\({\mathcal M}\)-gap conjecture and \(m\)-normal theories
type: Item
instance of: scholarly article
title: \({\mathcal M}\)-gap conjecture and \(m\)-normal theories
zbMATH Open document ID: 0914.03040
DOI: 10.1007/BF02773473
author: Ludomir Newelski
published in: Israel Journal of Mathematics
publication date: 1999-06-21T00:00:00Z
review text: Basic definitions and properties, connections with investigations in other papers, and the construction of small weakly minimal groups with an isolated generic type of arbitrarily large finite \({\mathcal M}\)-rank or of \({\mathcal M}\)-rank \(\infty\) are introduced. Of importance is the formulation of the \({\mathcal M}\)-gap conjecture: In a small superstable theory there are no types \(p\) such that \(\omega\leq{\mathcal M}(p)< \infty\). An \(m\)-normal theory satisfies the \({\mathcal M}\)-gap conjecture. So-called \(*\)-finite tuples play a similar role for the \(m\)-independence, as imaginaries do for the forking independence. Such special objects allow the author to define the notion of an \(m\)-normal theory more smoothly. Elements which are \(*\)-algebraic over a finite set of \(*\)-finite tuples are important for the description of \(m\)-independence. The proof of the fact that a 1-based superstable theory with few countable models is \(m\)-normal involves an analysis of traces of types under the few models assumption. Traces of complete types are traces of some formulas. The question arises whether any superstable theory with few countable models is \(m\)-normal. The examples considered in the introduction are groups. The additional properties of \({\mathcal M}\)-rank on generic types of a group are considered. Finally, a notion of independence induced by measure is considered. It is compared with the notion of \(m\)-independence.
Mathematics Subject Classification ID: 03C45
Mathematics Subject Classification ID: 03C65
Mathematics Subject Classification ID: 03C50
Mathematics Subject Classification ID: 20E99
Mathematics Subject Classification ID: 20A15
zbMATH DE Number: 1216498
zbMATH Keywords: stable theory
zbMATH Keywords: stationarizations of types
zbMATH Keywords: measure independence
zbMATH Keywords: \(*\)-algebraic element
zbMATH Keywords: type-definable group
zbMATH Keywords: generic subgroup
zbMATH Keywords: small weakly minimal groups
zbMATH Keywords: isolated generic type
zbMATH Keywords: \({\mathcal M}\)-rank
zbMATH Keywords: \({\mathcal M}\)-gap conjecture
zbMATH Keywords: superstable theory
zbMATH Keywords: \(m\)-normal theory
zbMATH Keywords: \(*\)-finite tuples
zbMATH Keywords: \(m\)-independence
zbMATH Keywords: countable models
zbMATH Keywords: traces of types
zbMATH Keywords: independence induced by measure
Wikidata QID: Q122945548
reviewed by: Hans-Jürgen Vogel
MaRDI profile type: Publication
cites work: Q3994342
cites work: Superstable groups
cites work: Q3801534
cites work: The classification of small weakly minimal sets. II
cites work: Q3775535
cites work: Forcing isomorphism II
cites work: A survey of basic stability theory, with particular emphasis on orthogonality and regular types
cites work: A proof of Saffe's conjecture
cites work: On type definable subgroups of a stable group
cites work: A model and its subset
cites work: Meager forking
cites work: ℳ-rank and meager types
cites work: On atomic or saturated sets
cites work: Q4895442
cites work: Geometry of *-Finite Types
cites work: Classification theory and the number of non-isomorphic models.
cites work: A proof of Vaught's conjecture for \(\omega\)-stable theories
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description: scientific article; zbMATH DE number 1216498