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 ιs and ι̂s of, respectively, De and DT (the Turing degrees) into Ds, and we show that the image of DT under ι̂s 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 ιs preserves the latter, as also other naturally arising properties such as that of totality or of being Γn, for Γ ∈ {Σ,Π,∆} and n > 0. We prove that the good enumeration and singleton degrees are immune and that the good Σ2 singleton degrees are hyperimmune. Finally we show that, for singleton degrees as < bs such that bs is good, any countable partial order can be embedded in the interval (as, bs).