Definition
Given a sheaf of monoids \(\mathcal M\) on a scheme \(X\), a log-ideal is a sheaf of monoid ideals \(\mathcal K\subseteq \mathcal M\). The sheaf of monoid ideals \(\mathcal K\) is said to be coherent if locally on \(X\), \(\mathcal K\) is generated by a finite set of sections.
An idealized log scheme is data \((X,\mathcal M_X, \alpha_X, \mathcal K_X)\) where \((X, \mathcal M_X, \alpha_X)\) is an ordinary log scheme, with \(\alpha_X:\mathcal M_X\to \mathcal O_X\) the structure map, and \(\mathcal K_X\subseteq \mathcal M_X\) a log-ideal such that \(\mathcal K_X\subseteq \alpha^{-1}_X(0)\). A morphism of idealized log schemes \(f:(X, \mathcal K_X)\to (Y, \mathcal K_X)\) is a morphism \(f:X\to Y\) of log schemes such that \[f^\flat(f^{-1}\mathcal K_X) \subseteq \mathcal K_X.\]
For divisorial log structures, \(\mathcal K_X\) will correspond to an ideal sheaf \(\mathcal I\subseteq \mathcal O_X\) so that the corresponding closed subscheme \(Z\) is a subset of the divisor defining the log structure.