## A characterization of the $$0$$-basis homogeneous bounding degrees.(English)Zbl 1213.03045

A countable model $$\mathcal A$$ has a $$\mathbf 0$$-basis if the types realized in $$\mathcal A$$ are uniformly computable, and $$\mathcal A$$ has a ($$\mathbf d$$-)decidable copy if there exists a model $$\mathcal B \cong \mathcal A$$ such that the elementary diagram of $$\mathcal B$$ is ($$\mathbf d$$-)computable. Let $$\mathbf d \leq\mathbf 0'$$ be any low$$_{2}$$ degree. In the paper under review, the author proves that there exists a homogeneous model $$\mathcal A$$ with a $$\mathbf 0$$-basis without a $$\mathbf d$$-decidable copy. This result extends a result by Goncharov, Millar and Peretyat’kin. The author also obtains an exact characterization of the $$\mathbf 0$$-basis homogeneous bounding $$\Delta _{2}^{0}$$ degrees. (A degree $$\mathbf d$$ is $$\mathbf 0$$-basis homogeneous bounding if any homogeneous $$\mathcal A$$ with a $$\mathbf 0$$-basis has a $$\mathbf d$$-decidable copy.)

### MSC:

 03C57 Computable structure theory, computable model theory 03D30 Other degrees and reducibilities in computability and recursion theory
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### References:

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