Definition
Let \(\alpha:P_X\to \mathcal O_X\) be a pre-log structure on \(X\). The log structure associated to \(P_X\) is the sheaf of monoids
\begin{align*} \mathcal M_X := \frac{P_X\oplus \mathcal O_X^\times}{\{(p, \alpha(p)^{-1}) ~|~ p\in \alpha^{-1}(\mathcal O^\times_X)\}} \end{align*}in addition to the morphism of sheaves of monoids \(\alpha_X:\mathcal M_X\to \mathcal O_X\) defined via
\begin{align*} \alpha_X(p,f):=\alpha(p)\cdot f. \end{align*}