The enumeration spectrum hierarchy of n‐families

Abstract

We introduce a hierarchy of sets which can be derived from the integers using countable collections. Such families can be coded into countable algebraic structures preserving their algorithmic properties. We prove that for different finite levels of the hierarchy the corresponding algebraic structures have different classes of possible degree spectra.

Degrees of Enumerations of Countable Wehner-Like Families

“…Fix a function g ≡ T ∅ such that the set C g = {σ ∈ ω <ω : σ g} of all finite strings that are not the prefix of g is c.e. Theorem 2.14 (see [6]). The family…”

“…In particular, if there exists a = a 1 ∩ a 2 , then {x : x ≤ a} is the degree spectrum of the family W(P ⊕ P, ε A 1 × ε A 2 ). Also, by using the methods of [1], a family with a spectrum of nonlow degrees was found in [6]. Recall that a degree x is said to be low if x = 0 .…”

“…In [6,7,14], Kalimullin and Faizrahmanov developed a general approach that allows one to obtain inversions of the spectra of algebraic structures by coding Wehner-like families into it. For this, the following notion was introduced, which generalizes the notion of a family of sets.…”

“…families in the case α = 1. The degree spectra of an α-family S is the class Sp S of all degrees enumerating S. As in the case of countable families, the degree spectrum of each α-family is also the spectrum of an algebraic structure (see [7]).…”

“…Applying n times the operator D to the 1-family from Theorem 2.15 relativized relative to the oracle ∅ (2n) , we obtain the following theorem. Theorem 3.6 (see [6,14]). For every n > 0, there is an n-family with degree spectrum Low 2n−1 = {x : x (2n−1) > 0 (2n−1) } of all nonlow 2n−1 -degrees.…”