Degree spectra and co-spectra of structures

Given a countable structure $\A$, we define the degree spectrum $DS(\A)$ of $\A$ to be the set of all enumeration degrees generated by the presentations of $\A$ on the natural numbers. The co-spectrum of $\A$ is the set of all lower bounds of $DS(\A)$. We prove some general properties of the degree spectra, which show that they behave with respect to their co-spectra very much like the cones of enumeration degrees. Among the results are the analogs of Selman's Theorem \cite{Selman}, the Minimal Pair Theorem and the existence of a quasi-minimal enumeration degree.

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