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

In the present paper, a submonoid of the well studied monoid POIₙ of all order-preserving partial injections on an n-element chain is studied. The set IOFₙᵖᵃʳ of all partial transformations in POIₙ which are fence-preserving as well as parity-preserving form a submonoid of POIₙ. We describe the Green's relations and ideals of IOFₙᵖᵃʳ. For each ideal of IOFₙᵖᵃʳ, we characterize the maximal subsemigroups. We will observe that there are three different types of maximal subsemigroups.