INITIAL SEGMENTS OF THE ENUMERATION DEGREES

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 (i.e., any jump that can be realized by enumeration degrees of the local structure); 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.

A NOTE ON INITIAL SEGMENTS OF THE ENUMERATION DEGREES

We show that no nontrivial principal ideal of the enumeration degrees is linearly ordered: in fact, below every nonzero enumeration degree one can embed every countable partial order. The result can be relativized above any total degree: if a,b are enumeration degrees, with a total, and a < b, then in the degree interval (a,b), one can embed every countable partial order.

On the jump classes of noncuppable enumeration degrees

We prove that for every Σ20 enumeration degree b there exists a noncuppable Σ20 degree a > 0e such that and . This allows us to deduce, from results on the high/low jump hierarchy in the local Turing degrees and the jump preserving properties of the standard embedding , that there exist Σ20 noncuppable enumeration degrees at every possible—i.e., above low1—level of the high/low jump hierarchy in the context of .

Bounding nonsplitting enumeration degrees

We show that every nonzero enumeration degree bounds a nonsplitting nonzero enumeration degree.

The distribution of properly Σ20 e-degrees

We show that for every enumeration degree a < 0′e there exists an e-degree c such that a ≤ c < 0′e, and all degrees b, with c ≤ b < 0′e, are properly Σ20.

Structural properties and Σ20 enumeration degrees

We prove that each Σ20 set which is hypersimple relative to ∅′ is noncuppable in the structure of the Σ20 enumeration degrees. This gives a connection between properties of Σ20 sets under inclusion and and the Σ20 enumeration degrees. We also prove that some low non-computably enumerable enumeration degree contains no set which is simple relative to ∅′.

Noncappable enumeration degrees below 0e′

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

Jumps of quasi-minimal enumeration degrees

Enumeration reducibility is a reducibility between sets of natural numbers defined as follows: A is enumeration reducible to B if there is some effective operation on enumerations which when given any enumeration of B will produce an enumeration of A. One reason for interest in this reducibility is that it presents us with a natural reducibility between partial functions whose degree structure can be seen to extend the structure of the Turing degrees of unsolvability. In [7] Friedberg and Rogers gave a precise definition of enumeration reducibility, and in [12] Rogers presented a theorem of Medvedev [10] on the existence of what Case [1] was to call quasi-minimal degrees. Myhill [11] also defined this reducibility and proved that the class of quasi-minimal degrees is of second category in the usual topology. As Gutteridge [8] has shown that there are no minimal enumeration degrees (see Cooper [3]), the quasi-minimal degrees are very much of interest in the study of the structure of the enumeration degrees. In this paper we define a jump operator on the enumeration degrees which was introduced by Cooper [4], and show that every complete enumeration degree is the jump of a quasi-minimal degree. We also extend the notion of a high Turing degree to the enumeration degrees and construct a “high” quasi-minimal enumeration degree—a result which contrasts with Cooper's result in [2] that a high Turing degree cannot be minimal. Finally, we use the Sacks' Jump Theorem to characterise the jumps of the co-r.e. enumeration degrees.