On injective partial Catalan monoids

Let $[n]$ be a finite chain $\{1,2,\dots ,n\}$, and let ${\mathcal{I}\mathcal{C}}_n$ be the semigroup consisting of all isotone and order-decreasing injective partial transformations on $[n]$. In addition, let ${\mathcal{Q}}_n^{\mathrm{\prime }}=\{\alpha \in {\mathcal{I}\mathcal{C}}_n:1\in ̸\text{Dom }\alpha \}$ be the subsemigroup of ${\mathcal{I}\mathcal{C}}_n$, consisting of all transformations in ${\mathcal{I}\mathcal{C}}_n$, each of whose domains does not contain $1$. For $1\le p\le n$, let $K(n,p)=\{\alpha \in {\mathcal{I}\mathcal{C}}_n:\mathrm{\mid }\text{Im }\alpha \mathrm{\mid }\le p\}$ and $M(n,p)=\{\alpha \in {\mathcal{Q}}_n^{\mathrm{\prime }}:\mathrm{\mid }\text{Im }\alpha \mathrm{\mid }\le p\}$ be the two-sided ideals of ${\mathcal{I}\mathcal{C}}_n$ and ${\mathcal{Q}}_n^{\mathrm{\prime }}$, respectively. Moreover, let ${RIC}_p(n)$ and ${RQ}_p^{\mathrm{\prime }}(n)$ denote the Rees quotients of $K(n,p)$ and $M(n,p)$, respectively. It is shown in this article that for any $S\in \{{\mathcal{R}\mathcal{I}\mathcal{C}}_p(n),K(n,p)\}$, $S$ is abundant; ${\mathcal{I}\mathcal{C}}_n$ is ample; and for any $S\in \{{\mathcal{Q}}_n^{\mathrm{\prime }},{\mathcal{R}\mathcal{Q}}_p^{\mathrm{\prime }}(n),M(n,p)\}$, $S$ is right abundant for all values of $n$, but not left abundant for $n\ge 2$. Furthermore, the ranks of the Rees quotients ${RIC}_p(n)$ and ${RQ}_p^{\mathrm{\prime }}(n)$ are shown to be equal to the ranks of the two-sided ideals $K(n,p)$ and $M(n,p)$, respectively. These ranks are found to be $\left(\genfrac{}{}{0px}{}{n}{p}\right)+(n-1)\left(\genfrac{}{}{0px}{}{n-2}{p-1}\right)$ and $\left(\genfrac{}{}{0px}{}{n}{p}\right)+(n-2)\left(\genfrac{}{}{0px}{}{n-3}{p-1}\right)$, respectively. In addition, the ranks of the semigroups ${\mathcal{I}\mathcal{C}}_n$ and ${\mathcal{Q}}_n^{\mathrm{\prime }}$ were found to be $2n$ and $n^2-3n+4$, respectively. Finally, we characterize all the maximal subsemigroups of ${\mathcal{I}\mathcal{C}}_n$ and ${\mathcal{Q}}_n^{\mathrm{\prime }}$.