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Splitting properties of r.e. sets and degrees1,2

Published online by Cambridge University Press:  12 March 2014

R. G. Downey  and
L. V. Welch
R. G. Downey
Affiliation:
Department of Mathematics, National University of Singapore, Kent Ridge, 0511, Singapore
L. V. Welch
Affiliation:
Department of Mathematics, Western Illinois University, Macomb, Illinois 61455
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Extract

A pair of r.e. sets A1, A2 are said to split an r.e. set A (written A1 Δ A2 = A) if A1 ∩ A2 = ∅ and A1 ∪ A2 = A. In the literature there are various results asserting certain splitting properties hold for all r.e. sets. For example Sacks' splitting theorem (cf. [So]) asserts that an r.e. nonrecursive set A may be split into a pair of Turing incomparable r.e. sets A1, A2, and Lachlan's splitting theorem [La5] asserts that A may be split into a pair of r.e. sets A1, A2 for which there exists an r.e. set B with B ⊕ A1, B ⊕ A2 <TA and with the infinum of the Turing degrees of B ⊕ A1 and B ⊕ A2 existing in the upper semilattice of r.e. degrees.

Two of the earliest observations establishing that splitting properties possessed by some r.e. sets are not possessed by others, are Lachlan's nondiamond theorem [La1] (and so, in particular, no complete r.e. set can be split nontrivially with degree theoretic inf 0) and the Yates-Lachlan construction of a minimal pair (a pair of nonzero r.e. degrees with inf 0). Other examples of this phenomenon, which are not obtained by interpreting degree theoretic results in the r.e. sets, are Lachlan's construction of nonmitotic r.e. sets [La2] (sets which cannot be split into a pair of r.e. sets of the same degree) and, later, Ladner's [Ld1, 2] and Ingrassia's [In] analysis of their degrees.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 51 , Issue 1 , March 1986 , pp. 88 - 109
Copyright
Copyright © Association for Symbolic Logic 1986

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Footnotes

1

The authors would like to thank Mike Ingrassia, Iraj Kalantari, Jeff Remmel and Michael Stob for many helpful discussions.

2

Some of the work presented here was carried out whilst the first author held a visiting position at Western Illinois University, was announced in abstract, and was presented to the Second Biennial Cambridge Logic Meeting, Boston, 1983.

References

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