  Given two elements $x,y$ of a semigroup $X$ we write $x\lesssim y$ if for
every homomorphism $\chi:X\to\{0,1\}$ we have $\chi(x)\le\chi(y)$. The
quasiorder $\lesssim$ is called the $binary$ $quasiorder$ on $X$. It induces
the equivalence relation $\Updownarrow$ that coincides with the least
semilattice congruence on $X$. In the paper we discuss some known and new
properties of the binary quasiorder on semigroups.
