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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.

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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.

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  • $\begingroup$ I’m sure I’m missing something (and maybe I used wrong terminology) but how does that get me a B r.e. in A (but not equivalent to A') such that B+A + 0' is equivalent to A'? Or so that B+A and 0' form a minimal paid over A? $\endgroup$
    – Peter Gerdes
    Commented Jul 13, 2020 at 15:43
  • $\begingroup$ Ohh, is your suggestion I choose e to be the index of a construction such that H_e(X) and H_j(X) always yield a minimal paid over X or non-trivially join to X' and now I get 0' to be one half of that pair. Hm, ok yah those are both uniform constructions. THANKS!! $\endgroup$
    – Peter Gerdes
    Commented Jul 13, 2020 at 15:52
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Just in case anyone else is a mere mortal and takes a moment to understand Slaman's answer here is a more spelled out version of his argument. I'll just do the capping side as it's the same argument for cupping

The construction of an r.e. minimal pair (for the other side an incompatible pair of r.e. degrees whose join is $0'$) is uniform. Therefore, there are hops $H_e$ and $H_j$ such that $H_e(X) \wedge H_j(X) = X$ (non-trivially). Now, by the theorem in the psudeojump paper we can invert a hop, i.e., we can find r.e. A such that $H_e(A)= 0'$ providing the desired $A$.

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