Goodness in the enumeration and singleton degrees

We investigate and extend the notion of a good approximation with respect to the enumeration (De) and singleton (Ds) 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 s and ^s of, respectively, De and DT (the Turing degrees) into Ds, and we show that the image of DT under ^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 s preserves the latter, as also other naturally arising properties such as that of totality or of being 0 n, for {, , } and n > 0. We prove that the good enumeration and singleton degrees are immune and that the good 0 2 singleton degrees are hyperimmune. Finally we show that, for singleton...

• Type: Journal
• Year: 2010
• Where: AML
• Authors: Charles M. Harris
• Keywords: AML 2010, Enumeration, Partial Order Embeddings, Retraceable Singleton Degrees