We show that every nonzero $\Sigma _{2}^{0}$ enumeration degree bounds a nonsplitting nonzero enumeration degree.

We show that every nonzero $${\Delta^{0}_{2}}$$ enumeration degree bounds the enumeration degree of a 1-generic set. We also point out that the enumeration degrees of 1-generic sets, below the first jump, are not downwards closed, thus answering a question of Cooper.

We show that in the language of {≤}, the Π₃-fragment of the first order theory of the $\Sigma _{2}^{0}$-enumeration degrees is undecidable. We then extend this result to show that the Π₃-theory of any substructure of the enumeration degrees which contains the $\Delta _{2}^{0}$-degrees is undecidable.

We show that there is a first order sentence φ(x; a, b, l) such that for every computable partial order.

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, XsA.

We give an alternative and more informative proof that every incomplete ${\Sigma^{0}_{2}}$ -enumeration degree is the meet of two incomparable ${\Sigma^{0}_{2}}$ -degrees, which allows us to show the stronger result that for every incomplete ${\Sigma^{0}_{2}}$ -enumeration degree a, there exist enumeration degrees x 1 and x 2 such that a, x 1, x 2 are incomparable, and for all b ≤ a, b = (b ∨ x 1 ) ∧ (b ∨ x 2 ).