### Abstract

Let X be a finite or infinite chain and let ${\mathcal{O}}(X)$ be the monoid of all endomorphisms of X. In this paper, we describe the largest regular subsemigroup of ${\mathcal{O}}(X)$ and Green’s relations on ${\mathcal{O}}(X)$. In fact, more generally, if Y is a nonempty subset of X and ${\mathcal{O}}(X,Y)$ is the subsemigroup of ${\mathcal{O}}(X)$ of all elements with range contained in Y, we characterize the largest regular subsemigroup of ${\mathcal{O}}(X,Y)$ and Green’s relations on ${\mathcal{O}}(X,Y)$. Moreover, for finite chains, we determine when two semigroups of the type ${\mathcal {O}}(X,Y)$ are isomorphic and calculate their ranks.

Original language | English |
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Pages (from-to) | 77-104 |

Number of pages | 28 |

Journal | Semigroup Forum |

Volume | 89 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1 Aug 2014 |

### Keywords

- Order-preserving
- Rank
- Restricted range
- Transformations

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## Cite this

*Semigroup Forum*,

*89*(1), 77-104. https://doi.org/10.1007/s00233-013-9548-x