**Definition.** A semigroup $X$ is called *$E$-separated* if for any distinct idempotents $x,y\in X$ there exists a homomorphism $h:X\to Y$ to a semilattice $Y$ such that $h(x)\ne h(y)$.


Observe that $X$ is $E$-separated if and only if the smallest semilattice congruence on $X$ is idempotent-separating. This seems to be an important notion, so I suggest that it could (and should) be studied in Theory of Semigroups, maybe under some different name. *Do you know any suitable references?*



$E$-separated semigroups
I finally found an answer to my own question: by an old (nontrivial) result of [Putcha and Weissglass](https://msp.org/pjm/1971/39-1/p21.xhtml), a semigroup $X$ is $E$-separated if and only if it is viable.


A semigroup $X$ is *viable* if for any elements $x,y\in X$ with $\{xy,yx\}\subseteq E(X)$ we have $xy=yx$.


However, up to my taste, the notion of viality is a bit less intuitive comparing to the equivalent notion of $E$-separatedness.


Another difference between those (equivalent) notions is that the viality is an internal property whereas the $E$-separatedness is external (defined with the help of external objects). But the $E$-separatedness also can be equivalently defined in internal terms: a semigroup is $E$-separated if the smallest semilattice congruence on $X$ separates idempotents.


Neither $E$-separatedness not the viablity are present in [this Wikipedia page](https://en.wikipedia.org/wiki/Special_classes_of_semigroups).
However they form an important class of semigroups.


