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 $1\leq k\leq n$, 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} &=& { \alpha\in \mathcal{O}{n}:\alpha\beta=\pi_{k} \mbox{ for some }\beta\in \mathcal{O}{n} \setminus{\pi{i}} }, \mathsf{R}{k} &=& { \alpha\in \mathcal{O}{n}:\gamma\alpha=\pi_{k} \mbox{ for some }\ \gamma\in \mathcal{O}{n}\setminus{\pi{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 $\mathcal{O}{n}$.