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

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 ${\pi }_k\in {\mathcal{O}}_n$ be the constant map defined by $x{\pi }_k=k$ for all $x\in X_n$. In this paper, we introduce and study the sets of left, right, and two-sided zero-divisors of ${\pi }_k$: \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 ${\mathcal{O}}_n$.