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.