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

Vítor H. Fernandes; Teresa M. Quinteiro

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arXiv · 1502.06097

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

Vítor H. Fernandes·Teresa M. Quinteiro

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Abstract

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 $\mathcal{C}_2$.

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Vítor H. Fernandes, Teresa M. Quinteiro. 2015-02-21. A note on bilateral semidirect product decompositions of some monoids of order-preserving partial permutations. https://arxiv.org/abs/1502.06097

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