关于某些保序部分置换幺半群的双边半直积分解的注记
A note on bilateral semidirect product decompositions of some monoids of order-preserving partial permutations
摘要：
本文考虑$\{1,\ldots,n\}$上所有单调部分置换的幺半群$\mathcal{PODI}_n$及其子幺半群$\mathcal{DP}_n$，$\mathcal{POI}_n$和$\mathcal{ODP}_n$，分别是所有部分等距的，所有保序部分置换的和所有保序部分等距的.我们证明了么半群$\mathcal{POI}_n$和$\mathcal{ODP}_n$都是它们的两个显著子么半群的双边半直积的直积，即广延变换和余广延变换的直积.此外，我们表明，$\mathcal{PODI}_n$是一个商的半直积的$\mathcal{POI}_n$和组$\mathcal{C}_2$的二阶，类似地，$\mathcal{DP}_n$是一个商的半直积的$\mathcal{ODP}_n$和$\mathcal{C}_2$。
英文摘要：
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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