An extension of the recursively enumerable Turing degrees

We embed the upper semilattice of r.e. Turing degrees into a slightly larger structure which is better behaved and more foundationally relevant. For P,Q ⊆ 2, we say P is Muchnik reducible to Q if for all Y ∈ Q there exists X ∈ P such that X is Turing reducible to Y . We let Pw denote the lattice of Muchnik degrees of nonempty Π1 subsets of 2. Pw is a countable distributive lattice with 0 and 1. Every countable distributive lattice is lattice embeddable in every nontrivial initial segment of Pw (Binns/Simpson). Every nonzero Muchnik degree in Pw is join reducible in Pw (Binns). We construct a natural upper semilattice embedding i of the r.e. Turing degrees into Pw. We have i(0) = 0 and i(0′) = 1. We show that Pw contains at least two specific, natural Muchnik degrees other than 0 and 1, viz., the Muchnik degree MLR of Martin-Lof random reals, and the Muchnik degree FPF of fixed point free functions. In Pw we have 0 < FPF < MLR < 1, and these Muchnik degrees correspond to subsystems of Z2 which have arisen in Reverse Mathematics. Moreover, FPF and MLR are incomparable with i(a) for all r.e. Turing degrees 0 < a < 0′.