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A splitting theorem for $n-REA$ degrees

Authors: Richard A. Shore and Theodore A. Slaman
Journal: Proc. Amer. Math. Soc. 129 (2001), 3721-3728
MSC (2000): Primary 03D25, 03D30; Secondary 03D55
Published electronically: April 25, 2001
MathSciNet review: 1860508
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Abstract: We prove that, for any $D$, $A$ and $U$ with $D>_{T}A\oplus U$ and r.e., in $A\oplus U$, there are pairs $X_{0},X_{1}$ and $Y_{0},Y_{1}$ such that $D\equiv_{T}X_{0}\oplus X_{1}$; $D\equiv_{T}Y_{0}\oplus Y_{1}$; and, for any $i$ and $j$ from $\{0,1\}$ and any set $B$, if $X_{i}\oplus A\geq_{T}B$ and $Y_{j}\oplus A\geq_{T}B$, then $A\geq_{T}B$. We then deduce that for any degrees $\mathbf{d}$, $\mathbf{a}$, and $\mathbf{b}$ such that $\mathbf{a}$and $\mathbf{b}$ are recursive in $\mathbf{d}$, $\mathbf{a}\not \geq _{T}\mathbf{b}$, and $\mathbf{d}$ is $n-REA$ into $\mathbf{a}$, $\mathbf{d}$can be split over $\mathbf{a}$ avoiding $\mathbf{b}$. This shows that the Main Theorem of Cooper (Bull. Amer. Math. Soc. 23 (1990), 151-158) is false.

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Additional Information

Richard A. Shore
Affiliation: Department of Mathematics, Cornell University, Ithaca, New York 14853

Theodore A. Slaman
Affiliation: Department of Mathematics, University of California, Berkeley, California 94720-3840

Keywords: Degrees, Turing degrees, recursively enumerable degrees, splitting theorems
Received by editor(s): November 15, 1999
Received by editor(s) in revised form: May 4, 2000
Published electronically: April 25, 2001
Additional Notes: The first author was partially supported by NSF Grant DMS-9802843.
The second author was partially supported by NSF Grant DMS-97-96121.
Communicated by: Carl G. Jockusch, Jr.
Article copyright: © Copyright 2001 American Mathematical Society

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