Definition:Ascending Chain Condition

Definition
Let $\struct {P, \le}$ be an ordered set.

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

Also see

 * Definition:Well-Founded Relation


 * Definition:Descending Chain Condition
 * Increasing Sequence in Ordered Set Terminates iff Maximal Element