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We study the global properties of ${\mathcal{D}}_n$, the Turing degrees of the n-r.e. sets. In Theorem 1.5, we show that the first order of ${\mathcal{D}}_n$ is not decidable. In Theorem 1.6, we show that for any two n and m with n < m, ${\mathcal{D}}_n$ is not a Σ1-substructure of ${\mathcal{D}}_m$. [ABSTRACT FROM AUTHOR]

We give an algorithm for deciding whether an embedding of a finite partial order ${\mathcal P}$ into the enumeration degrees of the ${\Sigma_2^0}$-sets can always be extended to an embedding of a finite partial order ${\mathcal Q} \supset {\mathcal P}$. [ABSTRACT FROM AUTHOR]