On certain semigroups of transformations with restricted range

Let X = {1, ..., n} be a finite chain and let POI(X) he the semigroup of all injective order-preserving partial transformations on X. For any nonempty subset Y of X, let POI(X, Y) be the subsemigroup of POI(X) of all transformations with range contained in Y. In this paper, we characterize Green's relations on POI(X,Y), and show that, the semigroup POI(X, Y) is left abundant but not right abundant when Y is a proper subset of X. Moreover, the cardinality and the rank of the semigroup POI(X,Y) is determined.