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

TL;DR

This paper investigates the algebraic structure of monoids of order-preserving partial permutations, showing they can be represented as quotients of bilateral semidirect products of specific submonoids, revealing their decomposition properties.

Contribution

It establishes new bilateral semidirect product decompositions for monoids of order-preserving partial permutations and related submonoids, advancing understanding of their algebraic structure.

Findings

Both $ ext{POI}_n$ and $ ext{ODP}_n$ are quotients of bilateral semidirect products of their submonoids.

$ ext{PODI}_n$ is a quotient of a semidirect product of $ ext{POI}_n$ and $ ext{C}_2$.

$ ext{DP}_n$ is a quotient of a semidirect product of $ ext{ODP}_n$ and $ ext{C}_2$.

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 $\mathcal{C}_2$.