Spectral Universality of Linear Orders with One Binary Relation

Zubkov, M. V.; Frolov, A. N.

doi:10.1134/s0037446620030076

Cited by 1 publication

(1 citation statement)

References 9 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…As usual, η denotes the (standard ordering of) rationals. Zubkov and Frolov [20] proved that for any countable graph G, there exists a binary relation R on (η, <) such that the spectrum DgSp (η,<) (R) is equal to the degree spectrum of the structure G. For further results on degree spectra of relations, we refer an interested reader to, e.g., Section 7 of [8].…”

Section: Related Workmentioning

confidence: 99%

Degrees of relations on canonically ordered natural numbers and integers

Wrocławski

Bazhenov

Kalociński

2023

Preprint

View full text Add to dashboard Cite

We investigate the degree spectra of computable relations on canonically ordered natural numbers (ω,<) and integers (ζ,<). As for (ω,<), we provide several criteria that fix the degree spectrum of a computable relation to all c.e. or to all Δ2 degrees; this includes the complete characterization of the degree spectra of so-called computable block functions that have only finitely many types of blocks. Compared to [1], we obtain a more general solution to the problem regarding possible degree spectra on (ω,<), answering the question whether there are infinitely many such spectra. As for (ζ,<), we prove the following dichotomy result: given an arbitrary computable relation R on (ζ,<), its degree spectrum is either trivial or it contains all c.e. degrees. This result, and the proof techniques required to solve it, extend the analogous theorem for (ω,<) obtained in [2], and provide initial insight to Wright's question whether such a dichotomy holds on computable ill-founded linear orders. This article is an extended version of [1]. MSC Classification: 03C57

show abstract

Section: Related Workmentioning

confidence: 99%