Degrees coded in jumps of orderings

Abstract

All structures to be considered here have universe ω, and all languages come equipped with Gödel numberings. If is a structure, then D(), the open diagram of , can be thought of as a subset of ω, and it makes sense to talk about the Turing degree deg(D()). This depends on the presentation as well as the isomorphism type of .For example, consider the ordering = (ω, <). For any B ⊆ ω, it is possible to code B in a copy of as follows: Let π be the permutation of ω such that for each n ∈ ω, π leaves 2n and 2n + 1 fixed if n ∈ B and switches 2n with 2n + 1 if n ∉ B. Let be the copy of such that ≃π. Then n ∈ B iff the sentence 2n < 2n + 1 is in D(). In §4, this idea will be used to show that for any structure that is not completely trivial, {deg(D()): ≃ } is closed upwards.It would be satisfying to have a way of assigning Turing degrees to structures such that the degree assigned to a given structure measured the recursion-theoretic complexity of the isomorphism type and was independent of the presentation. Jockusch suggested the following.

Turing degrees of isomorphism types of algebraic objects

“…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). Hirschfeldt, Khoussainov, Shore, and Slinko established in [15] that for every automorphically nontrivial structure A, there is a symmetric irreflexive graph, a partial order, a lattice, a ring, an integral domain of arbitrary characteristic, a commutative semigroup, or a 2-step nilpotent group whose degree spectrum coincides with DgSp(A).…”

“…On the other hand, Knight showed in [19] that for any Turing degree d, there is a nonstandard model of Peano Arithmetic with first jump degree d ′ . Knight also established that the only possible first jump degree for a linear order is 0 ′ .…”

Turing degrees of nonabelian groups

“…A structure M is automorphically trivial if there is a finite subset P of the domain M such that every permutation of M , whose restriction to P is the identity, is an automorphism of M. Knight [18] proved that for an automorphically nontrivial structure A, and a Turing degree x with x ≥ deg(A), there is a structure B ∼ = A such that deg(B) = x. That is, DgSp(A) is closed upwards.…”

“…For example, Richter [27] showed that a linear ordering, which is not isomorphic to a computable one, does not have a degree of its isomorphism type. Downey and Knight [8], building on the previous work of Knight [18] and Ash, Jockusch and Knight [2], established that for a Turing degree d with d ≥ 0 (α) , where α ≥ 1 is a computable ordinal, there is a linear ordering A whose αth jump degree is d, and such that A does not have βth jump degree for any β < α. The jump degrees have also been studied for torsion abelian groups by Oates [24], and for rank 1 torsion-free abelian groups by Coles, Downey and Slaman [6].…”

On the Complexity of the Successivity Relation in Computable Linear Orderings

“…degrees above any given nonzero c.e. degree, a result implicit in Downey [6,Theorem 7.5], using the method of Downey [6,Theorem 7.7], based on Downey and Moses [11], Downey and Knight [10], and Knight [22]; this result appears explicitly in Chubb, Frolov and Harizanov [4, Theorem 2.5], based on Downey and Moses [11].…”