On the groups with homogeneous theory
type: Item
instance of: scholarly article
title: On the groups with homogeneous theory
zbMATH Open document ID: 0921.03044
author: Hirotaka Kikyo
published in: Tsukuba Journal of Mathematics
publication date: 1999-07-25T00:00:00Z
review text: If \(2\leq m < n\) then a structure \(M\) is \((m,n)\)-homogeneous if for any \(\bar{a},\bar{b} \in M^n\), \(\bar{a}\) and \(\bar{b}\) have the same type if and only if corresponding \(m\)-subtuples have the same type. A complete theory is \((m,n)\)-homogeneous if all its subtuples are. The author discusses the conjecture that if \((M,\cdot)\) is a group (possibly with extra structure), then its theory is not \((m,n)\)-homogeneous for any \(m,n\) with \(2\leq m<n\). Handa has shown that no infinite abelian \(p\)-group can be interpreted in a theory which (for some \(m\)) is \((m,n)\)-homogeneous for all \(n>m\), and the above conjecture would imply Handa's result without the `abelian \(p\)-group' assumption. The author proves the conjecture for \(\omega\)-categorical theories, stable theories, o-minimal theories, and also shows that if \((M,\cdot)\) has simple theory then its theory is not \((2,3)\)-homogeneous.
Mathematics Subject Classification ID: 03C60
Mathematics Subject Classification ID: 03C50
zbMATH DE Number: 1247897
zbMATH Keywords: homogeneous structure
zbMATH Keywords: interpretable group
zbMATH Keywords: \(\omega\)-categorical theories
zbMATH Keywords: stable theories
zbMATH Keywords: o-minimal theories
zbMATH Keywords: homogeneous theories
reviewed by: H. D. Macpherson
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.21099/tkbjm/1496163601
OpenAlex ID: W633844197
DOI: 10.21099/TKBJM/1496163601
description: scientific article; zbMATH DE number 1247897