Not exactly what you are looking for, but there may be some connection. The following situation is frequent: a monoid $M$, an equivalence relation $R$ on $M$ which is compatible with the monoid operator, and $M/R$ is an idempotent monoid. Examples:
- Polynomials with a positive highest-degree coefficient, as an additive cancellative monoid; $P \mathbin R Q$ is $\operatorname{deg}(P) = \operatorname{deg}(Q)$.
- Functions $\mathbb{R}^+ \rightarrow \mathbb{R}^+$ as an additive cancellative monoid, with their equivalence relation on $+\infty$.
- Sets with infinite cardinal, operation is union, $A \mathbin R B$ is $\operatorname{card}(A) = \operatorname{card}(B)$.
- Geometric or topological sets, with union, $R$ is having the same dimension (for some dimensions, e.g. Hausdorff).
All these examples share the same pattern: the resulting idempotent monoid is a totally ordered set, and the operator is $\max$. One could build more complex examples with the same ingredients, e.g. multiple-variable polynomials, with $\operatorname{deg}$ being the set of highest-degree terms, but that looks artificial.
One possible question: given an idempotent monoid, can it always be considered as the quotient of a cancellative monoid?
