Product cones in dense pairs

TL;DR

This paper introduces the concept of product cones in dense pairs of o-minimal structures, proving their existence in real closed fields and non-existence in linear cases, thus answering a previously open question.

Contribution

It defines product cones in dense pairs and establishes their existence in real closed fields, providing a clear distinction from linear structures.

Findings

Product cones exist in dense pairs of o-minimal structures expanding real closed fields.

Product cones do not exist in linear o-minimal structures.

The paper settles a previously open question about the structure of dense pairs.

Abstract

Let $\mathcal M=\langle M, <, +, \dots\rangle$ be an o-minimal expansion of an ordered group, and $P\subseteq M$ a dense set such that certain tameness conditions hold. We introduce the notion of a `product cone' in $\widetilde{\mathcal M}=\langle \cal M, P\rangle$, and prove: if $\mathcal M$ expands a real closed field, then $\widetilde{\mathcal M}$ admits a product cone decomposition. If $\mathcal M$ is linear, then it does not. In particular, we settle a question from [10].