The hop $H\_e$ is defined by $H\_e(X) = X \oplus W\_e^{X}$. A 2-REA operator (or double hop) $J\_{\langle e,i\rangle}$ is defined by $J\_{\langle e,i\rangle}(X) = H\_e(H\_i(X))$


By a famous result from [Pseudo Jump Operators. I: The R. E. Case](https://doi.org/10.2307/1999041) by Jockusch and Shore it's known that for any hop $H\_e$ there is an r.e. set $A$ such that $H\_e(A) \equiv\_T 0'$. Or, as this holds uniformly, there is a computable function $f$ such that for any $e$ $H\_e(H\_{f(e)}(\emptyset)) \equiv\_T 0'$.


I'm wondering if this extends to 2-REA operators. In other words, given $J$ a 2-REA operator does there always exist a 2-REA set (i.e. $H\_e(H\_i(\emptyset))$ for some $e$, $i$) $A$ such that $J(A) \equiv\_T 0''$?


I'm guessing no. I bet there's a slick proof of this but it's not coming to me.


(Note that the claim holds if you allow $A$ to be an arbitrary set.)



Double Hop Inversion Theorem
Ohh, I think I was being dumb. There is a 2-REA operator $J$ such that $J(X) <\_T X'$ isn't of degree r.e. in $X$. Since $0''$ is of r.e. degree in every $X < 0''$ with $X' \geq\_T 0''$.


