Let ${\mathcal{X}}$ be a toric scheme and ${\mathcal{L}}$ a line bundle on ${\mathcal{X}}$. A toric structure on ${\mathcal{L}}$ is the choice of an element $z$ of the fibre ${\mathcal{L}}_{x_{0}}$, where $x_{0}\in{\mathcal{X}}_{\eta}$ is the distinguished point. A toric line bundle on ${\mathcal{X}}$ is a pair $({\mathcal{L}},z)$, where ${\mathcal{L}}$ is a line bundle over ${\mathcal{X}}$ and $v$ is a toric structure on ${\mathcal{L}}$. Frequently, when the toric structure is clear from the context, the element $z$ will be omitted from the notation and a toric line bundle will be denoted by the underlying line bundle. A toric section is a rational section that is regular and non vanishing over the principal open subset $X_{0}\subset{\mathcal{X}}_{\eta}$ and such that $s(x_{0})=z$. Exactly as in the case of toric varieties over a field, each $\mathbb{T}$-Cartier divisor defines a toric line bundle ${\mathcal{O}}(D)$ together with a toric section. When the $\mathbb{T}$-Cartier divisor comes from an H-lattice function $\psi$, the toric line bundle and toric section will be denoted ${\mathcal{L}}_{\psi}$ and $s_{\psi}$ respectively.