In this paper we introduce the concept of free ordered semigroups
as follows: If $(X,\le_X)$ is an ordered set, an ordered semigroup
$(F,.,\le_F)$ is said to be a free ordered semigroup over
$(X,\le_X)$, if there is an isotone mapping $\varepsilon :
(X,\le_X)\rightarrow (F, \le_F)$ satisfying the following
"universal" condition: for any ordered semigroup $(S,*,\le_S)$ and
any isotone mapping $f: (X,\le_X)\rightarrow (S,\le_S)$, there
exists a unique homomorphism $\varphi : (F,.,\le_F)\rightarrow
(S,*,\le_S)$ such that $\varphi\circ \varepsilon = f$. Basing on
the fact that the mapping $\varepsilon$ is reverse isotone, we
find relationships between the mappings $\varepsilon_1$ and
$\varepsilon_2$ which correspond to free ordered semigroups
$((F_1,.,\le_1), \varepsilon_1)$ and $((F_2,.,\le_2),
\varepsilon_2)$.