The degrees of r.e. sets without the universal splitting property
Rodney G. Downey · Transactions of the American Mathematical Society · 1985
It is shown that every nonzero r.e. degree contains an r.e. set without the universal splitting property. That is, if δ \delta is any r.e. nonzero degree, there exist r.e. sets ∅ > T B > T A \emptyset > {}_TB > {}_TA with deg ( A ) = δ \deg (A) = \delta such that if A 0 ⊔ A 1 {A_0} \sqcup {A_1} is an r.e. splitting of A A , then A 0 ≢ T B {A_0} ot \equiv {}_TB . Some generalizations are discussed.