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

Working toward showing the decidability of the $\forall \exists $ -theory of the ${\Sigma ^0_2}$ -enumeration degrees, we prove that no so-called Ahmad pair of ${\Sigma ^0_2}$ -enumeration degrees can join to ${\mathbf 0}_e'$.

Using properties of${\cal K}$-pairs of sets, we show that every nonzero enumeration degreeabounds a nontrivial initial segment of enumeration degrees whose nonzero elements have all the same jump asa. Some consequences of this fact are derived, that hold in the local structure of the enumeration degrees, including: There is an initial segment of enumeration degrees, whose nonzero elements are all high; there is a nonsplitting high enumeration degree; every noncappable enumeration degree is high; every nonzero low enumeration degree can be (...) capped by degrees of any possible local jump ; every enumeration degree that bounds a nonzero element of strictly smaller jump, is bounding; every low enumeration degree below a non low enumeration degreeacan be capped belowa. (shrink)