A semigroup is called globally idempotent when for any $x\in S$ there are $y,z\in S$ such that $x=yz$.
Is there a name for monoids whose every globally idempotent subsemigroup contains the identity element?
For example, the monoid $(\mathbb N,+)$ has this property, because if $S$ is a globally idempotent subsemigroup of $\mathbb N$, and $a$ is the smallest element of $S$, then $2a$ is the smallest element of $S+S$. Therefore $a=2a$, so $a=0$.
But there are many non-examples, even among groups.
Is anything known about such monoids?
Added: I'll explain why I'm asking about these. If $M$ is a monoid whose every globally idempotent subsemigroup contains the identity, then the natural order $\leq$ on idempotents of the power semigroup $P(M)$ coincides with $\supseteq$. And this seems like a very nice property.