What are the monoids in which every globally idempotent subsemigroup contains the identity element? 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.
 A: There is of course no such name, as this wouldn't bear much information about the semigroup.
Now, some beautiful paper, where the global idempotency really appears, and really means something:
Robertson, E. F.; Ruškuc, N.; Wiegold, J. Generators and relations of direct products of semigroups. Trans. Amer. Math. Soc. 350 (1998), no. 7, 2665–2685.
Also, if you are interested in power semigroups or finitary power semigroups, there is a whole thesis by Peter Gallagher (with very nice results about finitary power semigroups of groups, and some nice chacracterisation theorems and some really difficult to come up with examples!) from St Andrews.
Also, there is a series of papers by Volodymyr Mazorchuk on various power semigroups of finite transformation semigroups. Also check on the web some results by Mykola Rybak (I think from the journal "Algebra and Discrete Mathematics")
There is some work by Ash -- which is great if you like finite semigroups, you can find a lot about this in the recent book by Benjamin.
There is the most important result in this topic that there are two infinite non-isomorphic semigroups S and T, such that their finitary power semigroups are isomorphic.
Finally, the most important question in the topic is whether there are two non-isomorphic FINITE semigroups whose power semigroups would be isomorphic.
Also, just in case, my e-mail is: victor.maltcev@gmail.com
A: I don't know if there is a name for this. Note that this can never happen for a finite monoid which is not a group.  The minimal ideal of a finite semigroup is globally idempotent and if the finite monoid is not a group, this minimal ideal does not contain the identity. 
Afterthought. Your semigroup cannot have any non-identity idempotents if its globally idempotent subsemigroups contain the identity. 
