# 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 $\{x\in\nobreak\mathbb{Q}\,\mid\,0\leqslant\nobreak x\leqslant\nobreak 1\}$ 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 ChainFinite 2013-03-22 16:55:07 2013-03-22 16:55:07 lars_h (9802) lars_h (9802) 4 lars_h (9802) Definition msc 06A06 chain finite