Combinatorial results for zero-divisors regarding right zero elements of order-preserving transformations
Emrah Korkmaz, Hayrullah Ayık
TL;DR
The paper investigates the left, right, and two-sided zero-divisors of the right-zero elements $\pi_k$ in the semigroup $\mathcal{O}_{n}$ of order-preserving full transformations on $X_{n}$. It establishes explicit structural descriptions and cardinalities for the sets $\mathsf{L}_{k}$, $\mathsf{R}_{k}$, and $\mathsf{Z}_{k}$ for all $k$, ties these results to the interval-image subsemigroup $\mathcal{IO}_{n}$, and constructs minimal generating sets to determine ranks. Key contributions include exact size formulas, a natural isomorphism $\mathsf{R}_{1}\cong\mathsf{R}_{n}$, and explicit ranks for $\mathsf{R}_{1}$, $\mathsf{R}_{n}$, $\mathsf{Z}_{1}$, $\mathsf{Z}_{n}$, and $\mathsf{L}_{k}$, with detailed minimal generators drawn from families such as $\gamma_i$, $\beta_i$, $\xi_i$, and $\zeta_i$. These results deepen the algebraic and combinatorial understanding of zero-divisors in order-preserving semigroups and provide generating-set and rank data that can inform related graph-theoretic analyses and broader semigroup theory.
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 $π_{k}\in \mathcal{O}_{n}$ 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 $\mathcal{O}_{n}$.