The discontinuity of splitting in the recursively enumerable degrees (Q1902338)

We say: i) \({\mathbf b}> \mathbf{0}\) is noncuppable relative to \({\mathbf a}\) above \({\mathbf c}\) if \((\forall {\mathbf w}\geq {\mathbf c}) [{\mathbf a}\leq {\mathbf b} \vee {\mathbf w} \Rightarrow {\mathbf a}\leq {\mathbf w}]\). ii) \({\mathbf a}\), \({\mathbf b}\) are relatively noncuppable above \({\mathbf c}\) if \({\mathbf a}\), \({\mathbf b}\) are mutually noncuppable relative to each other above \({\mathbf c}\). iii) \({\mathbf a}\), \({\mathbf b}\) are relatively noncuppable if \({\mathbf a}\), \({\mathbf b}\) are relatively noncuppable above \textbf{0}. In this paper the authors prove the following: Theorem 2.2. Given r.e. degrees \({\mathbf c}\), \({\mathbf b}_0\) and \({\mathbf b}_1\) such that \({\mathbf c}<{\mathbf b}_i\) for all \(i<2\), there exist \({\mathbf a}_0\), \({\mathbf a}_1\) such that \({\mathbf a}_0\), \({\mathbf a}_1\) are relatively noncuppable above \({\mathbf c}\) and \({\mathbf c}<{\mathbf a}_i< {\mathbf b}_i\) for all \(i<2\). Theorem 2.4. There exists a relatively noncuppable pair \({\mathbf a}_0\), \({\mathbf a}_1\) such that \({\mathbf a}_0\vee {\mathbf a}_1\) is high. In fact, \({\mathbf a}_0\), \({\mathbf a}_1\) can be a minimal pair. Some open questions are also given.