Kalimullin Pairs of Σ2 Ω-enumeration Degrees

We study the notion of K-pairs in the local structure of the ωenumeration degrees. We introduce the notion of super almost zero sequences and investigate their structural properties. The study of degree structures has been one of the central themes in computability theory. Although the main focus has been on the structure of the Turing degrees and its local substructure, of the degrees below the first jump of the least degree, significant work has been done to examine the properties of an extension of the Turing degrees, the structure of the enumeration degrees. Enumeration reducibility introduced by Friedberg and Rogers [3] arises as a way to compare the computational strength of the positive information contained in sets of natural numbers. A set A is enumeration reducible to a set B if given any enumeration of the set B, one can effectively compute an enumeration of the set A. The induced structure of the enumeration degrees De is an upper semilattice with least element and jump operation. As we mention above, this structure can be viewed as an extension of the structure of the Turing degrees, due to an embedding ι : DT → De which preserves the order, the least upper bound and the jump operation. The local structure of the enumeration degrees, consisting of all degrees below the first enumeration jump of the least enumeration degree, Ge, can therefore in turn be seen as an extension of the local structure of the Turing degrees. The two structures, DT and De, as well as their local substructures, are closely related in algebraic properties, definability strength and in the techniques and methods used to study them. Results proved in one of the structures reveal properties of the other, methods used to study one of the structures suggest similar methods for the other and vice versa. A proof technique which arises from the study of the structure of the enumeration degrees is the use of the following notion. 0.1. Definition.[Kalimullin] A pair of sets of natural numbers A and B is a K-pair over a set U if there is a set W ≤e U such that: A×B ⊆W & A×B ⊆W. The notion of a K-pair over U , originally known as a U -e-ideal, was introduced and used by Kalimullin to prove the definability of the jump operation in the global structure De. In [7] Kalimullin proves that the property of being a K-pair over U is degree theoretic and first order definable in the global structure De. A pair of sets A and B form a K-pair over a set U if and only if their degrees a = de(A) and This research was supported by an BNSF Grant No. D002-258/18.12.08 and by a Marie Curie European Reintegration Grant No. 239193 within the 7th European Community Framework Programme.