The ideals of certain semigroups of transformations with restricted range

Let [Formula: see text] be a finite chain with [Formula: see text] elements and [Formula: see text] be the semigroup of all injective order-preserving partial transformations of [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 [Formula: see text], [Formula: see text] as well as [Formula: see text] on [Formula: see text] when [Formula: see text] is a proper subset of [Formula: see text], and describe the ideal structure of [Formula: see text].