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

 Title A note on bilateral semidirect product decompositions of some monoids of order-preserving partial permutations Publication Type Journal Article Year of Publication 2016 Authors Fernandes VH, Quinteiro TM Journal Bull. Korean Math. Soc. Volume 53 Pagination 495-506 Abstract In this note we consider the monoid $PODI_n$ of all monotone partial permutations on $\{1,łdots,n\}$ and its submonoids $DP_n$, $POI_n$ and $ODP_n$ of all partial isometries, of all order-preserving partial permutations and of all order-preserving partial isometries, respectively. We prove that both the monoids $POI_n$ and $ODP_n$ are quotients of bilateral semidirect products of two of their remarkable submonoids, namely of extensive and of co-extensive transformations. Moreover, we show that $PODI_n$ is a quotient of a semidirect product of $POI_n$ and the group $\mathcal{C}_2$ of order two and, analogously, $DP_n$ is a quotient of a semidirect product of $ODP_n$ and $\mathcal{C}_2$. URL http://arxiv.org/abs/1502.06097