The maximal subsemigroups of the ideals in a monoid of partial injections

Abstract

We study a submonoid of the well studied monoid \(POI_n\) of all order-preserving partial injections on an n-element chain. The set \(IOF_n^{par}\) of all partial transformations in \(POI_n\) which are fence-preserving as well as parity-preserving forms a submonoid of \(POI_n\). We describe Green’s relations and ideals of \(IOF_n^{par}\). For each ideal of \(IOF_n^{par}\), we characterize the maximal subsemigroups. We observe that there are three different types of maximal subsemigroups.