Defining new linear functions in tame expansions of the real ordered additive group

TL;DR

This paper investigates conditions under which certain tame expansions of ordered groups define new linear functions, extending the understanding of definability in o-minimal and semibounded structures.

Contribution

It introduces the notion of semibounded expansions for arbitrary ordered groups and characterizes when such expansions define new total linear functions.

Findings

Identifies conditions for defining new total linear functions in semibounded structures.

Provides examples where expansions define functions not originally definable in the base structure.

Extends the concept of semiboundedness from o-minimal to more general ordered groups.

Abstract

We explore \emph{semibounded} expansions of arbitrary ordered groups; namely, expansions that do not define a field on the whole universe. We introduce the notion of a \emph{semibounded} expansion of an arbitrary ordered group, extending the usual notion from the o-minimal setting. For $\mathcal{R}=( \mathbb{R}, <, +, \ldots)$, a semibounded o-minimal structure and $P\subseteq \mathbb{R}$ a set satisfying certain tameness conditions, we discuss under which conditions $(\mathcal R,P)$ defines total linear functions that are not definable in \mathcal{R}. Examples of such structures that does define new total linear functions include the cases when $\mathcal{R}$ is a reduct of $(\mathbb{R},<,+,\cdot_{\upharpoonright (0,1)^2},(x\mapsto \lambda x)_{\lambda\in I\subseteq \mathbb{R}})$, and $P= 2^\mathbb{Z}$, or $P$ is an iteration sequence (for any $I$) or $P=\mathbb{Z}$, for $I=\mathbb{Q}$.