A monoid \(S\) is said to be cancellative if whenever \(a + c = b + c\), \(a = b\). This terminology is more common in the study of commutative semi-groups (I think) but log geometers prefer calling such monoids integral. We use it here because the below lemma is Lemma 2.3 in [CHL24], and we match their terminology.
Lemma
Let \(A, B\) and \(C\) be cancellative monoids with morphisms \(\beta:A\to B\) and \(\gamma: A\to C\). Then
\begin{align*} B\oplus_A C \cong B\oplus C/\sim \end{align*}where \((b_1, c_1) \sim (b_2, c_2)\) whenever there is some \(s,t\in A\) such that \((b_1,c_1) + (\beta(s),-\gamma(s)) = (b_2, c_2) + (\beta(t),-\gamma(t))\). Note that this equation takes place in \((B\oplus C)^{gp}\), into which \(B\oplus C\) embeds by integrality.
Example
Let \(Q\) be an integral monoid with a morphism \(\mathbb N\to Q\) defined \(1\mapsto \rho\) and consider the diagonal morphism \(\mathbb N\to \mathbb N^2\) given \(n\mapsto (n, n)\). The fiber product \(\mathbb N^2 \oplus_{\mathbb N} Q\) plays a prominent role in the structure of log smooth curves; it is the ghost sheaf of the log structure etale locally near a node of a curve. In this case, the equivalence relation can be written as a condition on elements inside \(\mathbb Z^2\oplus Q^{gp}\):
We say that \((a,b,q) \sim (a',b',q')\) if there exist \(n,m\in \mathbb N\) such that
- \((a,b) + (n,n) = (a',b') + (m,m) \implies (a - a', b- b') = (m - n, m-n)\)
- \(q - n\rho = q' - m\rho = q - q' = (m - n)\rho\)
or in other words, if \(a - a' = b - b'\) and \(q - q' = (a - a')\rho\).