Lattice nonembeddings and initial segments of the recursively enumerable degrees

Abstract

This paper studies the structure of a general initial segment [0,a] (a¬= 0) of the upper semilattice R of r.e. degrees. In particular we address the question of what lattices may be embedded into all such segments. It turns out that some [0,a] may be very different from R since many lattices (and semilattices) embeddable into R are not embeddable into such [0,a]. The general theme seems to be that for a sufficiently close to 0', [0,a] is much more' distributive than is R