Ascending Chain Condition - Definition

Definition

A partially ordered set (poset) P is said to satisfy the ascending chain condition (ACC) if every strictly ascending sequence of elements eventually terminates. Equivalently, given any sequence

there exists a positive integer n such that

Similarly, P is said to satisfy the descending chain condition (DCC) if every strictly descending sequence of elements eventually terminates, that is, there is no infinite descending chain. Equivalently every descending sequence

of elements of P, eventually stabilizes.

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