Degrees of relations on canonically ordered natural numbers and integers

We investigate the degree spectra of computable relations on canonically ordered natural numbers \((\omega ,<)\) $(\omega ,<)$ and integers \((\zeta ,<)\) $(\zeta ,<)$ . As for \((\omega ,<)\) $(\omega ,<)$ , we provide several criteria that fix the degree spectrum of a computable relation to all c.e. or to all \(\Delta _2\) ${\mathrm{\Delta }}_2$ 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 Bazhenov et al. (in: LIPIcs, vol 219, pp 8:1–8:20, 2022), we obtain a more general solution to the problem regarding possible degree spectra on \((\omega ,<)\) $(\omega ,<)$ , answering the question whether there are infinitely many such spectra. As for \((\zeta ,<)\) $(\zeta ,<)$ , we prove the following dichotomy result: given an arbitrary computable relation R on \((\zeta ,<)\) $(\zeta ,<)$ , 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 \((\omega ,<)\) $(\omega ,<)$ obtained by Wright (Computability 7:349–365, 2018), 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 Bazhenov et al. (in: LIPIcs, vol 219, pp 8:1–8:20, 2022).