Grouplike and idempotent monoids Call a monoid group-like if it embeds into its group completion. There exists an obvious tension between group-like and idempotent monoids in that a group cannot contain non-trivial idempotent elements, so any idempotent elements of a monoid have to be trivialized by its group completion. Furthermore, almost all important examples of monoids are either one or the other. My question is, has this been formalized anywhere? Is there maybe a decomposition theorem for monoids into their grouplike and idempotent parts?
 A: 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; $R$ is $deg(P) = deg(Q)$.

*Functions $\mathbb{R}^+ \rightarrow \mathbb{R}^+$ as an additive cancellative monoid, with their equivalence relation on $+\infty$.

*Sets with infinite cardinal, operation is union, $R$ is $card(A) = card(B)$.

*Geometric or topologic sets, with union, $R$ is having 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 $deg$ 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?
