The lattice $M(\Lambda)$ can also be described as $M(\Lambda)=M\cap L_{\Lambda}^{\bot}$. Therefore, for a cone $\sigma\subset N_{\mathbb{R}}$, the notation just introduced agrees with the one in (4.4). Here, the polytope $\Lambda$ is contained in $N_{\mathbb{R}}$. By contrast, for a polyhedron $\Gamma\subset M_{\mathbb{R}}$, we follow Notation 3.103, so $M(\Gamma)=M\cap L_{\Gamma}$.