Given a (non-computable) $\Pi^0_2$ singleton $Y$ are there Turing incomparable $\Pi^0_2$ singletons $X_0, X_1$ with $Y \equiv_T X_0 \oplus X_1$?
What about the same question for arithmetic reducibility?
Splitting $\Pi^0_2$ Singletons?
Peter Gerdes
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