# locally finite poset

A poset $P$ is *locally finite ^{}* if every interval $[x,y]$ in $P$ is finite. For example, $\mathbb{Z}$ with the usual order is locally finite but not finite, while $\mathbb{Q}$ is neither.

Every locally finite poset is also chain finite, but the converse^{} does not hold. To see this, define a partial order^{} on $\mathbb{N}$ by the rule that
$k\le \mathrm{\ell}$ if and only if $k=0$ or $\mathrm{\ell}=1$. Thus $0$ is the minimum element, $1$ is the maximum element, and the remaining elements form an infinite^{} antichain^{}. Every bounded^{} chain in this poset is finite but the entire poset is an infinite interval, so the poset is chain finite but not locally finite.

Title | locally finite poset |
---|---|

Canonical name | LocallyFinitePoset |

Date of creation | 2013-03-22 14:09:15 |

Last modified on | 2013-03-22 14:09:15 |

Owner | mps (409) |

Last modified by | mps (409) |

Numerical id | 5 |

Author | mps (409) |

Entry type | Definition |

Classification | msc 06A99 |

Defines | locally finite |