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