Definition:Ascending Chain Condition

Definition
Let $\left({P, \leq}\right)$ be an ordered set.

Then $S$ is said to have the ascending chain condition (ACC) if every increasing sequence $x_1 \leq x_2 \leq x_3 \leq \cdots$ with $x_i \in P$ eventually terminates: there is $n \in \N$ such that $x_n = x_{n+1} = \cdots$.

Also see

 * Increasing Sequence in Ordered Set Terminates iff Maximal Element