Strongly determined types
type: Item
instance of: scholarly article
title: Strongly determined types
zbMATH Open document ID: 0934.03049
DOI: 10.1016/S0168-0072(99)00004-4
author: A. A. Ivanov
author: Dugald Macpherson
published in: Annals of Pure and Applied Logic
publication date: 2000-04-06T00:00:00Z
review text: The paper proposes a weak form of independence for a large class of theories \(T\), including several unstable examples. This independence is based on the notion of strongly determined extension of a type. Given a type \(p\) over \(A\), a strongly determined extension of \(p\) over \(A\) provides, for every model \(M\) of \(T\) containing \(A\), a type \(q\) over \(M\) extending \(p\) such that, if \(\vec{c}\) realizes \(q\), then every partial elementary map from \(M\) to \(M\) fixing \(\text{acl}^{\text{eq}} (A)\) pointwise is elementary over \(\vec{c}\). As said before, strongly determined types suggest an obvious notion of independence. It is noticed that this is just the forking independence in the stable setting, but may differ even for simple unstable theories (in fact, there are simple theories without strongly determined types over \(\emptyset\)). Furthermore, the axioms of symmetry, extension and local character may fail in the new setting. Theories with strongly determined types over any set include some natural unstable cases, such as weakly o-minimal theories. Likewise \(C\)-minimal structures admit strongly determined types, while, for \(P\)-minimal theories, only the existence of a strongly determined type over \(\emptyset\) is proved; for smoothly approximable theories and certain binary homogeneous structures, this existence result is stated over a singleton. On the other hand, some more artificial examples of structures which do not admit strongly determined types over any finite sets are provided. A possible statement of a Finite Equivalence Relation Theorem is discussed. Also developed is a notion of multiplicity for strongly determined types, leading to an analogue of \(\omega\)-stable theories in this setting. \textsl{Involved} finite simple groups in the sense of Hrushovski are treated. Finally, an application to covers of structures is provided.
Mathematics Subject Classification ID: 03C45
zbMATH DE Number: 1339985
zbMATH Keywords: independence
zbMATH Keywords: multiplicity
zbMATH Keywords: definable type
zbMATH Keywords: finite equivalence relation theorem
reviewed by: Carlo Toffalori
MaRDI profile type: Publication
cites work: Relations related to betweenness: their structure and automorphisms
cites work: Q3890672
cites work: The Small Index Property for Free Groups and Relatively Free Groups
cites work: Q3656407
cites work: \(\aleph _ 0\)-categorical, \(\aleph _ 0\)-stable structures
cites work: Q4026891
cites work: Q4355921
cites work: AFFINE COVERS OF LIE GEOMETRIES AND THE AMALGAMATION PROPERTY
cites work: \(p\)-adic and real subanalytic sets
cites work: Algebraic theories with definable Skolem functions
cites work: One-Dimensional p -Adic Subanalytic Sets
cites work: Structure of partially ordered sets with transitive automorphism groups
cites work: Finite covers with finite kernels
cites work: Counterexamples to a conjecture on relative categoricity
cites work: On the automorphism groups of finite covers
cites work: Q4356355
cites work: Some Constructions for Locally Finite Groups
cites work: Cell decompositions of \(C\)-minimal structures
cites work: A version of o-minimality for the p-adics
cites work: Countable homogeneous relational structures and ℵ0-categorical theories
cites work: Q3686704
cites work: The Small Index Property for ω‐Stable ω‐Categorical Structures and for the Random Graph
cites work: Cohomology of Structures and Some Problems of Ahlbrandt and Ziegler
cites work: Finitely axiomatizable ℵ1 categorical theories
cites work: Q4873972
cites work: ℵ0 -Categorical Structures Smoothly Approximated by Finite Substructures
cites work: Simple theories
cites work: Forking in Simple Unstable Theories
cites work: Binarity and \(\aleph_0\)-categoricity for variants of o-minimality
cites work: On the category of models of a complete theory
cites work: Q3801533
cites work: Autour De La Propriété Du Petit Indice
cites work: An introduction to forking
cites work: One-dimensional fibers of rigid subanalytic sets
cites work: Finite axiomatizability and theories with trivial algebraic closure
cites work: Q4309353
cites work: Weakly o-minimal structures and real closed fields
cites work: On variants of \(o\)-minimality
cites work: Q3041166
cites work: Simple unstable theories
cites work: Classification theory and the number of non-isomorphic models.
cites work: Reducts of the random graph
cites work: Algebraic types and automorphism groups
cites work: Generic Automorphisms of Homogeneous Structures
description: scientific article; zbMATH DE number 1339985