On the monoid of all order-decreasing partial transformations

Abstract

A partial transformation $\alpha$ on an $n$-element chain $\textbf{n}=\{1,\dots, n\}$ is called order-decreasing if $x\leq y$ implies $x\alpha\leq y\alpha$ for all $x,y\in \dom(\alpha)$. The set of all partial order-decreasing transformations on $\textbf{n}$ forms a monoid $\PD_n$. In this paper, we determine the maximal subsemigroups as well as the maximal idempotent generated subsemigroups of $\PD_n$. Furthermore, we investigate the abundance of the ideals of $\PD_n$, and  characterize the structure of the left (right) abundant principal ideal of $\PD_n$.