$\mathcal{D}$-maximal sets

Abstract

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.