A many-one degree is functional if it contains the index set of some class of partial recursive functions but does not contain an index set of a class of r.e. sets. We give a natural embedding of the r.e. m-degrees into the functional degrees of 0'. There are many functional degrees in 0' in the sense that every partial-order can be embedded. By generalizing, we show there are many functional degrees in every complete Turning degree. There is a natural tie (...) between the studies of index sets and partial-many-one reducibility. Every partial-many-one degree contains one or two index sets. (shrink)

The notion of superhigh computably enumerable (c.e.) degrees was first introduced by (Mohrherr in Z Math Logik Grundlag Math 32: 5–12, 1986) where she proved the existence of incomplete superhigh c.e. degrees, and high, but not superhigh, c.e. degrees. Recent research shows that the notion of superhighness is closely related to algorithmic randomness and effective measure theory. Jockusch and Mohrherr proved in (Proc Amer Math Soc 94:123–128, 1985) that the diamond lattice can be embedded into the c.e. tt-degrees (...) preserving 0 and 1 and that the two atoms can be low. In this paper, we prove that the two atoms in such embeddings can also be superhigh. (shrink)