We construct a class of relations on computable structures whose degree spectra form natural classes of degrees. Given any computable ordinal $\alpha$ and reducibility r stronger than or equal to m-reducibility, we show how to construct a structure with an intrinsically $\Sigma_\alpha$ invariant relation whose degree spectrum consists of all nontrivial $\Sigma_\alpha$ r-degrees. We extend this construction to show that $\Sigma_\alpha$ can be replaced by either $\Pi_\alpha$ or $\Delta_\alpha$.
"Realizing Levels of the Hyperarithmetic Hierarchy as Degree Spectra of Relations on Computable Structures." Notre Dame J. Formal Logic 43 (1) 51 - 64, 2002. https://doi.org/10.1305/ndjfl/1071505769