# order-preserving map

*Order-preserving map* from a poset $L$ to a poset $M$ is a function^{} $f$ such that

$$\forall x,y\in L:(x\ge y\u27f9f(x)\ge f(y)).$$ |

Order-preserving maps are also called *monotone functions* or *monotonic functions* or *order homomorphisms* or *isotone functions* or *isotonic functions*.

*Order-reversing map* from a poset $L$ to a poset $M$ is a function $f$ such that

$$\forall x,y\in L:(x\ge y\u27f9f(x)\le f(y)).$$ |

Order-reversing maps are also called *antitone functions*.

Title | order-preserving map |

Canonical name | OrderpreservingMap |

Date of creation | 2013-03-22 17:44:43 |

Last modified on | 2013-03-22 17:44:43 |

Owner | porton (9363) |

Last modified by | porton (9363) |

Numerical id | 10 |

Author | porton (9363) |

Entry type | Definition |

Classification | msc 06A06 |

Synonym | monotone function |

Synonym | monotonic function |

Synonym | order homomorphism |

Synonym | isotone function |

Synonym | isotonic function |

Synonym | order-preserving |

Synonym | isotone |

Synonym | isotonic |

Synonym | order-reversing |

Synonym | antitonic |

Synonym | antitone |

Related topic | Poset |

Related topic | LatticeHomomorphism |

Defines | monotonicity |