## Abstract

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}.

Original language | English |
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Pages (from-to) | 253-264 |

Number of pages | 12 |

Journal | Communications in Algebra |

Volume | 44 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2 Jan 2016 |

## Keywords

- Orientation-preserving
- Orientation-reversing
- Rank
- Restricted range
- Transformations