Product cones in dense pairs

Abstract

Let $\mathcal{M}=⟨M,<,+,\dots ⟩$ 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 $\tilde{\mathcal{M}}=⟨\mathcal{M},\mathcal{P}⟩$, and prove: if $\mathcal{M}$ expands a real closed field, then $\tilde{\mathcal{M}}$ admits a product cone decomposition. If $\mathcal{M}$ is linear, then it does not. In particular, we settle a question from [10].