Definition
A chart for a log structure \(\alpha_X:\mathcal M_X\to \mathcal O_X\) on a scheme \(X\) is a monoid morphism \(\varphi:\underline P\to \mathcal O_X\) where \(\underline P\) is the constant sheaf valued in the monoid \(P\) such that \(\varphi\) factors as \(\alpha_X\circ \beta\) for some \(\beta:\underline P \to \mathcal M_X\) which induces an isomorphism between \(\mathcal M_X\) and the log structure associated to \(\varphi:\underline P\to \mathcal O_X\).
If \(\alpha_X:\mathcal M_X\to \mathcal O_X\) possesses charts etale locally then we say the log structure is fine.