Abstract

This paper analyzes several properties of infima in Dn, the n-r.e. degrees. We first show that, for every n> 1, there are n-r.e. degrees a, b, and c, and an -r.e. degree x such that a < x < b, c and, in Dn, b c = a. We also prove a related result, namely that there are two d.r.e. degrees that form a minimal pair in Dn, for each n < ω, but that do not form a minimal pair in Dω. Next, we show that every low r.e. degree branches in the d.r.e. degrees. This result does not extend to the low2 r.e. degrees. We also construct a non-low r.e. degree a such that every r.e. degree b a branches in the d.r.e. degrees. Finally we prove that the nonbranching degrees are downward dense in the d.r.e. degrees