On Semigroups of Orientation-preserving Transformations with Restricted Range

Abstract

© 2016, Taylor & Francis Group, LLC. 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.