Ascending chain condition

Definition

Let $(P, \le)$ be a partially ordered set. The set $P$ satisfies the ascending chain condition if every sequence $a_1 \le a_2 \le a_3 \le \dots$ of elements of $P$ is eventually constant. That is, there is an integer $N$ with $a_n = a_N$ for all $n \ge N$.

Remarks

Equivalently, every nonempty subset of $P$ has a maximal element. This equivalence uses the axiom of dependent choice. The set $\mathbb{N}$ with its usual order does not satisfy the condition. With the reverse order, it does. For the set of the ideals of a ring, ordered by inclusion, the condition defines Noetherian rings. The ideals of $\mathbb{Z}$ satisfy the condition: a chain $(a_1) \subseteq (a_2) \subseteq \dots$ means $a_2 \mid a_1$, $a_3 \mid a_2$, and so on.

Explanations

L1 human

The condition says that you cannot climb forever. Every increasing chain stops growing after a finite number of steps. For the ideals of the integers, a strictly increasing chain like $(12) \subset (6) \subset (3) \subset (1)$ must end.