Definable quotients in locally o-minimal structures

TL;DR

This paper proves the existence of definable quotients in locally o-minimal structures under certain conditions, extending the understanding of quotient constructions in these mathematical frameworks.

Contribution

It establishes conditions for definable quotients in locally o-minimal structures, particularly for locally closed sets with proper group actions.

Findings

Definable quotients exist under specific technical conditions.

Conditions are satisfied for locally closed definable subsets with proper group actions.

Provides a framework for quotient constructions in locally o-minimal structures.

Abstract

Let $\mathcal F=(F, +. \cdot, <, 0, 1, \dots)$ be a definably complete locally o-minimal expansion of an ordered field. We demonstrate the existence of definable quotients of definable sets by definable equivalence relations when several technical conditions are satisfied. These conditions are satisfied when $X$ is a locally closed definable subset of $F^n$ and there is a definable proper action of a definable group $G$ on $X$.