A note on bilateral semidirect product decompositions of some monoids of order-preserving partial permutations

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