Completely mitotic r. e. degrees (Q1092895)

An r.e. set A is called mitotic if \(A=B\cup C\) with \(B\cap C=\emptyset\), B and C r.e. and \(A\equiv _ TB\equiv _ TC\). \textit{R. E. Ladner} [Trans. Am. Math. Soc. 184, 479-507 (1973; Zbl 0309.02045)] showed that there exists a completely mitotic r.e. degree: an r.e. \(\underset \tilde{} a\neq \underset \tilde{} 0\) such that all r.e. sets of degree \(\underset \tilde{} a\) are mitotic. Ingrassia in his thesis showed that degrees containing nonmitotic r.e. sets are dense. In this paper we examine the distribution of completely mitotic r.e. degrees. We show they can be low, high and \(low_ 2\)-low. Some negative results are established. For example, we show that there are initial segments containing no such degrees and these degrees cannot be low and promptly simple.