Isolated d.r.e. degrees are dense in r.e. degree structure (Q1354326)

A degree \({\mathbf a}\) is called d.r.e. if it contains a difference \(A-B\) of r.e. sets \(A\) and \(B\). A d.r.e. degree \({\mathbf a}\) is called isolated d.r.e. [introduced by \textit{S. B. Cooper} and \textit{X. Yi} in ``Isolated D.R.E. degrees'', to appear in Math. Logic Quartly] if it is a strongly minimal cover of an r.e. degree \({\mathbf d}\), i.e. \({\mathbf d}<{\mathbf a} \& \forall{\mathbf b}\in {\mathbf R} ({\mathbf b}\leq{\mathbf a}\Rightarrow{\mathbf b}\leq{\mathbf d})\). Cooper and Yi proved at first the existence of isolated d.r.e. degrees. In fact, they even showed that isolated d.r.e. degrees are weakly dense in the r.e. degrees in the sense that if \({\mathbf a}\), \({\mathbf b}\) are r.e. degrees such that \({\mathbf a}<{\mathbf b}\) and \({\mathbf b}\) is high, then there is an isolated d.r.e. degree \({\mathbf d}\) such that \({\mathbf a}<{\mathbf d}<{\mathbf b}\). The paper under review shows finally that isolated d.r.e. degrees are dense in the r.e. degrees, hence solves an open question of Cooper and Yi.