Maximal Regular Subsemibands of the Finite Order-Preserving Partial Transformation Semigroups $\mathcal {PO}(n,r)$

Abstract

Let $\mathcal {PO}_n$ be the semigroup of all order-preserving partial transformations on the finite set $X_n=\{1, 2,\ldots , n\}$. For $1\le r\le n-1$, set $\mathcal {PO}(n, r)=\{\alpha \in \mathcal {PO}_n: |\mathop {\text{ im }}\nolimits (\alpha )|\le r\}$. In this paper, we investigate the maximal regular subsemigroups and the maximal regular subsemibands of the semigroup $\mathcal {PO}(n,r)$. First, we completely describe the maximal regular subsemigroups of the semigroup $\mathcal {PO}(n,r)$, for $1\le r\le n-1$. Secondly, we show that, for $2\le r \le n-2$, any maximal regular subsemigroup of the semigroup $\mathcal {PO}(n,r)$ is a semiband and obtain that the maximal regular subsemigroups and the maximal regular subsemibands of the semigroup $\mathcal {PO}(n,r)$ coincide, for $2\le r\le n-2$. Finally, we obtain the complete classification of maximal regular subsemibands of the semigroup $\mathcal {PO}_n$.