The structure of the [formula omitted]-degrees contained within a single [formula omitted]-degree

Abstract

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