Relative categoricity in abelian groups. II
type: Item
instance of: scholarly article
title: Relative categoricity in abelian groups. II
zbMATH Open document ID: 1163.03018
author: Wilfrid Hodges
author: A. V. Yakovlev
published in: Annals of Pure and Applied Logic
publication date: 2009-04-16T00:00:00Z
review text: [For Part I by \textit{W. Hodges} see: S. B. Cooper et al. (eds.), Models and computability. Invited papers from the Logic colloquium '97, European meeting of the Association for Symbolic Logic, Leeds, UK, July 6--13, 1997. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 259, 157--168 (1999; Zbl 0958.03024).] Let \(L\) be a first-order language, and \(L(P)\) be \(L\) expanded with an additional unary relation symbol \(P\). Consider \(L(P)\)-structures \(A\) such that the set of elements \(A^p\) of \(A\) satisfying \(P(x)\) is an \(L\)-substructure of the (reduct to \(L\)) of \(A\). Such an \(A\) is said to be \((\kappa,\lambda)\)-categorical if for all models \(B\), \(C\) of its theory (the set of \(L(P)\)-sentences true in \(A\)), if furthermore \(|B^P|=|C^P|=\kappa\), \(|B|=|C|=\lambda\) and \(B^P=C^P\), then there is an isomorphism from \(B\) to \(C\) which extends \(B^P=C^P\). This paper considers the case of abelian groups, i.e., \(L\) is the language with \(+,-\) and \(0\), every considered \(L(P)\)-structure \(A\) is an abelian group and \(A^p\) is a subgroup of \(A\). Such an \(A\) is said to be an abelian group pair. The paper classifies the possible pairs \((\kappa,\lambda)\) for which an abelian group pair can be \((\kappa,\lambda)\)-categorical (the relative categoricity spectra). The authors show that for \(A^P\) infinite, the possible spectra are either all \((\kappa,\kappa)\) for \(\kappa\) an infinite cardinal, all \((\kappa,\lambda)\) where either \(\omega\leq \kappa < \lambda\) or \(\omega=\kappa=\lambda\), the single pair \((\omega,\omega)\) or, finally, all pairs such that \(\omega\leq \kappa < \lambda\). They furthermore give an algebraic description in each case. Finally, they also give a characterisation of relative categoricity spectra when \(A^p\) is finite and also when both \(A\) and \(A^p\) are finite. The proofs use methods from abelian group theory, model theory of abelian group and in some cases general model th…
reviewed by: Roger Villemaire
Mathematics Subject Classification ID: 03C35
Mathematics Subject Classification ID: 03C60
Mathematics Subject Classification ID: 20A15
Mathematics Subject Classification ID: 20K99
zbMATH DE Number: 5544176
zbMATH Keywords: model theory
zbMATH Keywords: abelian groups
zbMATH Keywords: group pairs
zbMATH Keywords: relative categoricity
MaRDI profile type: Publication
full work available at URL: j.apal.2008.03.007
OpenAlex ID: W2075614966
cites work: The elementary theory of abelian groups
cites work: Automorphisms of Bounded Abelian Groups
cites work: Q5609457
cites work: Q5673086
cites work: Q4003410
cites work: Q4938555
cites work: Q3882586
cites work: On $ω_1$-categorical theories of abelian groups
cites work: Categoricity in Power
cites work: Classification theory over a predicate. I
cites work: Classification theory and the number of non-isomorphic models.
cites work: Q4714214
cites work: Categoricity over P for first order T or categoricity for \(\phi\) \(\in {\mathcal L}_{\omega_ 1\omega}\) can stop at \(\aleph_ k\) while holding for \(\aleph_ 0,\dots ,\aleph_{k-1}\)
cites work: Abelian groups \(\aleph_ 0\)-categorical over a subgroup
DOI: 10.1016/J.APAL.2008.03.007
description: scientific article; zbMATH DE number 5544176