Definition
Let \(P\) be any monoid and \(k\) a field. A logarithmic point \((\Spec k, P)\) is a log scheme whose underlying scheme is \(\Spec k\) and whose log structure is \(\alpha:k^\times \oplus P\to k^\times\) given by \[\alpha(a,p) = \begin{cases}a & \text{when } p = 0 \\ 0 &\text{otherwise}\end{cases}.\] The standard log point is the log point \((\Spec k, \mathbb N)\).