We construct an incomplete 3-c.e. enumeration degree which is maximal among the n-c.e. enumeration degrees for every n with 3 # n # #. Consequently the n-c.e. enumeration degrees are not dense for any such n. We show also that no low n-c.e. e-degree can be maximal among the n-c.e. e-degrees, for 2 # n # #.

This paper continues the project, initiated in [ACK], of describing general conditions under which relative splittings are derivable in the lo-cal structure of the enumeration degrees, for which the Ershov hierarchy provides an informative setting. The main results below include a proof that any high total e-degree below 0 ′ e is splittable over any low e-degree below it, a non-cupping result in the high enumeration degrees which occurs at a low level of the Ershov hierarchy, and a ∅ ′′ ′-priority construction of a Π 0 1 e-degree unsplittable over a 3-c.e. e-degree below it. 1

Abstract. Given a class of structures K and n ∈ ω, we study the dichotomy between there being countably many n-back-and-forth equivalence classes and there being continuum many. In the latter case we show that, relative to some oracle, every set can be weakly coded in the (n − 1)st jump of some structure in K. In the former case we show that there is a countable set of infinitary Πn relations that captures all of the Πn information about the structures in K. In most cases where there are countably many n-back-and-forth equivalence classes, there is a computable description of them. We will show how to use this computable description to get a complete set of computably infinitary Πn formulas. This will allow us to completely characterize the relatively intrinsically Σ 0 n+1 relations in the computable structures of K, and to prove that no Turing degree can be coded by the (n − 1)st jump of any structure in K unless that degree is already below 0 (n−1). 1.

We prove that there exists a noncappable enumeration degree strictly below 0 0 e .