A splitting theorem for the Medvedev and Muchnik lattices
Stephen Binns · Mathematical logic quarterly · 2003
Abstract This is a contribution to the study of the Muchnik and Medvedev lattices of non‐empty Π01 subsets of 2ω. 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, ifP > 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.