A Jump Inversion Theorem for the Degree Spectra

Abstract

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 𝒜 is contained in the set of the jumps of the degrees in some spectrum ℬ then there exists a spectrum 𝒞 such that 𝒞⊆ℬ and 𝒜 is equal to the set of the jumps of the degrees in 𝒞.