Jumps of orderings

Abstract:

Here it is shown that for each recursive ordinal $\alpha \geqslant 2$ and each Turing degree ${\mathbf {d}} > {{\mathbf {0}}^{(\alpha )}}$, there is a linear ordering ${\mathbf {A}}$ such that ${\mathbf {d}}$ is least among the $\alpha$th jumps of degrees of (open diagrams of) isomorphic copies of ${\mathbf {A}}$ and for $\beta < \alpha$, the set of $\beta$th jumps of degrees of copies of ${\mathbf {A}}$ has no least element.