On Semigroups of Orientation-preserving Transformations with Restricted Range

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