Splitting properties and jump classes

Abstract

We show that the promptly simple sets of Maass form a filter in the lattice ℰ of recursively enumerable sets. The degrees of the promptly simple sets form a filter in the upper semilattice of r.e. degrees. This filter nontrivially splits the high degrees (a is high ifa′=0″). The property of prompt simplicity is neither definable in ℰ nor invariant under automorphisms of ℰ. However, prompt simplicity is easily shown to imply a property of r.e. sets which is definable in ℰ and which we have called the splitting property. The splitting property is used to answer many questions about automorphisms of ℰ. In particular, we construct lowd-simple sets which are not automorphic, answering a question of Lerman and Soare. We produce classes invariant under automorphisms of ℰ which nontrivially split the high degrees as well as all of the other classes of r.e. degrees defined in terms of the jump operator. This refutes a conjecture of Soare and answers a question of H. Friedman.