Effectively and Noneffectively Nowhere Simple Sets

R. Shore proved that every recursively enumerable (r.e.) set can be split into two (disjoint) nowhere simple sets. Splitting theorems play an important role in recursion theory since they provide information about the lattice E of all r.e. sets. Nowhere simple sets were further studied by D. Miller and J. Remmel, and we generalize some of their results. We characterize r.e. sets which can be split into two (non)effectively nowhere simple sets, and r.e. sets which can be split into two r.e. non-nowhere simple sets. We show that every r.e. set is either the disjoint union of two effectively nowhere simple sets or two noneffectively nowhere simple sets. We characterize r.e. sets whose every nontrivial splitting is into nowhere simple sets, and r.e. sets whose every nontrivial splitting is into effectively nowhere simple sets. R. Shore proved that for every effectively nowhere simple set A, the lattice L∗(A) is effectively isomorphic to E∗, and that there is a nowhere simple set A such that L∗(A) is not effectively isomorphic to E∗. We prove that every nonzero r.e. Turing degree contains a noneffectively nowhere simple set A with the lattice L∗(A) effectively isomorphic to E∗.