Spectra of highn and non-lown degrees

Abstract

We survey known results on spectra of structures and on spectra of relations on computable structures. asking when the set of all high n degrees can be such a spectrum, and likewise for the set of nonlow n degrees. We then repeat these questions specifically for linear orders and for relations on the computable dense linear order Q. New results include realizations of the set of nonlow n Turing degrees as the spectrum of a relation on Q for all n ≥ 1, and a realization of the set of nonlow n Turing degrees as the spectrum of a linear order whenever n ≥ 2. The state of current knowledge is summarized in a table in the concluding section.

Limitwise monotonic sequences and degree spectra of structures

“…For instance, Miller [20] 3276 I. KALIMULLIN, B. KHOUSSAINOV, AND A. MELNIKOV built a linear order L such that DegSp(L) contains a Δ 0 2 degree a if and only if a > 0. Similar examples can be found in other classes of structures (see, e.g., [15], [9] and [21]).…”

Computability and Uncountable Linear Orders Ii: Degree Spectra

“…Within the ω-setting, there are several well-known and widely used theorems stating that if an order-type λ is a-ω-computable (for some fixed theorem-dependent degree a), then κ • λ is ω-computable (for some fixed theorem-dependent order-type κ). For example, the following theorem has been used to exhibit linear orders having spectra exactly the non-low n degrees for n ≥ 2 (see [8]) and to exhibit linear orders having arbitrary α th jump degree (see [4]).…”

“…A Nonlow n Spectrum. For n ≥ 2, there are countable linear orderings whose degree spectrums consist of the nonlow n ω-degrees [8]. For n = 1, though, while it is known (see [9]) that the collection of nonlow ω-degrees is a degree spectrum, it is yet unknown if it is the degree spectrum of a linear order.…”

Low5 Boolean Subalgebras and Computable Copies

“…In doing so, we attend to one detail. For each n, using our C-oracle, we define the interval [−2n − 2, −2n − 1) to lie in D iff n ∈ C. Clearly, it remains easy to make D doubly dense in (−∞, 0) after this is done, and by doing so, we ensure that C ≤ T D. The rest of the construction can then be devoted to making (B, D) ∼ = (B, A) with D ≤ T C. Now C (4) can be expressed by a Σ C 4 formula, say ∃a∀b∃c∀dR(n, a, b, c, d), with R ≤ T C. Notice that by [3,Corollary 5.14], we may assume that this R has the property that for each n, there is at most one value of a satisfying ∀b∃c∀dR (n, a, b, c, d). We also have a reduction, uniform in n, a, and b:…”