Classifying totally categorical groups (and others)
type: Item
instance of: scholarly article
title: Classifying totally categorical groups (and others)
zbMATH Open document ID: 0854.03038
DOI: 10.1016/0168-0072(95)00012-7
author: Katrin Tent
published in: Annals of Pure and Applied Logic
publication date: 1997-01-13T00:00:00Z
review text: The author studies the structure of groups definable in a unidimensional, one-based theory \(T\) in which every minimal type is locally finite; any totally categorical theory satisfies these conditions. A connected group \(\wedge\)-definable in \(T\) is said to be basic; by Hrushovski-Pillay's theorem, every basic group is abelian. A basic group is said to be irreducible if its generic is; here an element of a group is called irreducible if there are no \(b,c \in \text{acl} (a)\) of smaller rank with \(a \in \text{acl} (bc)\). Main result: for every irreducible basic \(H\), there exists an irreducible basic supergroup \(G\) of \(H\) in \(\text{acl}^{\text{eq}} (H)\) -- the so-called modular closure of \(H\) -- such that, for any \(n < \omega\), every connected subgroup of \(G^n\) is the connected component of a subgroup linearly defined over the ring of quasi-endomorphisms of \(G\). Thus \(H\) is a subgroup of a \(\wedge\)-definable essentially pure module, which is moreover ``almost'' quantifier eliminable. The results extend the analysis of \textit{S. Buechler} from his preprint [Vaught's conjecture for superstable theories of finite rank (1992)] to the case of a unidimensional, one-based theory with all minimal types locally finite. The key point is the fact that there is a bound for the ranks of irreducibles algebraic in the group. Buechler used the few models assumption for that; the author uses instead the local finiteness assumption -- the modular closure is found as a group with irreducible generics of maximal rank. She gives an example which shows that the local finiteness assumption cannot be omitted. For a totally categorical \(H\) of \(U\)-rank 2, an explicit construction for the modular closure of \(H\) is given.
Mathematics Subject Classification ID: 03C60
Mathematics Subject Classification ID: 03C35
Mathematics Subject Classification ID: 20A15
zbMATH DE Number: 908457
zbMATH Keywords: unidimensional theory
zbMATH Keywords: definable group
zbMATH Keywords: one-based theory
zbMATH Keywords: totally categorical theory
zbMATH Keywords: basic group
zbMATH Keywords: modular closure
zbMATH Keywords: essentially pure module
zbMATH Keywords: irreducibles
zbMATH Keywords: local finiteness assumption
reviewed by: Oleg Belegradek
MaRDI profile type: Publication
full work available at URL: 0168-0072(95)00012-7
OpenAlex ID: W2047909419
cites work: Q3994342
cites work: Elimination of quantifiers for modules
cites work: On one-based theories
cites work: Locally modular theories of finite rank
cites work: Vaught's conjecture for superstable theories of finite rank
cites work: Q3784054
cites work: Unidimensional theories are superstable
cites work: Q3775535
cites work: Relation entre le rang U et le poids
cites work: Uncountable theories that are categorical in a higher power
cites work: A survey of basic stability theory, with particular emphasis on orthogonality and regular types
cites work: Q3028990
cites work: Model theory of modules
description: scientific article; zbMATH DE number 908457