Combinatorial results for zero-divisors regarding right zero elements of order-preserving transformations

Abstract

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 $1\le k\le n$, let $\ddot{I}{€}_k\in {\mathcal{O}}_n$ be the constant map defined by $x\ddot{I}{€}_k=k$ for all $x\in X_n$. In this paper, we introduce and study the sets of left, right, and two-sided zero-divisors of $\ddot{I}{€}_k$: \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 $1\le k\le n$. Furthermore, we compute the ranks of ${\mathsf{R}}_1$,\, ${\mathsf{R}}_n$,\, ${\mathsf{Z}}_1$,\, ${\mathsf{Z}}_n$ and ${\mathsf{L}}_k$ for each $1\le k\le n$, because these are significant subsemigroups of ${\mathcal{O}}_n$.