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? EDIT: Wait I've solved that one with an arithmeticly minimal singleton.
It seems to me that every $\Delta^0_2$-set $X$ is a $\Pi^0_2$-singleton. Namely, $X$ satisfies the statement "for all $n$, $n\in X$ implies the $\Pi^0_2$-condition for membership in $X$ and $n\notin X$ implies the $\Pi^0_2$-condition for membership in the complement of $X$." Then, Sacks's example of a set of minimal Turing degree below $0'$ (hence $\Delta^0_2$) is a $\Pi^0_2$-singleton whose Turing degree cannot be split into two smaller degrees.
