On certain semigroups of transformations with restricted range

Let [Formula: see text] be a finite chain and let [Formula: see text] be the semigroup of all injective order-preserving partial transformations on [Formula: see text]. For any nonempty subset [Formula: see text] of [Formula: see text], let [Formula: see text] be the subsemigroup of [Formula: see text] of all transformations with range contained in [Formula: see text]. In this paper, we characterize Green’s relations on [Formula: see text], and show that the semigroup [Formula: see text] is left abundant but not right abundant when [Formula: see text] is a proper subset of [Formula: see text]. Moreover, the cardinality and the rank of the semigroup [Formula: see text] is determined.