Degree Spectra of Relations on a Cone

Degree Spectra of Relations on a Cone Synopsis

Let $mathcal A$ be a mathematical structure with an additional relation $R$. The author is interested in the degree spectrum of $R$, either among computable copies of $mathcal A$ when $(mathcal A,R)$ is a "e;natural"e; structure, or (to make this rigorous) among copies of $(mathcal A,R)$ computable in a large degree d. He introduces the partial order of degree spectra on a cone and begin the study of these objects. Using a result of Harizanov-that, assuming an effectiveness condition on $mathcal A$ and $R$, if $R$ is not intrinsically computable, then its degree spectrum contains all c.e. degrees-the author shows that there is a minimal non-trivial degree spectrum on a cone, consisting of the c.e. degrees.

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