥ T A ≅ r ( L ) . This combined with Theorem 7 yieldsthe following characterization of bi-embeddability spectra of linear orderings. Corollary 9.
Let L be a linear ordering. (1) If η ↪ L , then η is a b.e. basis for L , i.e., DgSp ≈ ( L ) = DgSp ≅ ( η ) = { d ∶ d ≥ } , (2) if L is scattered, then its Hausdorff rank is a b.e. basis for L , i.e., DgSp ≈ ( L ) = DgSp ≅ ( r ( L )) . Montalb´ans result shows that the bi-embeddability spectra of linear orderings ofhyperarithmetic Hausdorff rank always have a minimum – the computable degree.However, this is not the case for all linear orderings.
Proposition 10.
Let L be a linear ordering with ω CK1 ≤ r ( L ) ≤ ω ω CK1 . Then DgSp ≈ ( L ) = DgSp ≅ ( ω CK1 ) does not contain a least element.Proof. That
DgSp ≈ ( L ) = DgSp ≅ ( ω CK1 ) follows from Corollary 9 because r ( L ) < ω CK1 and therefore every linear ordering of order type ω CK1 can compute a linearordering of order type r ( L ) .Goncharov, Knight, Harizanov, and Shore [24] characterized the degrees thatcompute maximal well ordered initial segments of the Harrison ordering which hasorder type ω C K ( + η ) . Let H be the family of these degrees. They showed that H coincides with the family of degrees that compute a Π path through Kleene’s O .The family of this degrees on the other hand does not contain a minimal element,in particular, it contains a minimal pair of Turing degrees.Now, clearly every maximal well ordered initial segment of ω CK1 ( + η ) is of ordertype ω CK1 and therefore H ⊆ DgSp ≅ ( ω C K ) . Note that this already implies that DgSp ≅ ( ω CK1 ) does not contain a least element as ω CK1 does not have a computablecopy and H contains a minimal pair. Nevertheless, we show the other inclusion aswell, i.e., DgSp ≅ ( ω C K ) ⊆ H . To see this let A be of order type ω CK1 ; uniformlypartition ω in disjoint infinite, coinfinite sets A i , and fix a computable ordering ≤ I of order type 1 + η on the natural numbers. Since our sequence of sets A i is uniformand computable we get computable bijections f i ∶ A i → A . Define an ordering ≤ on ω by x ≤ y ⇔ ( x, y ∈ A i and f ( x ) ≤ A f ( y )) or ( x ∈ A i , y ∈ A j , i ≠ j and i ≤ I j ) . The ordering defined by ≤ has order-type ω CK1 ( + η ) and deg ( ≤ ) = deg ( A ) . There-fore, DgSp ≅ ( ω CK1 ) ⊆ H . (cid:3) I-EMBEDDABILITY SPECTRA AND BASES OF SPECTRA 9 Strongly locally finite graphs
A graph G is strongly locally finite if it is the disjoint union of finite graphs,or, equivalently, if all of its connected components are finite. In what follows let F = ⟨ F i ⟩ i ∈ ω be a Friedberg enumeration of the finite connected graphs. We mayassume without loss of generality that F is such that we can compute the size ∣ F i ∣ of every graph F i uniformly in i . Given x ∈ G , let [ x ] G be the atomic diagram ofthe component of x and denote by ⌜[ x ] G ⌝ the number i such that ⌜[ x ] G ⌝ = F i (if G is clear from the context we omit the subscript).The trace of a graph is the set of indices of finite graphs embeddable into G , i.e., tr ( G ) = { i ∶ F i ↪ G } . The components of G form a preordering P G under embeddability, i.e., for x, y ∈ G [ x ] ≤ P G [ y ] ∶ ⇔ [ x ] ↪ [ y ] . We denote by c ( G ) the set of components of G , i.e., c ( G ) = { i ∶ F i is isomorphic to a component of G } . A component of G is open if it belongs to an infinite ascending chain of P G , andopen ( G ) is the subset of c ( G ) containing all open components of G .We first state some computability theoretic facts about the relations introducedabove. Proposition 11.
Given a strongly locally finite graph G and x, y ∈ G , (1) y ∈ [ x ] G , tr ( G ) are Σ G , (2) and [ x ] ↪ [ y ] , ∣[ x ]∣ ≤ ∣[ y ]∣ , [ x ] ≅ [ y ] , c ( G ) are Σ G .Proof. Ad item (1): For x ∈ G , [ x ] G is definable by the following Σ formula. y ∈ [ x ] G ⇔ ⋁ n ∈ ω ∃ u , . . . u n ⋀ ≤ i ≠ j ≤ n u i Eu j Given x ∈ G with ∣[ x ]∣ = n , let D ([ x ])( x , . . . , x n ) be the formula obtained byreplacing every constant in the atomic diagram of [ x ] by a variable. Note thatgiven n we can computably define D ([ x ])( x , . . . , x n ) and that for F i we can obtain n computably. Thus the trace of G is definable by the following Σ formula. x ∈ tr ( G ) ⇔ ∃ x , . . . , x n D ( F x )( x . . . x n ) Ad item (2): In general, given x ∈ G the size of its component [ x ] is Σ as ∣[ x ]∣ = n ⇔ ∃ x , . . . x n ⋁ ≤ i ≤ n x i ∈ [ x ] ∧ ∀ y ( ⋁ ≤ i ≤ n x i ≠ y → ⋁ ≤ i ≤ n ¬ x i Ey ) . Then [ x ] ↪ [ y ] ⇔ ⋁ n ∈ ω ∣[ x ]∣ = n ∧ ∃ y , . . . y n D ∃ ([ x ])( y , . . . , y n ) , which is Σ . Thus also ∣[ x ]∣ ≤ ∣[ y ]∣ is Σ and [ x ] ≅ [ y ] ⇔ [ x ] ↪ [ y ] ∧ [ y ] ↪ [ x ] as [ x ] and [ y ] are finite; hence, it is also Σ . By definition, x ∈ c ( G ) if and only if ∃ y ∈ G F x ≅ [ y ] which by the above arguments is Σ . (cid:3) Definition 6.
A graph G is open-ended if every component of G is open.We say that a graph S G is the skeleton of G if S G ≅ ⋃ i ∈ tr (G) F i . It is not hard tosee that two bi-embeddable graphs A , B have the same trace, and thus the sameskeleton. For open-ended strongly locally finite graphs the skeletons form a basis. Theorem 12.
Let G be an open-ended strongly locally finite graph, then S G is ab.e. basis of G .Proof. We first show that G and S G are bi-embeddable given that G is open-ended.Given enumerations of the components of G and S G , say we have defined an em-bedding µ on the first s components of the enumeration of G and want to define itfor the component with index s + G is open-ended, so is S G ; thus, there is a component which is disjoint from the range of µ and in whichthe component with index s + µ accordingly. It is then not hardto see that in the limit µ is an embedding of G in S G . By the same argument wecan embed S G in G .By Proposition 6, it remains to show that S G is minimal with respect to Muchnikreducibility, i.e., that every A ≈ G computes a copy of S G . By Proposition 11, tr ( A ) is Σ A . Let W A e = tr ( A ) and W A e,s the approximation to W A e at stage s . We constructthe copy of S G in stages. At every stage s check if any i < s enters W A e,s and if sobuild a component isomorphic to F i using elements bigger than s not yet usedduring the construction. As the construction is A -computable and tr ( A ) = tr ( G ) ,the constructed structure is an A -computable copy of S G . (cid:3) Notice that we can reformulate Theorem 12 as follows. For any open-endedgraph G , we have thatDgSp ≈ ( G ) = { deg ( Y ) ∶ tr ( G ) is c.e. in Y } . This is close to the definition of enumeration degree of a structure S as given byMontalb´an [25] in the spirit of Knight [26]. Definition 7.
A structure S has enumeration degree X ⊆ ω if the following holdsDgSp ≅ ( S ) = { deg ( Y ) ∶ X is c.e. in Y } . Related to this is the notion of the jump degree of a structure.
Definition 8.
A structure S has jump degree X ⊆ ω if deg ( X ) is the least degreein DgSp ′≅ ( S ) = { d ′ ∶ d ∈ DgSp ( A )} The set DgSp ′≅ ( S ) is often called the jump spectrum of S .Coles, Downey, and Slaman [27] showed that for any set X ⊆ ω the set { d ′ ∶ X is c.e. in d } has a minimum. It follows from this that a structure has jumpdegree if it has enumeration degree. We assume without loss of generality that no i may enter W e,s at a stage s smaller than i I-EMBEDDABILITY SPECTRA AND BASES OF SPECTRA 11
Examples of classes of structures always having an enumeration degree are alge-braic fields (see Frolov, Kalimullin, and Miller [28]) and connected, finite-valence,pointed graphs (see Steiner [29]). Bi-embeddability spectra of open-ended graphsare therefore similar to isomorphism spectra of structures in these classes.
Theorem 13. (1)
For every X ⊆ ω there is an open ended graph G such that tr ( G ) ≡ e X . (2) For all open-ended G , DgSp ′≈ ( G ) = { d ′ ∶ d ∈ DgSp ≈ ( G )} is a cone of degrees.Proof. The idea of the proof is similar to that given in [28, Corollary 1].(1) Let X ⊆ ω and define G to be the graph conisting of a cycle of length n forevery n ∈ X .We have tr ( G ) ≡ e X . Indeed, to enumerate X from an enumerationof tr ( G ) , enumerate tr ( G ) and for every x ∈ tr ( G ) check in a c.e. way if F x is a cycle. If so enumerate the length of the cycle. Clearly this is anenumeration of X . On the other hand given an element x ∈ X , consider thetrace of the cycle of length x and enumerate it. By Proposition 11 this isc.e. Thus, given an enumeration of X we can produce an enumeration of tr ( G ) .(2) Given an open-ended G , by the above mentioned result by Coles, Downey,and Slaman [27], the set of jumps of degrees enumerating tr ( G ) has a min-imum. By Theorem 12 this is DgSp ′≈ ( G ) . (cid:3) Corollary 14.
There is a open-ended graph such such that
DgSp ≈ ( G ) does nothave a least element.Proof. Take X ⊆ ω to be non-total. It follows that the set of Turing degreesenumerating X does not have a least element. Then by (1) of Theorem 13 and theobservation after Theorem 12 we get that there is G such that DgSp ≈ ( G ) = { deg ( Y ) ∶ tr ( G ) is c.e. in Y } = { deg ( Y ) ∶ X is c.e. in Y } . Therefore DgSp ≈ ( G ) does not have a least element. (cid:3) It is immediate from the construction in item (1) that G ≡ T tr ( G ) ≡ T D . Thus G has enumeration degree with respect to its bi-embeddability type and, as it is b.e.trivial, also with respect to its isomorphism type.5. Open question
We close by stating a few open question which make for interesting furtherresearch.
Question . Is there a bi-embeddability spectrum that is not an isomorphism spec-trum and vice versa?
Question . Is there a structure having finite bi-embeddability basis greater than1?
Question . Is there a structure having countable bi-embeddability basis?
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Institute of Discrete Mathematics and Geometry, Technische Universit¨at Wien,Wiedner Hauptstraße 8-10/104 Vienna, Austria
E-mail address : [email protected] Institute of Discrete Mathematics and Geometry, Technische Universit¨at Wien,Wiedner Hauptstraße 8-10/104 Vienna, Austria
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