Branching in the {\Sigma^0_2} -enumeration degrees: a new perspective

We give an alternative and more informative proof that every incomplete -enumeration degree is the meet of two incomparable -degrees, which allows us to show the stronger result that for every incomplete -enumeration degree a, there exist enumeration degrees x ₁ and x ₂ such that a, x ₁, x ₂ are incomparable, and for all b  ≤  a, b  =  (b ∨ x ₁ ) ∧ (b ∨ x ₂ ). The first author would like to thank her advisor, Andrea Sorbi, whose guidance made this paper possible. The second author has been supported by a Marie Curie Incoming International Fellowship of the European Community FP6 Program under contract number MIFI-CT-2006-021702.