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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2507.04900 |
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Table of Contents:
- For any positive integer $n$, let $\mathcal{O}_{n}$ be the semigroup of all order-preserving full transformations on $X_{n}=\{1<\cdots <n\}$. For any $1\leq k\leq n$, let $π_{k}\in \mathcal{O}_{n}$ be the constant map defined by $xπ_{k}=k$ for all $x\in X_{n}$. In this paper, we introduce and study the sets of left, right, and two-sided zero-divisors of $π_{k}$: \begin{eqnarray*} \mathsf{L}_{k} &=& \{ α\in \mathcal{O}_{n}:αβ=π_{k} \mbox{ for some }β\in \mathcal{O}_{n} \setminus\{π_{k}\} \}, \mathsf{R}_{k} &=& \{ α\in \mathcal{O}_{n}:γα=π_{k} \mbox{ for some }\ γ\in \mathcal{O}_{n}\setminus\{π_{k}\} \}, \ \mbox{and} \ \mathsf{Z}_{k}=\mathsf{L}_{k}\cap \mathsf{R}_{k}. \end{eqnarray*} We determine the structures and cardinalities of $\mathsf{L}_{k}$, $\mathsf{R}_{k}$ and $\mathsf{Z}_{k}$ for each $1\leq k\leq n$. Furthermore, we compute the ranks of $\mathsf{R}_{1}$,\, $\mathsf{R}_{n}$,\, $\mathsf{Z}_{1}$,\, $\mathsf{Z}_{n}$ and $\mathsf{L}_{k}$ for each $1\leq k\leq n$, because these are significant subsemigroups of $\mathcal{O}_{n}$.