The structure of the s-degrees contained within a single e-degree

For any enumeration degree a let Ds a be the set of s-degrees contained in a. We answer an open question of Watson by showing that if a is a nontrivial Σ2-enumeration degree, then Ds a has no least element. We also show that every countable partial order embeds into Ds a. Finally, we construct Σ 0 2-sets A and B such that B ≤e A but for every X ≡e B, X s A.