Locally o-minimal structures
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 \((\mathbb R,+,<,\mathbb Z)\).
scientific article; zbMATH DE number 6090092
scientific article