chain finite

A poset is said to be chain finite if every chain with bothmaximal (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 beunbounded (http://planetmath.org/UpperBound) above or below.$\mathbb{Q}$ with the standard order relation is not chain finite,since for example$\{x\in \mathbb{Q}\mid \mathrm{\hspace{0.17em}0}⩽x⩽1\}$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).