A splitting theorem for \(n\)-REA degrees (Q2750871)

The main result of this paper is that every \(n\)-REA degree \(\mathbf d\) is splittable over a given degree \({\mathbf a}\) avoiding a further degree \({\mathbf b}\) (for precise definitions see below). The motivation for this result is to disprove Cooper's claim that this result would fail for some 2-REA degree. Cooper used this claim in preliminary publications [\textit{S. B. Cooper}, ``Definability and global degree theory'', Lect. Notes Log. 2, 25-45 (1993; Zbl 0797.03044); On a conjecture of Kleene and Post, Department of Pure Mathematics, Leeds University, Preprint Series No. 7 (1993)] for the proof of the important result, that the jump is definable within the upper semilattice of all Turing degrees. Later, Cooper published a corrected version [\textit{S. B. Cooper}, ``On a conjecture of Kleene and Post'', Math. Logic Quarterly 47, 3-33 (2001; Zbl 0977.03023)] of his paper; the authors [``Defining the Turing jump'', Math. Res. Lett. 6, 711-722 (1999; Zbl 0958.03029)] also proved the same result. NEWLINENEWLINENEWLINEThe paper is based on the following result. Let \(\mathbf a\), \(\mathbf u\) and \(\mathbf d\) be Turing degrees such that \({\mathbf a} \cup {\mathbf u} < {\mathbf d}\) and \(\mathbf d\) is recursively enumerable relative to the join \({\mathbf a} \cup {\mathbf u}\). Then there are two pairs \(({\mathbf x}_0,{\mathbf x}_1)\), \(({\mathbf y}_0,{\mathbf y}_1)\) of Turing degrees such that their joins \({\mathbf x}_0 \cup {\mathbf x}_1\) and \({\mathbf y}_0 \cup {\mathbf y_1}\) are both equal to \({\mathbf d}\) while on the other hand every pair \(({\mathbf x}_i,{\mathbf y}_j)\) \((i,j \in \{0,1\})\) is a minimal pair relative to \(\mathbf a\), that is, every Turing degree \(\mathbf b\) with \({\mathbf b} \leq {\mathbf a} \cup {\mathbf x}_i\) and \({\mathbf b} \leq {\mathbf a} \cup {\mathbf y}_j\) satisfies \({\mathbf b} \leq {\mathbf a}\). NEWLINENEWLINENEWLINEThe authors use this result to show that every \(n\)-REA degree (and in particular every 2-REA degree) \(\mathbf d\) is splittable over any degree \(\mathbf a\) avoiding \(\mathbf b\). That is, every \(n\)-REA degree \(\mathbf d\) satisfies that, for all degrees \(\mathbf a\) and \(\mathbf b\) with \({\mathbf a}, {\mathbf b} < {\mathbf d}\) and \({\mathbf b} \not\leq {\mathbf a}\), there are degrees \({\mathbf d}_0,{\mathbf d}_1\) such that \({\mathbf d} = {\mathbf d}_0 \cup {\mathbf d}_1\), \({\mathbf d}_0 \not\geq {\mathbf b}\) and \({\mathbf d}_1 \not\geq {\mathbf b}\). Recall that a degree \(\mathbf d\) is \(n\)-REA iff there are degrees \({\mathbf c}_0,\ldots,{\mathbf c}_n\) such that \({\mathbf 0} = {\mathbf c}_0\), \({\mathbf c}_n = {\mathbf d}\) and every \(m<n\) satisfies that \({\mathbf c}_m \leq {\mathbf c}_{m+1}\) and \({\mathbf c}_{m+1}\) is recursively enumerable relative to \({\mathbf c}_m\).