# antichain

A subset $A$ of a poset $$ is an *antichain ^{}* if no two elements are comparable. That is, if $a,b\in A$ then $a{\nless}_{P}b$ and $b{\nless}_{P}a$.

A *maximal antichain* of $T$ is one which is maximal.

In particular, if $$ is a tree then the maximal antichains are exactly those antichains which intersect every branch, and if the tree is splitting then every level is a maximal antichain.

Title | antichain |
---|---|

Canonical name | Antichain |

Date of creation | 2013-03-22 12:52:25 |

Last modified on | 2013-03-22 12:52:25 |

Owner | Henry (455) |

Last modified by | Henry (455) |

Numerical id | 6 |

Author | Henry (455) |

Entry type | Definition |

Classification | msc 05C05 |

Classification | msc 03E05 |

Related topic | TreeSetTheoretic |

Related topic | Aronszajn |

Defines | antichain |

Defines | maximal antichain |