Chain in a poset

Let $(P, \le)$ be a poset (Partial order and partially ordered set). A subset $C \subseteq P$ is a chain if any two of its elements are comparable: for all $x, y \in C$, either $x \le y$ or $y \le x$.

Equivalently, $C$ is a chain if the restriction of $\le$ to $C$ is a total order on $C$.

• The empty set is a chain, and so is every singleton, both vacuously. This is not a technicality to be waved past: the empty chain is exactly what forces a chain-complete poset to have a least element (Chain-complete poset), and that least element is the starting point of the Bourbaki–Witt construction (Bourbaki–Witt fixed point theorem). A convention that quietly excludes the empty chain has to reintroduce the same content as a separate hypothesis.
• A chain need not be finite, need not be countable, and need not have a largest element. In the power set of $\mathbb{N}$ ordered by inclusion, the sets $\{1, \dots, n\}$ for $n \ge 1$ form a chain with no largest element.
• "Chain" is a property of a subset, not of the ambient poset. The whole poset $P$ is a chain exactly when $\le$ is a total order.