On injective partial Catalan monoids

TL;DR

This paper studies the algebraic structure of certain semigroups of order-preserving, injective partial transformations on finite chains, revealing their abundance properties, ranks, and maximal subsemigroups.

Contribution

It introduces and analyzes the properties, ranks, and maximal subsemigroups of injective partial Catalan monoids and their related Rees quotients, providing new algebraic insights.

Findings

Semigroups are abundant and ample under certain conditions.

Ranks of Rees quotients are explicitly calculated.

Maximal subsemigroups are characterized.

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 $\mathcal{Q}^{\prime}_{n} = \{\alpha \in \mathcal{IC}_{n} : \, 1\not \in \text{Dom } \alpha\}$ 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 $K(n,p) = \{\alpha \in \mathcal{IC}_{n} : \, |\text{Im }\, \alpha| \leq p\}$ and $M(n,p) = \{\alpha \in \mathcal{Q}^{\prime}_{n} : \, |\text{Im } \, \alpha| \leq p\}$ 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 $\mathcal{Q}^{\prime}_{n}$.