  A semigroup 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
$h(x)\ne h(y)$. Developing results of Putcha and Weissglass, we characterize
$E$-separated semigroups via certain commutativity properties of idempotents of
$X$. Also we characterize $E$-separated semigroups in the class of
$\pi$-regular $E$-semigroups.
