A splitting theorem for $n-REA$ degrees

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.