## On the jump classes of noncuppable enumeration degrees.(English)Zbl 1215.03056

The upper semilattice of enumeration degrees (e-degrees) is generated by the relation ‘$$A$$ is enumeration reducible to $$B$$’ $$(A \leq _{\text{e}} B)$$, which holds iff there is an effective procedure that takes any enumeration of $$B$$ to an enumeration of $$A$$. The jump of the e-degree of $$A$$ is defined as the e-degree of the characteristic function of the set $$\{e\mid e \in\Phi^A_e\}$$. The Turing degrees are embedded in the e-degrees by a map that takes any set’s Turing degree to the e-degree of its characteristic function. This preserves jump. Thus ‘high$$_m$$’, ‘low$$_n$$’, and ‘cuppable’ may be ascribed in the natural way to e-degrees.
A main theorem shows that for every $$\Sigma^0_2$$ e-degree b there is a noncuppable $$\Sigma^0_2$$ e-degree $${\mathbf a}>{\mathbf{0}}_{\text{e}}$$ with $${\mathbf b}'\leq_{\text{e}}{\mathbf a}'$$ and $${\mathbf a}''\leq_{\text{e}}{\mathbf b}''$$. It follows that for every $$m \geq 0$$ and $$n \geq 1$$ there are noncuppable e-degrees $${\mathbf x},{\mathbf y}< {\mathbf{0}}'_{\text{e}}$$ such that $${\mathbf x}$$ is high$$_{m+1}$$ but not high$$_m$$ and $${\mathbf y}$$ is low$$_{n+1}$$ but not low$$_n$$. Moreover, there is a noncuppable $${\mathbf z}<{\mathbf{0}}'_{\text{e}}$$ with the property that $${\mathbf{0}}^n _{\text{e}}< {\mathbf z}^n< {\mathbf{0}}^{n+1}_{\text{e}}$$ for all $$n$$.

### MSC:

 03D25 Recursively (computably) enumerable sets and degrees 03D30 Other degrees and reducibilities in computability and recursion theory

### Keywords:

enumeration reducibility; Turing degrees; noncuppable
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### References:

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