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Degree theoretical splitting properties of recursively enumerable sets

Published online by Cambridge University Press:  12 March 2014

Klaus Ambos-Spies  and
Peter A. Fejer
Klaus Ambos-Spies*
Affiliation:
Lehrstuhl Informatik II, Universität Dortmund, 4600 Dortmund 50, Bundesrepublik Deutschlands
Peter A. Fejer
Affiliation:
Department of Mathematics and Computer Science, University of Massachusetts at Boston, Boston, Massachusetts 02125
*
Fachbereich Informatik, Universität Oldenburg, D-2900 Oldenburg, West Germany
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A recursively enumerable splitting of an r.e. set A is a pair of r.e. sets B and C such that A = B ∪ C and B ∩ C = ⊘. Since for such a splitting deg A = deg B ∪ deg C, r.e. splittings proved to be a quite useful notion for investigations into the structure of the r.e. degrees. Important splitting theorems, like Sacks splitting [S1], Robinson splitting [R1] and Lachlan splitting [L3], use r.e. splittings.

Since each r.e. splitting of a set induces a splitting of its degree, it is natural to study the relation between the degrees of r.e. splittings and the degree splittings of a set. We say a set A has the strong universal splitting property (SUSP) if each splitting of its degree is represented by an r.e. splitting of itself, i.e., if for deg A = b ∪ c there is an r.e. splitting B, C of A such that deg B = b and deg C = c. The goal of this paper is the study of this splitting property.

In the literature some weaker splitting properties have been studied as well as splitting properties which imply failure of the SUSP.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 53 , Issue 4 , December 1988 , pp. 1110 - 1137
Copyright
Copyright © Association for Symbolic Logic 1988

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References

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