Abstract. Π0 1 classes are important to the logical analysis of many parts of mathematics. The Π0 1 classes form a lattice. As with the lattice of computably enumerable sets, it is natural to explore the relationship between this lattice and the Turing degrees. We focus on an analog of maximality, or more precisely, hyperhypersimplicity, namely the notion of a thin class. We prove a number of results relating automorphisms, invariance, and thin classes. Our main results are an analog of Martin’s work on hyperhypersimple sets and high degrees, using thin classes and anc degrees, and an analog of Soare’s work demonstrating that maximal sets form an orbit. In particular, we show that the collection of perfect thin classes (a notion which is definable in the lattice of Π0 1 classes) forms an orbit in the lattice of Π01 classes; and a degree is anc iff it contains a perfect thin class. Hence the class of anc degrees is an invariant class for the lattice of Π0 1 classes. We remark that the automorphism result is proven via a ∆0 3 automorphism, and demonstrate that this complexity is necessary. 1.

Unter den Linden 6

The purpose of this communication is to announce some recent results on the computably enumerable sets. There are two disjoint sets of results; the first involves invariant classes and the second involves automorphisms of the computably enumerable sets. What these results have in common is

Abstract. 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 � A are automorphic via Ψ, then they are automorphic via Λ where Λ ↾ L ∗ (A) =ΨandΛ↾E ∗ (A) is∆0 3. We give an algebraic description of when an arbitrary set �A is in the orbit of a computably enumerable set A. We construct the first example of a definable orbit which is not a ∆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 is in the same orbit as A, then they are in the same ∆0 6-orbit and, furthermore, we provide a classification of when two simple sets are in the same orbit. 1.

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