A splitting theorem for the Medvedev and Muchnik lattices

This is a contribution to the study of the Muchnik and Medvedev lattices of non-empty Π1 subsets of 2 ω. In both these lattices, any non-minimum element can be split, i.e. it is the non-trivial join of two other elements. In fact, in the Medvedev case, if P >M Q, then P can be split above Q. Both of these facts are then generalised to the embedding of arbitrary finite distributive lattices. A consequence of this is that both lattices have decidible ∃-theories.