We prove an algebraic extension theorem for the computably enumerable sets, epsilon. Using this extension theorem and other work we then show if A and (A) over cap are automorphic via Psi, then they are automorphic via Lambda where Lambda up arrow epsilon*( A) = Psi and Lambda up arrow epsilon*(A) is Delta(0)(3). We give an algebraic description of when an arbitrary set (A) over cap is in the orbit of a computably enumerable set A. We construct the first example of a definable orbit which is not a Delta(0)(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 (A) over cap is in the same orbit as A, then they are in the same. Delta(0)(6)-orbit and, furthermore, we provide a classification of when two simple sets are in the same orbit.
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