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

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