Intervals of the Lattice of Computably Enumerable Sets and Effective Boolean Algebras

We prove that each interval of the lattice E of c.e. sets under inclusion is either a boolean algebra or has an undecidable theory. This solves an open problem of Maass and Stob 9]. We develop a method to prove undecidability by interpreting ideal lattices, which can also be applied to degree structures from complexity theory. We also answer a question left open in 6] by giving an example of a non-deenable subclass of E which has an arithmetical index set and is invariant under automorphisms.