Bi‐embeddability spectra and bases of spectra

Fokina, Ekaterina B.; Rossegger, Dino; Mauro, Luca San

doi:10.1002/malq.201800056

Cited by 7 publications

(5 citation statements)

References 30 publications

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“…Informally speaking, the poset CR(L ) contains all computable copies of L , up to computable bi-embeddability. This observation provides a connection to the recent study of bi-embeddability spectra [14] and computable bi-embeddable categoricity [15,16].…”

Section: Pr(s ) = ({Deg Pr (A )supporting

confidence: 63%

Computable Reducibility for Computable Linear Orders of Type ω

2022

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We study computable reducibility for computable isomorphic copies of the standard ordering of natural numbers. Following Andrews and Sorbi, we isolate the class of self-full degrees inside the induced degree structure Ω. We show that, over an arbitrary degree from Ω, there exists an infinite antichain of self-full degrees. This fact implies that the poset Ω has continuum many automorphisms. We prove that any non-self-full degree from Ω has no minimal covers, which implies that, inside Ω, the self-full degrees are precisely those elements that have a minimal cover. Bibliography: 18 titles.The paper studies computable reducibility for computable linear orders. Let R and S be binary relations on the set of natural numbers ω. The relation R is computably reducible to S (denoted by R c S) if there is a total computable function f (x) such thatIn this case, one also says that the function f is a computable reduction from R to S.The systematic study of computable reducibility for positive (computably enumerable) equivalence relations was initiated by Ershov [1] who constructed one of the first examples of a universal positive equivalence. Further developments led to sophisticated classifications for various * To whom the correspondence should be addressed.

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Section: Pr(s ) = ({Deg Pr (A )supporting

confidence: 63%

Computable Reducibility for Computable Linear Orders of Type ω

2022

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“…Knight [16] showed that isomorphism spectra of automorphically trivial structures contain only one Turing degree and that the isomorphism spectra of automorphically nontrivial structures are upwards closed in the Turing degrees. In [7] the authors showed that if A is automorphically trivial and B ≈ A, then B ∼ = A. Thus, as every biembeddability and elementary bi-embeddability spectrum is a union of isomorphism spectra, Knight's result carries over to this setting.…”

Section: Definition 44 a Category C Is Degree Invariant If For Every ...mentioning

confidence: 99%

“…Recently, researchers initiated the study of degree spectra with respect to other model theoretic equivalence relations such as bi-embeddability [7], elementary bi-embeddability [20], elementary equivalence [1–3], or equivalence [8]. One of the main goals in this line of research is to distinguish these equivalence relations with respect to the degree spectra they realize.…”

Section: Introductionmentioning

confidence: 99%

Degree Spectra of Analytic Complete Equivalence Relations

Rossegger¹

2021

J. symb. log.

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We study the bi-embeddability and elementary bi-embeddability relation on graphs under Borel reducibility and investigate the degree spectra realized by these relations. We first give a Borel reduction from embeddability on graphs to elementary embeddability on graphs. As a consequence we obtain that elementary bi-embeddability on graphs is a $\boldsymbol {\Sigma }^1_1$ complete equivalence relation. We then investigate the algorithmic properties of this reduction. We obtain that elementary bi-embeddability on the class of computable graphs is $\Sigma ^1_1$ complete with respect to computable reducibility and show that the elementary bi-embeddability and bi-embeddability spectra realized by graphs are related.

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“…Degree spectra with respect to another natural equivalence relation, that of bi-embeddability, are considered in [4].…”

Section: Introductionmentioning

confidence: 99%

Degree spectra of structures relative to equivalences

Semukhin¹,

Turetsky²,

Fokina³

2019

Algebra Logika

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A standard way to capture the inherent complexity of the isomorphism type of a countable structure is to consider the collection of all Turing degrees relative to which a given structure has a computable isomorphic copy. This set is called the degree spectrum of structure. Similarly, to characterize the complexity of models of a theory, one may consider the collection of all degrees relative to which the theory has a computable model. In this case we get the spectrum of the theory. In this paper we generalize these two notions to arbitrary equivalence relations. For a structure A and an equivalence relation E, we define the degree spectrum DgSp(A, E) of A relative to E to be the set of all degrees capable of computing a structure B that is E-equivalent to A. Then the standard degree spectrum of A is DgSp(A, ∼ =) and the degree spectrum of the theory of A is DgSp(A, ≡). We consider the relations ≡ Σn (A ≡ Σn B iff the Σ n theories of A and B coincide) and study degree spectra with respect to ≡ Σn .

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Bi‐embeddability spectra and bases of spectra

Cited by 7 publications

References 30 publications

Computable Reducibility for Computable Linear Orders of Type ω

Computable Reducibility for Computable Linear Orders of Type ω

Degree Spectra of Analytic Complete Equivalence Relations

Degree spectra of structures relative to equivalences

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