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- Publisher
- Springer Journals
- Copyright
- Copyright © 2010 by Springer-Verlag
- Subject
- Mathematics; Algebra; Mathematics, general; Mathematical Logic and Foundations
- ISSN
- 0933-5846
- eISSN
- 1432-0665
- DOI
- 10.1007/s00153-010-0192-9
- Publisher site
- See Article on Publisher Site
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).
Journal
Archive for Mathematical Logic – Springer Journals
Published: Jun 19, 2010