Elementary Bi-embeddability Spectra of Structures

Rossegger, Dino

doi:10.1007/978-3-319-94418-0_35

Cited by 6 publications

(7 citation statements)

References 17 publications

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“…Indeed, one can show that the usual encoding using “bouquet graphs” (cf. ) has a computable bi‐embeddable copy. However, if we consider the graph

scriptG

obtained by taking the disjoint unions of the graphs

G ({n} \oplus F)

for

{n} \oplus F \in F

we obtain a b.e.…”

Section: Be Trivialitymentioning

confidence: 99%

“…There are also other spectra known to be bi‐embeddability spectra. In , Rossegger observed that for all computable successor ordinals α and β,

{d : {boldd}^{(α)} \geq {bold0}^{(β)}}

is the bi‐embeddability spectrum of a structure; he constructed b.e. trivial structures having such spectra.…”

Section: Be Trivialitymentioning

confidence: 99%

“…In terms of the above definition, the theory spectrum of T is the spectrum of a model

scriptA

of T under elementary equivalence,

{prefixDgSp}_{\equiv} false(scriptA false)

. Recently, Rossegger investigated elementary bi‐embeddability spectra of structures,

{prefixDgSp}_{≊} false(scriptA false)

. Two structures are elementary bi‐embeddable if either is elementary embeddable in the other.…”

Section: Introductionmentioning

confidence: 99%

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Bi‐embeddability spectra and bases of spectra

Fokina

Rossegger

Mauro

2019

Mathematical Logic Qtrly

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We study degree spectra of structures with respect to the bi‐embeddability relation. The bi‐embeddability spectrum of a structure is the family of Turing degrees of its bi‐embeddable copies. To facilitate our study we introduce the notions of bi‐embeddable triviality and basis of a spectrum. Using bi‐embeddable triviality we show that several known families of degrees are bi‐embeddability spectra of structures. We then characterize the bi‐embeddability spectra of linear orderings and study bases of bi‐embeddability spectra of strongly locally finite graphs.

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“…Indeed, one can show that the usual encoding using “bouquet graphs” (cf. ) has a computable bi‐embeddable copy. However, if we consider the graph

scriptG

obtained by taking the disjoint unions of the graphs

G ({n} \oplus F)

for

{n} \oplus F \in F

we obtain a b.e.…”

Section: Be Trivialitymentioning

confidence: 99%

“…There are also other spectra known to be bi‐embeddability spectra. In , Rossegger observed that for all computable successor ordinals α and β,

{d : {boldd}^{(α)} \geq {bold0}^{(β)}}

is the bi‐embeddability spectrum of a structure; he constructed b.e. trivial structures having such spectra.…”

Section: Be Trivialitymentioning

confidence: 99%

“…In terms of the above definition, the theory spectrum of T is the spectrum of a model

scriptA

of T under elementary equivalence,

{prefixDgSp}_{\equiv} false(scriptA false)

. Recently, Rossegger investigated elementary bi‐embeddability spectra of structures,

{prefixDgSp}_{≊} false(scriptA false)

. Two structures are elementary bi‐embeddable if either is elementary embeddable in the other.…”

Section: Introductionmentioning

confidence: 99%

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Bi‐embeddability spectra and bases of spectra

Fokina

Rossegger

Mauro

2019

Mathematical Logic Qtrly

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“…We therefore jump invert using pairs of well-orderings. This technique has recently been used in [Chi+09;Baz17;Ros18]. In what follows we fix a path P through Kleene's O and identify computable ordinals with their notations on this path.…”

Section: Examples Of Bi-embeddable Categoricity Spectramentioning

confidence: 99%

Degrees of bi-embeddable categoricity of equivalence structures

et al. 2018

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We investigate the complexity of embeddings between bi-embeddable structures. In analogy with categoricity spectra, we define the biembeddable categoricity spectrum of a structure A as the family of Turing degrees that compute embeddings between any computable bi-embeddable copies of A; the degree of bi-embeddable categoricity of A is the least degree in this spectrum (if it exists). We extend many known results about categoricity spectra to the case of bi-embeddability. In particular, we exhibit structures without degree of bi-embeddable categoricity, and we show that every degree d.c.e above 0 (α) for α a computable successor ordinal and 0 (λ) for λ a computable limit ordinal is a degree of bi-embeddable categoricity. We also give examples of families of degrees that are not bi-embeddable categoricity spectra.

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“…Recently, researchers initiated the study of degree spectra with respect to other model theoretic equivalence relations such as bi-embeddability [FRS19], elementary bi-embeddability [Ros18], elementary equivalence [AM15; And+17; AK13], or Σ n equivalence [FRS17]. 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

2020

Preprint

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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 this 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 Σ Σ Σ 1 1 complete equivalence relation. We then investigate the algorithmic properties of this reduction to show that every bi-embeddability spectrum of a graph is the jump spectrum of an elementary bi-embeddability spectrum of a graph.

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Elementary Bi-embeddability Spectra of Structures

Cited by 6 publications

References 17 publications

Bi‐embeddability spectra and bases of spectra

Bi‐embeddability spectra and bases of spectra

Degrees of bi-embeddable categoricity of equivalence structures

Degree spectra of analytic complete equivalence relations

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