# chain finite

A poset is said to be *chain finite* if every chain with both
maximal (http://planetmath.org/MaximalElement) and minimal element is finite.

$\mathbb{Z}$ with the standard order relation is chain finite,
since any infinite subset of $\mathbb{Z}$ must be
unbounded^{} (http://planetmath.org/UpperBound) above or below.
$\mathbb{Q}$ with the standard order relation is not chain finite,
since for example
$\{x\in \mathbb{Q}\mid \mathrm{\hspace{0.17em}0}\u2a7dx\u2a7d1\}$
is infinite and has both a maximal element $1$ and a minimal element $0$.

Chain finiteness is often used to draw conclusions^{} about an order from information about its covering relation (or equivalently, from its Hasse diagram^{}).

Title | chain finite |
---|---|

Canonical name | ChainFinite |

Date of creation | 2013-03-22 16:55:07 |

Last modified on | 2013-03-22 16:55:07 |

Owner | lars_h (9802) |

Last modified by | lars_h (9802) |

Numerical id | 4 |

Author | lars_h (9802) |

Entry type | Definition |

Classification | msc 06A06 |

Defines | chain finite |