Extensions Theorems, Orbits, and Automorphisms of the Computably Enumerable Sets

We prove an algebraic extension theorem for the computably enumerable sets, E . Using this extension theorem and other work we then show: If A and  are automorphic via Ψ then they are automorphic via Λ where Λ ↾ L∗(A) = Ψ and Λ ↾ E∗(A) is ∆ 3 . We give an algebraic description of when an arbitrary set  is in the orbit of a computably enumerable set A. We construct the first example of a definable orbit which is not a ∆ 3 orbit. We conclude with some results which restrict the ways one can increase the complexity of orbits. For example, we show that if A is simple and  is in the same orbit as A then they are in the same ∆ 6 -orbit and furthermore we provide a classification of when two simple sets are in the same orbit.