In both cases, the fact that (a) implies (b) and that (b) implies (c) is clear. The fact that (c) implies (d) follows from equation (4.93). The fact that (1d) implies (1a) is [KKMS73, §IV.3(k)].

Finally, we prove that (2d) implies (2a). Let $\psi$ be an H-lattice concave function. Each pair $(m,l)\in{\widetilde{M}}$ defines a rational section $\varpi^{l}\chi^{m}s_{\psi}$ of $D_{\psi}$. The section is regular if and only if the function $m(u)+l$ lies above $\psi$. Moreover, for a polyhedron $\Lambda\in\Pi$, this section does not vanish on ${\mathcal{X}}_{\Lambda}$ if and only if $\psi(u)=m(u)+l$ for all $u\in\Lambda$. Therefore, the affine pieces of the graph of $\psi$ define a set of global sections that generate $\mathcal{O}(D_{\psi})$. ∎