\(\mathcal{D}\)-maximal sets (Q2795914)

The study of the structure \({\mathcal E}=(\{A\subseteq\omega:A \text{ is computably enumerable (c.e)} \}, \subseteq)\) occupies a very important part of the theory of computability, in particular, the study of the orbits of its elements. \textit{R. I. Soare} proved in his seminal work [Ann. of Math. (2) 100, 80--120, (1994; Zbl 0253.02040)] that maximal sets form an orbit in \({\mathcal E}\). In this interesting paper under review, the authors consider \({\mathcal D}\)-maximal sets, which are generalizations of maximal sets, introduced in [\textit{E. Herrmann} and \textit{M. Kummer}, J. Symb. Log. 59, No. 1, 60--72 (1994; Zbl 0799.03046)]: a noncomputable c.e. set \(A\) is \({\mathcal D}\)-maximal iff for all c.e. sets \(V\supseteq A\) there exists a c.e. set \(D\) disjoint from \(A\) such that either \(V=A\cup D\) or \(V\cup D=\omega\). Given a c.e. set \(A\), let \({\mathcal L}(A)=(\{A\cup W: W\in{\mathcal E}\},\subseteq)\) be the principal filter generated by \(A\). Then let \({\mathcal D}(A)=\{B:B\in{\mathcal L}(A)\text{ and } B-A\text{ is c.e.}\}\). The authors define the notion of generating sets for \({\mathcal D}(A)\). This notion produces ten classes. They prove that each c.e. set is classifiable into one of these classes and that there are complete and incomplete \({\mathcal D}\)-maximal sets in each of these classes. They also get that the \({\mathcal D}\)-maximal sets of some of these classes form a single orbit and that there are infinitely many orbits of \({\mathcal D}\)-maximal sets of some of the other classes.