Meager forking
type: Item
instance of: scholarly article
title: Meager forking
zbMATH Open document ID: 0817.03017
DOI: 10.1016/0168-0072(94)90028-0
author: Ludomir Newelski
published in: Annals of Pure and Applied Logic
publication date: 1995-01-09T00:00:00Z
review text: Let \(T\) be a stable theory. Recall that if \(A\) is a finite set and \(\varphi(x)\) is a formula, then the set of all strong types over \(A\) consistent with \(\varphi(x)\) is always closed, and open when \(\varphi(x)\) does not fork over \(A\). If \(P\) is a closed set of strong types over \(A\), then forking is said to be meager on \(P\) if and only if for every formula \(\varphi(x)\) forking over \(A\), the set of \(r\in P\) consistent with \(\varphi(x)\) is nowhere dense in \(P\). Correspondingly a notion of meager regular type is proposed. First it is shown that a meager regular type is locally modular. Recall that if \(T\) is superstable and \(p\) is a non- trivial locally modular regular type, then over some finite set \(A\) there is a definable Abelian group \(G\) with locally modular generics non- orthogonal to \(p\). If \(p\) is meager, then also the generics of \(G\) are meager. Now assume that \(T\) has \(< 2^{\aleph_ 0}\) countable models and \(G\) is a definable Abelian group with locally modular regular generics. Under these conditions, a counterpart of Saffe's conjecture is proved. More precisely, it is shown that, if \(A\) is finite, \(a\in G\) is generic over \(A\) and \(q= \text{tp}(a/A)\), then one of the following conditions holds. (1) There is \(b\in G\) such that \(b\) is generic over \(A\), \(b\in a+ S\) and \(\text{tp}(b/A)\) is isolated. (2) There are finitely many realizations \(a_ 0, \dots, a_{n-1}\) of \(q\) such that, for every \(b\models q\), there is \(i< n\) such that \(\text{stp}(b)\) is realized in \(a_ i+ S\). Using these results, the conjecture of vanishing multiplicities is proved for a superstable \(T\), saying that if \(T\) has \(< 2^{\aleph_ 0}\) countable models, then every complete type is multiplically stable. As a final application, a question concerning topological stability of pseudo-types is treated.
Mathematics Subject Classification ID: 03C45
zbMATH DE Number: 706878
zbMATH Keywords: superstable theory
zbMATH Keywords: stable theory
zbMATH Keywords: meager regular type
zbMATH Keywords: definable Abelian group with locally modular regular generics
zbMATH Keywords: counterpart of Saffe's conjecture
zbMATH Keywords: conjecture of vanishing multiplicities
zbMATH Keywords: topological stability of pseudo-types
reviewed by: Carlo Toffalori
MaRDI profile type: Publication
full work available at URL: 0168-0072(94)90028-0
OpenAlex ID: W4213192561
cites work: Q3994342
cites work: The geometry of weakly minimal types
cites work: Q3801534
cites work: The classification of small weakly minimal sets. II
cites work: Q3784054
cites work: Kueker's conjecture for stable theories
cites work: Unidimensional theories are superstable
cites work: A dichotomy theorem for regular types
cites work: Uncountable theories that are categorical in a higher power
cites work: Abelian groups with modular generic
cites work: Superstable theories with few countable models
cites work: Weakly minimal formulas: A global approach
cites work: A proof of Saffe's conjecture
cites work: On type definable subgroups of a stable group
cites work: Scott analysis of pseudotypes
cites work: ℳ-rank and meager types
cites work: Classification theory and the number of non-isomorphic models
Recommended article: Fundamentals of forking
Recommended article: A Generalization of Forking
Recommended article: Around stable forking
Recommended article: Meager forking and \(m\)-independence
Recommended article: Stable forking and imaginaries
Recommended article: Forking in VC-minimal theories
Recommended article: Q4764628
Recommended article: The fork calculus
Recommended article: Forking in Simple Unstable Theories
Recommended article: Measures and forking
description: scientific article; zbMATH DE number 706878