  In this note we consider the monoid $\mathcal{PODI}_n$ of all monotone
partial permutations on $\{1,\ldots,n\}$ and its submonoids $\mathcal{DP}_n$,
$\mathcal{POI}_n$ and $\mathcal{ODP}_n$ of all partial isometries, of all
order-preserving partial permutations and of all order-preserving partial
isometries, respectively. We prove that both the monoids $\mathcal{POI}_n$ and
$\mathcal{ODP}_n$ are quotients of bilateral semidirect products of two of
their remarkable submonoids, namely of extensive and of co-extensive
transformations. Moreover, we show that $\mathcal{PODI}_n$ is a quotient of a
semidirect product of $\mathcal{POI}_n$ and the group $\mathcal{C}_2$ of order
two and, analogously, $\mathcal{DP}_n$ is a quotient of a semidirect product of
$\mathcal{ODP}_n$ and $\mathcal{C}_2$.
