A poset is said to be chain finite if every chain with both maximal 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 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).