Maximal Regular Subsemibands of Finite Orientation-Preserving Partial Transformation Semigroup

Abstract

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