An ordered semigroup $S$ is called left (resp. right) simple if  $S$ does
not contain proper left (resp. right) ideals. In particular, a
$poe$-semigroup $S$ is left (resp. right) simple if and only if $S$ does
not contain proper left (resp. right) ideal elements. Each left (resp.
right) ideal element is a bi-ideal element. An ordered semigroup $S$ is
left and right simple if and only if $S$ does not contain proper
bi-ideals. In particular, a $poe$-semigroup $S$ is left and right simple
if and only if $S$ does not contain proper bi-ideal elements.