Every incomplete computably enumerable truth-table degree is branching

If r is a reducibility between sets of numbers, a natural question to ask a bout the structure Y-r of the r-degrees containing computably enumerable se ts is whether every element not equal to the greatest one is branching (i.e ., the meet of two elements strictly above it). For the commonly studied re ducibilities. the answer to this question is known except for the case of t ruth-table (tt) reducibility. In this paper, we answer the question in the tt case by showing that every tt-incomplete computably enumerable truth-tab le degree a is branching in Y-tt. The fact that every Turing-incomplete com putably enumerable truth-table degree is branching has been known for some time. This fact can be shown using a technique of Ambos-Spies and, as notic ed by Nies, also follows from a relativization of a result of Degtev. We gi ve a proof here using the Ambos-Spies technique because it has not yet appe ared in the literature. The proof uses an infinite injury argument. Our mai n result is the proof when a is Turing-complete but tt-incomplete. Here we are able to exploit the Turing-completeness of a in a novel way to give a f inite injury proof.