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

In the present paper, a submonoid of the well studied monoid $PO{I}_n$ of all order-preserving partial injections on an $n$-element chain is studied. The set $IO{F}_n^{par}$ of all partial transformations in $PO{I}_n$ which are fence-preserving as well as parity-preserving form a submonoid of $PO{I}_n$. We describe the Green’s relations and ideals of $IO{F}_n^{par}$. For each ideal of $IO{F}_n^{par}$, we characterize the maximal subsemigroups. We will observe that there are three different types of maximal subsemigroups.