Let Xn be a chain with n elements (n ∈ ℕ), and let 𝒪𝒫n be the monoid of all orientation-preserving transformations of Xn. In this article, for any nonempty subset Y of Xn, we consider the subsemigroup 𝒪𝒫n(Y) of 𝒪𝒫n of all transformations with range contained in Y: We describe the largest regular subsemigroup of 𝒪𝒫n(Y), which actually coincides with its subset of all regular elements. Also, we determine when two semigroups of the type 𝒪𝒫n(Y) are isomorphic and calculate their ranks. Moreover, a parallel study is presented for the correspondent subsemigroups of the monoid 𝒪ℛn of all either orientation-preserving or orientation-reversing transformations of Xn.
