Definable Functions to Quotients in Ordered Abelian Groups

TL;DR

This paper investigates the structure of definable functions from ordered abelian groups to their quotients, showing they are uniformly piecewise linear or boolean combinations of linear functions under certain conditions.

Contribution

It establishes uniform piecewise linearity and boolean combination structure for definable functions into quotients in ordered abelian groups, extending understanding of their definable families.

Findings

Definable functions to quotients are uniformly piecewise linear.

Functions into certain quotients are boolean combinations of linear functions.

Results apply to families of functions over definable convex subgroups.

Abstract

In this paper we study definable families of functions from an ordered abelian group into various naturally arising definable quotients. We show that for an ordered abelian group $G$ and definable family of convex subgroups $\{D\}_{D\in\mathcal{D}}$, any definable family of functions $\{f_D\} _{D\in\mathcal{D}}$ with $f_D:G^d\rightarrow\frac{G}{D}$ is uniformly piecewise linear; for a prime $p$, integers $s,r\geq 1$, and groups $D^{[p^s]}$ defined later, if $f_D:G^d\rightarrow\frac{G}{D+p^rG}$ or $f_D:G^d\rightarrow\frac{G}{D^{[p^s]}+p^rG}$ we instead obtain that the definable family of functions is uniformly piecewise a boolean combination of linear functions to quotients by subgroups which are uniformly definable from $D$.