The maximal subsemigroups of some semigroups of partial injections

We consider the structure of the ideals of the semigroup of all order-preserving partial injections as well as of the semigroup of all order-preserving or order-reversing partial injections on an n-element set. The main result of the presentation gives the characterization of the maximal subsemigroups of the ideals of the mentioned above two semigroups.