On Semigroups of Orientation-preserving Transformations with Restricted Range

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