Locally o-minimal open core

TL;DR

This paper characterizes when the open core of a definably complete expansion of a densely ordered abelian group is locally o-minimal, linking it to properties of definable closed subsets being either discrete or containing an interval.

Contribution

It provides a precise criterion for local o-minimality of the open core in terms of the structure's definable closed sets.

Findings

Open core is locally o-minimal iff definable closed sets are discrete or contain an interval

Establishes a necessary and sufficient condition for local o-minimality in this context

Connects properties of definable closed sets to local o-minimality

Abstract

We demonstrate that the open core of a definably complete expansion of a densely linearly ordered abelian group is locally o-minimal if and only if any definable closed subset of $R$ is either discrete or contains a nonempty open interval. Here, the notation $R$ denotes the universe of the original structure.