A Jump Inversion Theorem for the Degree Spectra

In the present article, we continue the study of the properties of the spectra of structures as sets of degrees initiated in [11]. Here, we consider the relationships between the spectra and the jump spectra. Our first result is that every jump spectrum is also a spectrum. The main result sounds like a Jump inversion theorem. Namely, we show that if a spectrum $\mathcal A$ is contained in the set of the jumps of the degrees in some spectrum $\mathcal B$ then there exists a spectrum $\mathcal C$ such that $\mathcal {C \subseteq B}$ and $\mathcal A$ is equal to the set of the jumps of the degrees in $\mathcal C$.