La classification élémentaire de groupes abeliens
type: Item
instance of: scholarly article
title: La classification élémentaire de groupes abeliens
zbMATH Open document ID: 0533.20025
author: Isidore Fleischer
published in: Expositiones Mathematicae
publication date: 1984-01-01T00:00:00Z
review text: The author characterizes the classes of abelian groups which are elementarily equivalent in the sense of logic. The answer is found on p. 28: If \(\phi\) (x) is a formula of the theory in one free variable x such that \(\phi\) (x) and \(\phi\) (y) imply \(\phi\) (x-y) then \(H=\{a\in A:\quad \phi(a)\quad holds\quad in\quad A\}\) is a subgroup of A for any abelian group A. If K is another subgroup determined by some formula then the cardinalities of the quotients H/\(H\cap K\) provide the crucial invariants. Two abelian groups A and B are elementarily equivalent if and only if each quotient H/\(H\cap K\) in A has the same finite cardinality as the corresponding quotient in B or both are infinite. It is actually not necessary to consider all such quotients H/\(H\cap K\) but only those with H and K in the subgroup lattice \({\mathcal L}(A)\) which is generated by A and \(\{\) 0\(\}\) under the operations \(\cap\), \(+\), and those of taking images and preimages under the endomorphisms \(x\mapsto nx\) where n is a natural number. This description is handy, e.g. for any non-zero divisible torsion-free group A we have \({\mathcal L}(A)=\{A,0\}\) and hence any two such groups are elementarily equivalent. In addition to the above result it is shown that the first order theory of abelian groups is decidable. The paper is well-written and supplies informally the logical concepts required.
Mathematics Subject Classification ID: 20K27
Mathematics Subject Classification ID: 20A15
zbMATH DE Number: 3845840
zbMATH Keywords: elementarily equivalent abelian groups
zbMATH Keywords: subgroup lattice
zbMATH Keywords: divisible torsion-free group
zbMATH Keywords: first order theory of abelian groups
reviewed by: Adolf Mader
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3845840