Product cones in 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].