Pseudojump inversion in special r. b. $Π^0_1$ classes

The Jump Inversion Theorem says that for every real $A \ge_T 0'$ there is a real $B$ such that $A \equiv_T B' \equiv_T B \oplus 0'$. A known refinement of this theorem says that we can choose $B$ to be a member of any special $Π^0_1$ subclass of $\{0,1\}^ω$. We now consider the possibility of analogous refinements of two other well-known theorems: the Join Theorem -- for all reals $A$ and $Z$ such that $A \ge_T Z \oplus 0'$ and $Z >_T 0$, there is a real $B$ such that $A \equiv_T B' \equiv_T B \oplus 0' \equiv_T B \oplus Z$ -- and the Pseudojump Inversion Theorem -- for all reals $A \ge_T 0'$ and every $e \in ω$, there is a real $B$ such that $A \equiv_T B \oplus W_e^B \equiv_T B \oplus 0'$. We show that in these theorems, $B$ can be found in some special $Π^0_1$ subclasses of $\{0,1\}^ω$ but not in others.

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