Splitting properties of r.e. sets and degrees
R. G. Downey, Lawrence Welch · Journal of Symbolic Logic · 1986
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