We show that the Q-degree of a hyperhypersimple set includes an infinite collection of Q 1-degrees linearly ordered under ${\leq_{Q_1}}$ with order type of the integers and consisting entirely of hyperhypersimple sets. Also, we prove that the c.e. Q 1-degrees are not an upper semilattice. The main result of this paper is that the Q 1-degree of a hemimaximal set contains only one c.e. 1-degree. Analogous results are valid for ${\Pi_1^0}$ s 1-degrees.

A splitting of an r.e. set A is a pair A1, A2 of disjoint r.e. sets such that A1 A2 = A. Theorems about splittings have played an important role in recursion theory. One of the main reasons for this is that a splitting of A is a decomposition of A in both the lattice, , of recursively enumerable sets and in the uppersemilattice, R, of recursively enumerable degrees . Thus splitting theor ems have been used to obtain results about (...) the structure of , the structure of R, and the relationship between the two structures. Furthermore it is fair to say that questions about splittings have often generated important new technical developments in recursion theory. In this article we survey most of the results and techniques associated with splitting properties of r.e. sets in ordinary recursion theory. (shrink)