On Semigroups of Orientation-preserving Transformations with Restricted Range - Fernandes V. (2017)
© 2016, Taylor & Francis Group, LLC. Let X n be a chain with n elements (n ∈ ℕ), and let & #x1D4AA; & #x1D4AB; 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 & #x1D4AA; & #x1D4AB; n (Y) of & #x1D4AA; & #x1D4AB; n of all transformations with range contained in Y: We describe the largest regular subsemigroup of & #x1D4AA; & #x1D4AB; n (Y), which actually coincides with its subset of all regular elements. Also, we determine when two semigroups of the type & #x1D4AA; & #x1D4AB; n (Y) are isomorphic and calculate their ranks. Moreover, a parallel study is presented for the correspondent subsemigroups of the monoid & #x1D4AA;ℛ n of all either orientation-preserving or orientation-reversing transformations of X n .