Abstract
We investigate and extend the notion of a good approximation with respect to the enumeration \({({\mathcal D}_{\rm e})}\) and singleton \({({\mathcal D}_{\rm s})}\) degrees. We refine two results by Griffith, on the inversion of the jump of sets with a good approximation, and we consider the relation between the double jump and index sets, in the context of enumeration reducibility. We study partial order embeddings \({\iota_s}\) and \({\hat{\iota}_s}\) of, respectively, \({{\mathcal D}_{\rm e}}\) and \({{\mathcal D}_{\rm T}}\) (the Turing degrees) into \({{\mathcal D}_{\rm s}}\) , and we show that the image of \({{\mathcal D}_{\rm T}}\) under \({\hat{\iota}_s}\) is precisely the class of retraceable singleton degrees. We define the notion of a good enumeration, or singleton, degree to be the property of containing the set of good stages of some good approximation, and we show that \({\iota_s}\) preserves the latter, as also other naturally arising properties such as that of totality or of being \({\Gamma^0_n}\) , for \({\Gamma \in \{\Sigma,\Pi,\Delta\}}\) and n > 0. We prove that the good enumeration and singleton degrees are immune and that the good \({\Sigma^0_2}\) singleton degrees are hyperimmune. Finally we show that, for singleton degrees a s < b s such that b s is good, any countable partial order can be embedded in the interval (a s, b s).
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Research supported by EPSRC research grant No. EP/G000212, Computing with Partial Information: Definability in the Local Structure of the Enumeration Degrees.
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Harris, C.M. Goodness in the enumeration and singleton degrees. Arch. Math. Logic 49, 673–691 (2010). https://doi.org/10.1007/s00153-010-0192-9
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DOI: https://doi.org/10.1007/s00153-010-0192-9