For finite semigroups (and thus monoids), the Krohn–Rhodes theorem gives a decomposition into (simple) groups and aperiodic semigroups (subsemigroup of the flip-flop).

For finite semigroups (and thus monoids), the Krohn–Rhodes theorem 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).