Definable homomorphisms of abelian groups in o-minimal structures
type: Item
instance of: scholarly article
title: Definable homomorphisms of abelian groups in o-minimal structures
zbMATH Open document ID: 0949.03033
DOI: 10.1016/S0168-0072(99)00016-0
published in: Annals of Pure and Applied Logic
publication date: 2000-12-03T00:00:00Z
review text: The setting is a definable structure \({\mathcal M}\) (which might be stable) in an o-minimal structure \({\mathcal N}\), with a pair of abelian groups \(A\), \(B\) definable in \({\mathcal M}\). The main results are that if there is in \({\mathcal M}\) an infinite definable set of definable homomorphisms \(A\rightarrow B\), then in \({\mathcal M}\) there is a definable infinite field, whose additive group is definably isomorphic in \({\mathcal M}\) both to a definable subgroup of \(B\) and to a definable section of \(A\). Furthermore, the group \({\mathcal H}\) of all \({\mathcal M}\)-definable homomorphisms \(A\rightarrow B\) has a definable subgroup \(H\) such that every \({\mathcal N}\)-definable subgroup of \({\mathcal H}\) lies in the union of finitely many cosets of \(H\). Among various applications, it is shown that any \({\mathcal N}\)-definable group which is not abelian-by-finite interprets an infinite field. There are also applications to definably compact groups: for example, if \(A,B\) are definable abelian groups (in \({\mathcal N}\)) with \(A\) definably compact, then there is no infinite definable family of homomorphisms \(A\rightarrow B\), or \(B\rightarrow A\). The methods are model-theoretic. The authors work with an infinitary notion of definability for groups and rings, and work with infinitesimals (with respect to a large elementary extension) and dimension arguments.
Mathematics Subject Classification ID: 03C64
Mathematics Subject Classification ID: 03C60
Mathematics Subject Classification ID: 22C05
Mathematics Subject Classification ID: 20K45
zbMATH DE Number: 1398636
zbMATH Keywords: o-minimal structure
zbMATH Keywords: definably compact group
zbMATH Keywords: definable group
zbMATH Keywords: definable infinite field
zbMATH Keywords: infinitesimals
zbMATH Keywords: dimension
author: Ya'acov Peterzil
author: S. Starchenko
reviewed by: H. D. Macpherson
MaRDI profile type: Publication
full work available at URL: s0168-0072(99)00016-0
OpenAlex ID: W1966141759
Wikidata QID: Q127363875
description: scientific article; zbMATH DE number 1398636