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

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