Chain

Definition

Let $(P, \le)$ be a partially ordered set. A subset $C$ of $P$ is a chain when, for all $a, b \in C$, $a \le b$ or $b \le a$.

Remarks

So a chain is a subset on which $\le$ is a total order. In the power set of $\mathbb{N}$, the sets $\{0\} \subseteq \{0, 1\} \subseteq \{0, 1, 2\} \subseteq \dots$ form a chain. Chains are the key notion in Zorn's lemma and in the Hausdorff maximal principle.

Explanations

L1 human

A chain is a part of the order that lies on one line. In the divisors of $12$, the set $\{1, 2, 4, 12\}$ is a chain, but $\{2, 3\}$ is not.