Complementing below recursively enumerable degrees (Q1093627)

This paper contains two main results about r.e. degrees. First, it is shown that for any nonzero low r.e. degree \(\underset \tilde{} c\) and any r.e. degree \(\underset \tilde{} a\) not below \(\underset \tilde{} c\), there exists a minimal degree below \(\underset \tilde{} a\) and incomparable with \(\underset \tilde{} c\). Second, the authors construct a specific r.e. degree \(\underset \tilde{} a\) such that a complementation theorem like that of Posner for the degrees below \(\underset \tilde{} 0'\) does not hold for the degrees below \(\underset \tilde{} a\). In fact, they build a particular r.e. degree \(\underset \tilde{} b\) between \(\underset \tilde{} 0\) and \(\underset \tilde{} a\) such that no \(\underset \tilde{} c\) can both cap \(\underset \tilde{} b\) to \(\underset \tilde{} 0\) and cup \(\underset \tilde{} b\) to \(\underset \tilde{} a\).