Generalized stability of torsion-free abelian groups
type: Item
instance of: scholarly article
title: Generalized stability of torsion-free abelian groups
zbMATH Open document ID: 1278.03072
author: M. A. Rusaleev
published in: Algebra and Logic
publication date: 2012-12-17T00:00:00Z
review text: In 1969, Shelah introduced the notion of stable theories that then initiated an intensive research on this subject. One instance is \((P,a)\)-stability, which is defined by adding to a language a unary predicate symbol and adding to types the condition of being algebraically closed for that predicate. The main theorem of the article states that any complete theory for a torsion-free abelian group is in fact \((P,a)\)-stable. The author proves this result using quantifier elimination down to positive primitive formulas and thus showing that the question of being \((P,a)\)-stable reduces to asking if a system of linear equations with integer coefficients has a solution in the algebraic closure of constants involved in the system. In the end, the author also gives an example of an abelian group that has a \((P,s)\)-unstable theory.
Mathematics Subject Classification ID: 03C45
Mathematics Subject Classification ID: 20K20
zbMATH DE Number: 6116185
zbMATH Keywords: stable theories
zbMATH Keywords: abelian groups
zbMATH Keywords: quantifier elimination
reviewed by: Lutz Strüngmann
MaRDI profile type: Publication
full work available at URL: s10469-011-9130-0
OpenAlex ID: W1994001117
cites work: Stable theories
cites work: Categoricity in Power
cites work: Classification theory and the number of non-isomorphic models
cites work: Q4436302
DOI: 10.1007/S10469-011-9130-0
description: scientific article; zbMATH DE number 6116185