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

We study a submonoid of the well studied monoid P O I n of all order-preserving partial injections on an n-element chain. The set I O F n par of all partial transformations in P O I n which are fence-preserving as well as parity-preserving forms a submonoid of P O I n . We describe Green's relations and ideals of I O F n par . For each ideal of I O F n par , we characterize the maximal subsemigroups. We observe that there are three different types of maximal subsemigroups. [ABSTRACT FROM AUTHOR]