On injective partial Catalan monoids

Abstract

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 ${\mathcal{Q}}_n^{\prime }=\{\hat{I}\pm \in {\mathcal{I}\mathcal{C}}_n:1\notin \text{Dom }\hat{I}\pm \}$ 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 $K(n,p)=\{\hat{I}\pm \in {\mathcal{I}\mathcal{C}}_n:|\text{Im }\hat{I}\pm |\le p\}$ and $M(n,p)=\{\hat{I}\pm \in {\mathcal{Q}}_n^{\prime }:|\text{Im }\hat{I}\pm |\le p\}$ be the two-sided ideals of ${\mathcal{I}\mathcal{C}}_n$ and ${\mathcal{Q}}_n^{\prime }$, respectively. Moreover, let ${RIC}_p(n)$ and ${RQ}_p^{\prime }(n)$ denote the Rees quotients of $K(n,p)$ and $M(n,p)$, respectively. It is shown in this article that for any \( S \in \{ \mathcal{RIC}_{p}(n), K(n,p) \} \), \( S \) is abundant; \( \mathcal{IC}_{n} \) is ample; and for any \( S \in \{ \mathcal{Q}^{\prime}_{n}, \mathcal{RQ}^{\prime}_{p}(n), M(n,p) \} \), \( S \) is right abundant for all values of \( n \), but not left abundant for \( n \geq 2 \). Furthermore, the ranks of the Rees quotients ${RIC}_p(n)$ and ${RQ}_p^{\prime }(n)$ 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 $\left(\genfrac{}{}{0ex}{}{n}{p}\right)+(n-1)\left(\genfrac{}{}{0ex}{}{n-2}{p-1}\right)$ and $\left(\genfrac{}{}{0ex}{}{n}{p}\right)+(n-2)\left(\genfrac{}{}{0ex}{}{n-3}{p-1}\right)$, respectively. In addition, the ranks of the semigroups ${\mathcal{I}\mathcal{C}}_n$ and ${\mathcal{Q}}_n^{\prime }$ 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 ${\mathcal{Q}}_n^{\prime }$.