Arithmetic non-trivial 2-l.u.b Remember a degree $\mathbb{d}$ is the $n$-lub of $\mathbb{c}_j$ in the Turing degrees if it is the least element (not merely a minimal element) set of $\mathbb{c}^{(n)}$ such that  $\mathbb{c}$ computes every $\mathbb{c}_j$.  It is non-trivial if it's not the $n$-th jump of a finite join of the degrees $\mathbb{c}_n$.
Enderton, Putnam and Sacks have proved $0^{\omega}$ is the (non-trivial) 2-lub of $0^n$.  Are there any arithmetic degrees that are non-trivial 2-lub?
If so can they be arithmetic?  What about an upper bound  $\mathbb{c}$ of $\mathbb{c}_j$ with $\mathbb{c}$ of r.e. degree and  $\mathbb{c}^{(2)}$ a non-trivial 2-lub of $\mathbb{c}_n$?
 A: Yes, it turns out that there is such an arithmetic (indeed 3-REA) non-trivial 2-lub.  In fact, $\emptyset'''$ is such a degree.  Consider the construction I give here in answer to this question.
This construction builds an $\omega$-REA degree $A$ such that $A^{[\leq n]}$ is low for every $n$ but such that $A$ computes $\emptyset'''$ [1].  Let ${\bf{a}}_n$ be the degree of $A^{[\leq n]}$.  We have ${\bf{a}}_n \leq_T \emptyset'$ and ${\bf{a}}''_n \equiv_T \emptyset''$.  The first fact provides an upper bound ($\emptyset'$) whose double jump is  $\emptyset'''$ and the second fact shows that $\emptyset'''$ is not the double jump of any finite join of the degrees ${\bf{a}}_n$.  All that remains is to verify that the double jump of any other upper bound computes $\emptyset'''$.
Now suppose ${\bf d}$ computes each ${\bf{a}}_n$.  I claim that ${\bf d}''$ can compute $A$.  This follows because given an index for $A^{[\leq n]}$ as a ${\bf d}$ computable set we can (by uniformity of the REA construction) ${\bf d}$ computably recover an index for $A^{[\leq n +1]}$ as a ${\bf d}$-r.e. set and, since $A^{[\leq n +1]}$ is ${\bf d}$ computable we can recover an index for $A^{[\leq n +1]}$ as a ${\bf d}$ computable set computably in ${\bf d}''$.  Thus, if ${\bf d}$ is an upper bound of ${\bf{a}}_n$ it follows that ${\bf d}''$ computes $\emptyset'''$.
[1]  The construction of $A$ ensures that if for each $x$ $\lim_{t\to\infty} p_e(e,s,t)$ exists and $\lim_{t\to\infty} p_e(e,0,t) \neq \lim_{t\to\infty} p_e(e,1,t)$ then $A$ can determine if $\lim_{s\to\infty}\lim_{t\to\infty} p_e(e,s,t)$ exists.  But, by appropriate choice of $e$ this lets us answer the membership question in $\emptyset'''$.

Note that this also answer the question of whether I can have an r.e. degree whose double jump is a 2-lub.  However, I'm unsure if this extends to incomplete r.e. degrees but I'm guessing it might with a bit more care in the constructions.
