Hierarchies of torsion-free Abelian groups
type: Item
instance of: scholarly article
title: Hierarchies of torsion-free Abelian groups
zbMATH Open document ID: 0625.03026
DOI: 10.1007/BF01978886
author: N. G. Khisamiev
published in: Algebra and Logic
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/187346
review text: Using the arithmetic hierarchy of sets one can define the arithmetic hierarchy of Abelian groups. Let N be the set of all natural numbers, and let X denote one of the symbols \(\Sigma^ 0_ n\), \(\Pi^ 0_ n\), \(\Delta^ 0_ n\), where \(n\geq 1\). A map \(\nu\) of the set N onto the Abelian group A is said to be an enumeration of this group. The pair (A,\(\nu)\) is called an X-group if for any numbers x and y one can effectively find the number of the element \(\nu x+\nu y\) and the set \(\{<x,y>|\nu x=\nu y\}\) is an X-set. The group A is said to be X- representable, if there exists an enumeration \(\nu\) of the group A, such there the pair (A,\(\nu)\) is an X-group. We note that in a different terminology \(\Delta^ 0_ 1\)-representable groups are called constructivizable groups. We call the class of X-representable groups the X-class. The scheme of inclusions between these classes is the following: \(\Delta^ 0_ 1\subseteq \Sigma^ 0_ 1\subseteq \Delta^ 0_ 2\), \(\Delta^ 0_ 1\subseteq \Pi^ 0_ 2\subseteq \Delta^ 0_ 2\) etc. In this paper we investigate this hierarchy for the class of torsion-free Abelian groups.
Mathematics Subject Classification ID: 03D45
Mathematics Subject Classification ID: 20K20
Mathematics Subject Classification ID: 20K27
Mathematics Subject Classification ID: 03D55
zbMATH DE Number: 4014696
zbMATH Keywords: arithmetic hierarchy of Abelian groups
zbMATH Keywords: constructivizable groups
zbMATH Keywords: torsion-free Abelian groups
MaRDI profile type: Publication
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OpenAlex ID: W2045531999
description: scientific article; zbMATH DE number 4014696