For *finite* semigroups (and thus monoids), the [Krohn–Rhodes theorem](https://en.wikipedia.org/wiki/Krohn%E2%80%93Rhodes_theory) gives a decomposition into (simple) groups and aperiodic semigroups (subsemigroups of the flip-flop, which is idempotent). However, the decomposition is more complicated than a separation into a “group-like part” and an “idempotent part”; it is in terms of iterated wreath products (or alternatively, iterated semidirect products, I guess).