On injective partial Catalan monoids
F. S. Al-Kharousi, A. Umar, M. M. Zubairu
TL;DR
This paper investigates the injective partial Catalan monoid $\mathcal{IC}_{n}$ and its subsemigroup $\mathcal{Q}^{\prime}_{n}$ on a finite chain, focusing on abundance, ampleness, and rank-related invariants. It develops the starred Green's relation framework for these semigroups and their Rees quotients, establishing $\mathcal{IC}_{n}$ as ample and $\mathcal{Q}^{\prime}_{n}$ as right abundant, while also proving $\mathcal{IC}_{n}$ and $\mathcal{Q}^{\prime}_{n}$ are $\mathcal{J}$-trivial and non-regular except at idempotents. The authors compute exact rank formulas for the Rees quotients $\mathrm{RIC}_{p}(n)$ and $\mathrm{RQ}^{\prime}_{p}(n)$, equate these with the ranks of $K(n,p)$ and $M(n,p)$, and determine the global ranks $\mathrm{rank}(\mathcal{IC}_{n})=2n$ and $\mathrm{rank}(\mathcal{Q}^{\prime}_{n})=n^{2}-3n+4$. They also classify all maximal subsemigroups of $\mathcal{IC}_{n}$ and $\mathcal{Q}^{\prime}_{n}$, obtaining explicit combinatorial descriptions and counts ($2n$ and $n^{2}-3n+4$ respectively). Overall, the work advances the structural understanding of injective partial Catalan-type transformation semigroups and provides concrete generating sets and invariants for further study.
Abstract
Let $[n]$ be a finite chain $\{1, 2, \ldots, n\}$, and let $\mathcal{IC}_{n}$ be the semigroup consisting of all isotone and order-decreasing injective partial transformations on $[n]$. In addition, let $\mathcal{Q}^{\prime}_{n} = \{α\in \mathcal{IC}_{n} : \, 1\not \in \text{Dom } α\}$ be the subsemigroup of $\mathcal{IC}_{n}$, consisting of all transformations in $\mathcal{IC}_{n}$, each of whose domains does not contain $1$. For $1 \leq p \leq n$, let $K(n,p) = \{α\in \mathcal{IC}_{n} : \, |\text{Im }\, α| \leq p\}$ and $M(n,p) = \{α\in \mathcal{Q}^{\prime}_{n} : \, |\text{Im } \, α| \leq p\}$ be the two-sided ideals of $\mathcal{IC}_{n}$ and $\mathcal{Q}^{\prime}_{n}$, respectively. Moreover, let ${RIC}_{p}(n)$ and ${RQ}^{\prime}_{p}(n)$ 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 ${RQ}^{\prime}_{p}(n)$ 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 $\binom{n}{p}+(n-1)\binom{n-2}{p-1}$ and $\binom{n}{p}+(n-2)\binom{n-3}{p-1}$, respectively. In addition, the ranks of the semigroups $\mathcal{IC}_{n}$ and $\mathcal{Q}^{\prime}_{n}$ were found to be $2n$ and $n^{2}-3n+4$, respectively. Finally, we characterize all the maximal subsemigroups of $\mathcal{IC}_{n}$ and $\mathcal{Q}^{\prime}_{n}$.