Goodness in the enumeration and singleton degrees

We investigate and extend the notion of a good approximation with respect to the enumeration ${({mathcal D}_{rm e})}We investigate and extend the notion of a good approximation with respect to the enumeration (D_e){({mathcal D}_{rm e})} and singleton (D_s){({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 i_s{iota_s} and [^(i)]_s{hat{iota}_s} of, respectively, D_e{{mathcal D}_{rm e}} and D_T{{mathcal D}_{rm T}} (the Turing degrees) into D_s{{mathcal D}_{rm s}} , and we show that the image of D_T{{mathcal D}_{rm T}} under [^(i)]_s{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 i_s{iota_s} preserves the latter, as also other naturally arising properties such as that of totality or of being G⁰_n{Gamma^0_n} , for G ? {S,P,D}{Gamma in {Sigma,Pi,Delta}} and n > 0. We prove that the good enumeration and singleton degrees are immune and that the good S⁰₂{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).