Completely Mitotic r.e. Degrees

An r.e. set A is called mitotic if there exist a pair of disjoint r.e. sets Al, A2 with A, UA, = A (in this case we write AI UA, =A) such that AI =,.A2 sTA. We refer to such a splitting as a mitotic splitting of A. Lachlan [lo] was the first person to show that not all, r.e. sets are mitotic. More extensive investigations into (non)mitoticity were provided by Ladner [ll, 121 who constructed various types of nonmitotic r.e. sets. He also proved the following very interesting theorem: An r.e. set A is mitotic iff A is autoreducible where A is called autoreducible (Trachtenbrot [20]) if there is a functional @ such that, for all x, @(A U {x}; X) = A(x). Following Ladner’s investigations, there have been several other results concerning the existence of nonmitotic r.e. sets. One example is Ingrassia’s [8] result that the degrees containing nonmitotic r.e. sets are dense in R, the r.e. degrees. The interest in Ingrassia’s result is that nonmitotic r.e. sets do not live in all nonzero r.e. degrees. The most difficult of Ladner’s results establishes this. That is, in [12] Ladner constructed a completely mifofic nonzero r.e. degree a, where a is completely mitotic if all of its r.e. elements are mitotic. Our goal in this paper is to investigate the class of completely mitotic degrees. Save for Ladner’s one construction of a low,-low (as P. Cohen observed in [12]) completely mitotic r.e. degree there are no other existence theorems for these degrees. In particular one of the main open questions here was whether or not there exist (even) low nonzero completely mitotic degrees. Ambos-Spies and Fejer [2] have shown that Ladner’s construction cannot be used to answer this, since his construction automatically gives a cont@o~~ r.e. degree (namely an r.e. degree consisting of a single r.e. wtt-degree). In [2] they showed that if a # 0 is low and contiguous then a contains a nonmitotic r.e. set.