Relative Splittings of in the-Enumeration Degrees

Summary

Summary. We prove some results on the degrees. We show that splits in the e-degrees above any incomplete degree; on the other hand, one can find a splitting of into e-degrees b and d, such that for any degree

Introduction

A set A is enumeration reducible (e-reducible) to a set B if there exists a recursively enumerable (r. e.) set (called in this context an enumeration operator, or, simply, an e-operator) such that

The structure De of the e-degrees is the structure of the equivalence classes (called e-degrees) of sets of numbers under the equivalence relation generated by the preordering the e-degree of A is denoted by the symbol. De is in fact an uppersemilattice with least element where any r. e. set W: the partial ordering relation of will be denoted by

Throughout the paper we will refer to some fixed acceptable numbering of the (r. e.) sets; we therefore obtain a corresponding listing of the e-operators. We will consider finite recursive approximations to the r.e. sets (in the sense of [20, p.16]). We get corresponding finite recursive approximations to the e-operators. We will indicate by the approximation to defined by where is some fixed finite recursive approximation to K.

We adopt the usual notational conventions: see for instance Soare [20], and Cooper [9]. Given a set F, let and. Thus, We sometimes identify a given finite set with its canonical index, thus writing for instance to denote the number where u is the canonical index of F. If B and then if then

It is known ([16], [19]) that the e-degrees extend the structure of the Turing degrees: the mapping defined by (where and denote the Turing degree and the characteristic function of A, respectively) defines an order theoretic embedding preserving joins and least element.

The degrees

McEvoy ([15]) defines a jump operation on. Of particular interest is the structure (it is known that. It is shown in [8] that S is dense and S coincides with the structure of the (in fact, if and only if). One can distinguish several substructures of S. In particular, the correspond, under the above mentioned embedding, to the r. e. Turing degrees a is a if and only if for some r. e.