Show that in Example 8.2, (5), $\overline{\mathcal{M}}_{X,x}=P$ if $\dim\sigma=\dim M_{\mathbb{R}}$ and $x$ is the unique zero-dimensional torus orbit of $X$. More generally,

$$\overline{\mathcal{M}}_{X,x}={{\tau}^{\scriptscriptstyle\vee}\cap N\over\tau^{\perp}\cap N}=\operatorname{Hom}_{monoid}(\tau\cap M,\mathbb{N}),$$

when $x\in X$ is in the torus orbit corresponding to a face $\tau$ of $\sigma$. In particular, $\tau$ can be recovered as $\operatorname{Hom}_{monoid}(\overline{\mathcal{M}}_{X,x},\mathbb{R}_{\geq 0})$, where $\mathbb{R}_{\geq 0}$ is the additive monoid of non-negative real numbers. ∎