Some recent research directions in the computably enumerable sets

As suggested by the title, this paper is a survey of recent results and questions on the collection of computably enumerable sets under inclusion. This is not a broad survey but one focused on the author's and a few others' current research.Comment: to appear in the book "The Incomputable" in the Springer/CiE book serie

D\mathcal{D}-maximal sets

Soare proved that the maximal sets form an orbit in $\mathcal{E}$. We consider here $\mathcal{D}$-maximal sets, generalizations of maximal sets introduced by Herrmann and Kummer. Some orbits of $\mathcal{D}$-maximal sets are well understood, e.g., hemimaximal sets, but many are not. The goal of this paper is to define new invariants on computably enumerable sets and to use them to give a complete nontrivial classification of the $\mathcal{D}$-maximal sets. Although these invariants help us to better understand the $\mathcal{D}$-maximal sets, we use them to show that several classes of $\mathcal{D}$-maximal sets break into infinitely many orbits.Comment: Submitted. This version was revised during the fall of 201

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

We prove an algebraic extension theorem for the computably enumerable sets, $\mathcal{E}$. Using this extension theorem and other work we then show if $A$ and $\hat{A}$ are automorphic via $\Psi$ then they are automorphic via $\Lambda$ where \Lambda \restriction \L^*(A) = \Psi and \Lambda \restriction \E^*(A) is $\Delta^0_3$. We give an algebraic description of when an arbitrary set \Ahat is in the orbit of a \ce 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 $\hat{A}$ 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.Comment: Comments as of Aug 31, 05: This is now the final final version of the paper. Another section, 5.3, was added to the paper. No other change were made. This section was added to allow a clean clear inferface with the sequel. Comments as of March 31, 05: This is now the final version of this paper. (Section 7 was rewritten. A few other lemmas were added.