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 G. 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 G. 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 .

Restrictions on the degree spectra of algebraic structures

“…By [8,9], instead of all algebraic structures A , it suffices to consider only undirected graphs V, R without loops (i.e., the binary relation R is symmetric and irreflexive), where the carrier V ⊆ ω is an arbitrary set and the computable relation R is fixed so that ω, R is a random graph. For example, we can assume that R(m, n) dfn ⇐⇒ at least one of the numbers m 2 n and n 2 m is odd (which is equivalent to n ∈ D m or m ∈ D n for the canonical numbering {D n } n∈ω of all finite sets).…”

“…There are many interesting papers dedicated to the study of the spectra of various algebraic structures (see [1][2][3][4][5][6][7][8][9][10], for example). In particular, it follows from [2] that for an arbitrary degree a the set {x : x ≥ a} is the spectrum of some algebraic structure.…”

Turing degrees of isomorphism types of algebraic objects

“…If a structure A is automorphically trivial, then all isomorphic copies of A have the same Turing degree. It was shown in [14] that if the language is finite, then that degree must be 0. On the other hand, Knight [19] proved that for an automorphically nontrivial structure A, we have that DgSp(A) is closed upwards, that is, if b ∈ DgSp(A) and d > b, then d ∈ DgSp(A).…”

Turing degrees of nonabelian groups

“…That is, DgSp(A) is closed upwards. On the other hand, for an automorphically trivial structure, all isomorphic copies have the same Turing degree, and in the case of a finite language that degree must be 0 (see [13]). Harizanov, Knight and Morozov [12] showed that, while for every automorphically trivial structure M we have Richter [27] established the following general criterion for the existence of a structure whose isomorphism type has an arbitrary Turing degree.…”