Source: J. Symbolic Logic
Volume 72, Issue 1
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''.
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