Abstract

This paper (see Theorem 6 below) describes general conditions under which relative splittings and specified diamond embeddings are derivable in the local structure of the enumeration degrees. In so doing, three underlying themes are touched on - that of characterising the context for the Turing degrees provided via enumeration reducibility; secondly, general questions of definability and the role of splitting and nonsplitting; and also (emerging from the techniques) the description of new relationships between information content and degree theoretic structure. Jockusch [1968] introduced the notion of a semirecursive set: A ⊆ ω is semirecursive if there is a computable f : ω² → ω for which, for all x, y: (i) f(x, y) = x or f(x, y) = y, (ii) x ∈ A ∨ y ∈ A ⇒ f(x, y) ∈ A. The paper begins with a result...

Citations