The jump operator on the \(\omega \)-enumeration degrees (Q1032630)

The upper semilattice \(\mathcal{D}_{\omega}\) of \(\omega\)-enumeration degrees and the jump operator in the \(\omega\)-enumeration degrees was introduced in [\textit{I. N. Soskov}, ``The \(\omega\)-enumeration degrees'', J. Log. Comput. 17, No. 6, 1193--1214 (2007; Zbl 1174.03018)]. A set \textbf{A} is enumeration reducible to a set \textbf{B} if there is a c.e. set \(\Phi\) such that: \[ \mathbf n \in \mathbf A \Leftrightarrow \exists \mathbf u(\langle \mathbf n,\mathbf u \rangle \in \Phi \, \& \, D_{\mathbf u} \subseteq \mathbf B), \] where \(D_{\mathbf u}\) denotes the finite set with canonical index \textbf{u.} The following results are obtained in the paper under review: 1. The enumeration degrees are first-order definable in the structure \(\mathcal{D}_{\omega}'\) of the \(\omega\)-enumeration degrees augmented by the jump operator. 2. The groups of the automorphisms of \(\mathcal{D}_{\omega}'\) and of the enumeration degrees \(\mathcal{D}_{e}\) are isomorphic. 3. Let \(\phi\) be an automorphisms of \(\mathcal{D}_{e}'\). Then \(\phi(x)= x\) for all \(x\) above \(\mathbf 0_e^{(4)}\). In the second part of this paper, the jumps of the \(\omega\)-enumeration degrees below \(\mathbf 0_{\omega}'\) are investigated. The ideal of almost zero degrees is defined and the authors obtain a natural characterization of the class \(H\) of the \(\omega\)-enumeration degrees below \(\mathbf 0_{\omega}'\) which are high \(n\) for some \(n\) (Theorem 5.8) and of the class \(L\) of the \(\omega\)-enumeration degrees below \(\mathbf 0_{\omega}'\) which are low \(n\) for some \(n\) (Theorem 5.9).