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Definable Functions to Quotients in Ordered Abelian Groups

Harper Wells

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 $D$.

Definable Functions to Quotients in Ordered Abelian Groups

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 and definable family of convex subgroups , any definable family of functions with is uniformly piecewise linear; for a prime , integers , and groups defined later, if or 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 .

Paper Structure

This paper contains 10 sections, 29 theorems, 85 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and Notation
  3. Definable functions from an ordered abelian group into a quotient
  4. Functions to quotients by definable convex subgroups
  5. Functions to quotients by subgroups of the form $\alpha^{[p^s]}+p^rG$, with $\alpha\in \mathcal{S}_n$ for some $n$
  6. Functions to quotients by subgroups of the form $D+p^rG$ with $D\leq G$ a definable convex subgroup
  7. Examples
  8. Patterns in $\textrm{dim} _{\mathbb{F}_p}\frac{\alpha^{[p^s]}+pG}{\alpha^{[p^{s+1}]}+pG}$
  9. An example of a definable function that is not piecewise linear
  10. Uniformity for families of subgroups of the form $D+p^rG$

Key Result

Theorem 1.1

Suppose either $H = D$ or $H= D+p^rG$ for some $D\leq G$ convex and $r\geq 1$. Also suppose that $d,e\in\mathbb{N}_{>0}$, and $f:G^d\rightarrow (\frac{G}{H})^{(e)}$ is a $B$-definable function, where by $X^{(e)}$ we mean subsets of $X$ of size $e$. Then we can find some $k\in \mathbb{N}$, $\textrm{a

Theorems & Definitions (58)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • proof
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 4.1
  • proof
  • Lemma 5.1
  • proof
  • ...and 48 more