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A splitting theorem for degrees

Authors:	Richard A Shore Theodore A Slaman

Institution:	Department of Mathematics, Cornell University, Ithaca, New York 14853 ; Department of Mathematics, University of California, Berkeley, California 94720-3840

Abstract:	We prove that, for any , and 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 and from $\{0,1\}$ and any set , 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 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.

Keywords:	Degrees Turing degrees recursively enumerable degrees splitting theorems

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