Oriented partial permutation semigroups with restricted range

Let [Formula: see text] be a finite chain with [Formula: see text] elements [Formula: see text], and let [Formula: see text] be the semigroup of all either orientation-preserving or orientation-reversing injective partial transformations of [Formula: see text]. In this paper, for any nonempty subset [Formula: see text] of [Formula: see text], we consider the subsemigroup [Formula: see text] of [Formula: see text] of all transformations with range contained in [Formula: see text]. We study the regularity and describe Green’s relations on [Formula: see text]. Moreover, we calculate the rank and describe the ideal structure of [Formula: see text].