On variants of \(o\)-minimality
type: Item
instance of: scholarly article
title: On variants of \(o\)-minimality
zbMATH Open document ID: 0858.03039
DOI: 10.1016/0168-0072(95)00037-2
author: Dugald Macpherson
author: Charles Steinhorn
published in: Annals of Pure and Applied Logic
publication date: 1996-10-13T00:00:00Z
review text: A \(C\)-structure \((M,C)\) consists of a structure \(M\) along with a ternary relation \(C(x;y,z)\) on the domain of \(M\), where the domain of \(M\) has at least 2 elements and \(C\) satisfies certain additional axioms. Let \((T,\leq)\) be an infinite tree in which there are incomparable elements above any node of the tree. The set \(M\) of maximal chains of \(T\) with the ternary relation \(C(x;y,z)\) induced on \(M\) by requiring that \(y\) and \(z\) branch above where \(x\) and \(y\) branch turns out to be a very representative \(C\)-structure. A \(C\)-minimal structure is an expansion \({\mathcal M} = (M,C, \dots)\) of a \(C\)-structure \((M,C)\) such that for every \({\mathcal N} \equiv {\mathcal M}\), every definable subset of \(N\) is quantifier-free definable in \((N,C)\). This paper is an investigation of \(C\)-minimal structures as an interesting variant of \(o\)-minimality. This choice is influenced by work of Adeleke and Neumann on Jordan groups. The authors develop some general model theory for \(C\)-minimal structures. They go on to study \(C\)-minimal groups and fields. The paper ends with some other variants of \(o\)-minimality.
Mathematics Subject Classification ID: 03C50
Mathematics Subject Classification ID: 03C60
zbMATH DE Number: 908472
zbMATH Keywords: \(C\)-minimal structure
zbMATH Keywords: variant of \(o\)-minimality
zbMATH Keywords: \(C\)-minimal group
zbMATH Keywords: \(C\)-minimal fields
zbMATH Keywords: Jordan groups
reviewed by: J. M. Plotkin
MaRDI profile type: Publication
full work available at URL: 0168-0072(95)00037-2
OpenAlex ID: W2051492734
Wikidata QID: Q61848372
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DBLP publication ID: journals/apal/MacphersonS96
description: scientific article; zbMATH DE number 908472