Let 0 be the one-one degree
consisting of the infinite recursive sets whose complements are also infinite. In this
paper are studied initial segments of the one-one degrees ≧ 0. A characterization is
stated of the order types of those initial segments with greatest member which are at
the same time initial segments of the many-one degrees. It is shown that any
finite initial segment with greatest member is a lattice, and that any finite
recursively enumerable initial segment with greatest member is a distributive
lattice.