Spectra of structures and relations

Abstract

We consider embeddings of structures which preserve spectra: if g:ℳ →𝒮 with 𝒮 computable, then ℳ should have the same Turing degree spectrum (as a structure) that g(ℳ) has (as a relation on 𝒮). We show that the computable dense linear order ℒ is universal for all countable linear orders under this notion of embedding, and we establish a similar result for the computable random graph 𝔖. Such structures are said to be spectrally universal. We use our results to answer a question of Goncharov, and also to characterize the possible spectra of structures as precisely the spectra of unary relations on 𝔖. Finally, we consider the extent to which all spectra of unary relations on the structure ℒ may be realized by such embeddings, offering partial results and building the first known example of a structure whose spectrum contains precisely those degrees c with c' ≥_T 0''.