A SPLITTING THEOREM FOR n−REA DEGREES

We prove that, for any D, A and U with D >T A ⊕ U and r.e. in A⊕ U , there are pairs X0, X1 and Y0, Y1 such that D ≡T X0 ⊕X1; D ≡T Y0 ⊕ Y1; and, for any i and j from {0, 1} and any set B, if Xi ⊕A ≥T B and Yj ⊕ A ≥T B then A ≥T B. We then deduce that for any degrees d, a, and b such that a and b are recursive in d, a 6≥T b, and d is n − REA in a, d can be split over a avoiding b. This shows that the Main Theorem of Cooper [1990] and [1993] is false.