Isomorphism types of Rogers semilattices for families from different levels of the arithmetical hierarchy

Abstract

We investigate differences in isomorphism types for Rogers semilattices of computable numberings of families of sets lying in different levels of the arithmetical hierarchy.Among the many possible applications of the theory of generalized computable numberings propounded in [1], a particularly interesting and popular one is to arithmetical numberings, that is, numberings of families of arithmetical sets. When considering a family A of Σ 0 n -sets, generalized computable numberings can be characterized as follows: a numbering α of A is generalized computable if and only if the setIn what follows, such a numbering α will merely be referred to as Σ 0 n -computable. We recall that if α and β are numberings of the same family of objects, then α is said to be reducibleAn equivalence class of a numbering α (depending of course on the collection of numberings under study) will be denoted by the symbol deg(α). Since is a preordering relation, ≡ is an equivalence relation. If A is a family of Σ 0 n -sets, then ≡ partitions the set Com 0 n (A) of all Σ 0 n -computable numberings into equivalence classes, thus generating a degree structure, denoted by R 0 n (A) and called the Rogers semilattice of A.In this paper we continue research on isomorphism types of Rogers semilattices, started in [2]. We are interested in differences between elementary theories and isomorphism types at different arithmetical levels. In [3,4] it was shown that for every fixed level of the arithmetical hierarchy, there exist infinitely many families with pairwise different elementary theories for their Rogers semilattices. In [2] we established that, for every n, the isomorphism type of the Rogers semilattice of some Σ 0 n+5 -computable family B is different from the isomorphism type of the Rogers semilattice R 0 n+1 (A) of an arbitrary Σ 0 n+1 -computable family A. In this paper we improve on this result by showing that, for every n, the isomorphism type of the Rogers semilattice of any non-trivial (i.e., non one-element) Σ 0 n+4 -computable family B is different from the isomorphism type of the Rogers semilattice R 0 n+1 (A) of any Σ 0 n+1 -computable family A. For the unexplained terminology and notation relative to computability theory, our main references are in [5][6][7]. We will use the term computable function to connote completely defined computable functions. For the main concepts and notions of the theory of numberings and computable Boolean algebras, we ask the *

Arithmetical D-degrees

“…The last statement strengthens a result of [3]; together with the results of [4], it allows us to expand the class of semilattices which are principal ideals or segments in the Roger semilattices of arithmetical numberings. This implies, in particular, a strengthening of the results on the difference of the isomorphism types of the Roger semilattices of the arithmetical numberings of the various levels of the arithmetical hierarchies presented in [5,6]. § 1.…”

“…Then it was proved in [6] that if m ≥ n + 3 then R 0 n+1 (F ) and R 0 m+1 (G ) are either not isomorphic or trivial. Corollary 5 gives a stronger result: comparing with [6], the "level difference" is reduced from 3 to 2.…”

Weak reducibility of computable and generalized computable numberings

“…When A = ∅, we come to a classical notion of computable numberings (see [4]), and when A = ∅ (n+1) , we have a notion of Σ 0 n+2 -computable numberings (see, for example, [1,2,5,6]). We call a family computable (A-computable, Σ 0 n+2 -computable), if it possesses a computable (A-computable, Σ 0 n+2 -computable) numbering.…”

Isomorphism types of Rogers semilattices in the analytical hierarchy

“…Let n be a non-zero natural number. Badaev, Goncharov, and Sorbi [20] proved that for any m ≥ n + 3, any non-trivial Rogers Σ 0 m -semilattice is not isomorphic to a Rogers Σ 0 n -semilattice. Podzorov [21] obtained a generalization of this theorem: he showed that a similar result holds for m ≥ n + 2.…”