The right (resp. left) simple and the simple ordered
semigroups play an important role in the structure of
ordered semigroups. In this note we prove that there is no
essential difference between the right (resp. left) simple
and the right (resp. left) 0-simple ordered semigroups.
In this respect, we prove that an ordered groupoid $S$
without zero is right (resp. left) simple if and only if
the ordered groupoid $S^0$ arising from $S$ by the
adjunction of a zero is right (resp. left) 0-simple.
Moreover, an ordered semigroup $S$ with a zero element
0 is right (resp. left) 0-simple if and only if the set
$S\setminus \{0\}$ is a right (resp. left) simple subsemigroup
of $S$. The sufficient condition holds in ordered
groupoids, in general. That is, if $S$ is an ordered
groupoid with zero and if the set $S\setminus \{0\}$ is a right
(resp. left) simple subgroupoid of $S$, then $S$ is
right (resp. left) 0-simple.