Genericity and Non-bounding in the Enumeration degrees

On Kalimullin pairs

We study Kalimullin pairs, a definable class (of pairs) of enumeration degrees that has been used to give first-order definitions of other important classes and relations, including the enumeration jump and the total enumeration degrees. We show that the global definition of Kalimullin pairs is also valid in the local structure of the enumeration degrees, giving a simpler local definition than ...

Enumeration 1-genericity in the Local Enumeration Degrees

We discuss a notion of forcing that characterises enumeration 1genericity and we investigate the immunity, lowness and quasiminimality properties of enumeration 1-generic sets and their degrees. We construct an enumeration operator ∆ such that, for any A, the set ∆ is enumeration 1-generic and has the same jump complexity as A. We also prove that every nonzero ∆ 2 degree bounds a nonzero enumer...

Bounding nonsplitting enumeration degrees

Bounding and nonbounding minimal pairs in the enumeration degrees

We show that every nonzero ∆ 2 e-degree bounds a minimal pair. On the other hand, there exist Σ 2 e-degrees which bound no minimal pair.

A Generic Set That Does Not Bound a Minimal Pair

The structure of the semi lattice of enumeration degrees has been investigated from many aspects. One aspect is the bounding and nonbounding properties of generic degrees. Copestake proved that every 2-generic enumeration degree bounds a minimal pair and conjectured that there exists a 1-generic set that does not bound a minimal pair. In this paper we verify this longstanding conjecture by cons...