Is there an r.e. set $A$ such that 0’ is cuppable relative to $A$? What about cappable?


This is equivalent to asking if there is an r.e. $A$ such that 0’ is one half of a pair of $A$ r.e. non-$A$ computable sets whose meet is $A$ and similarly if there is an $A$ such that 0’ can be (non-trivially) joined to $A'$ via an $A$ r.e. set.


I’m pretty sure I’ve seen results on this and I’d hazard a guess they might even be in Odifreddi but it’s really hard to search for since 0’ turns up lots of false positives as do cupping and capping plus you have to decide to search for capping, cappable or minimal pair. If this has been asked here before I apologize but same problem.



Cupping and capping for 0’ relative to a recursively enumerable set
You could look at the Jockusch and Shore papers on pseudo jump operators. They showed that for every $e$ there is an r.e. $A$ such that $A+W^A\_e$ is Turing equivalent to $0’$. So $0’$ can have the behaviors that you mentioned relative to r.e. sets.


