Product cones in dense pairs

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