Jump inversions of algebraic structures and Σ‐definability

Faĭzrahmanov, M. Kh.; Kach, Asher M.; Kalimullin, I. Sh.; Montalbán, Antonio; Puzarenko, V. G.

doi:10.1002/malq.201800015

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Be Triviality1

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“…Soskov gave an example of an isomorphism spectrum of a structure

scriptA

such that

{prefixDgSp}_{≅} false(scriptA false) \subseteq false{boldd : boldd \geq 0^{false(ω false)} false}

and showed that no structure has

{d : {boldd}^{(ω)} \in {prefixDgSp}_{≅} false(scriptA false)}

as its isomorphism spectrum . Faizrahmanov, Kach, Kalimullin, and Montalbán recently showed that no structure realizes the family

{d : {boldd}^{(ω)} \geq {bolda}^{(ω)}}

for

a \geq {bold0}^{(ω)}

as its isomorphism spectrum .…”

Section: Be Trivialitymentioning

confidence: 99%

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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“…Soskov gave an example of an isomorphism spectrum of a structure