Locally o-minimal structures
type: Item
instance of: scholarly article
title: Locally o-minimal structures
zbMATH Open document ID: 1259.03051
author: Tomohiro Kawakami
author: Kota Takeuchi
author: Akito Tsuboi
author: Hiroshi Tanaka
published in: Journal of the Mathematical Society of Japan
publication date: 2012-10-02T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.jmsj/1343133743
Mathematics Subject Classification ID: 03C64
zbMATH DE Number: 6090092
zbMATH Keywords: local o-minimality
zbMATH Keywords: cell decomposition
zbMATH Keywords: simple products
MaRDI profile type: Publication
OpenAlex ID: W1966671634
cites work: Q4392286
cites work: Geometric categories and o-minimal structures
cites work: Definable Sets in Ordered Structures. II
cites work: Weakly o-minimal structures and real closed fields
cites work: A guide to classical and modern model theory
cites work: Geometry of subanalytic and semialgebraic sets
cites work: Q5442085
cites work: Notes on local o-minimality
cites work: Weakly o-minimal nonvaluational structures
DOI: 10.2969/JMSJ/06430783
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review text: This paper continues the study of locally o-minimal structures, following the work of \textit{C. Toffalori} and \textit{K. Vozoris} [Math. Log. Q. 55, No. 6, 617--632 (2009; Zbl 1222.03045)]. A structure expanding a dense linear order without endpoints \((M,<)\) is said to be locally o-minimal if for any definable set \(A\subseteq M\) and any point \(a\in M\) there is an open interval \(I\) around \(a\) such that \(I\cap A\) is a finite union of points and intervals. Such a structure is strongly o-minimal if for each \(a\) there is an interval \(I\) such that \(I\cap A\) is a finite union of points and intervals for any definable \(A\). The authors also introduce a notion of uniformly locally o-minimal.NEWLINENEWLINEThe first main result is a characterization of strong local o-minimality in terms of the o-minimality of certain local structures on intervals. (These local structures, as the authors call them, are normally known as induced structures.) The authors then introduce a notion of simple product and show several results on the simple product of two locally o-minimal structures. Using these results the authors obtain a characterization of locally o-minimal expansions of \((\mathbb R,+,<,\mathbb Z)\).
description: scientific article; zbMATH DE number 6090092