Ascending chain condition - Definition and Overview

In mathematics, a poset P is said to satisfy the ascending chain condition (ACC) if every ascending chain a₁ ≤ a₂ ≤ ... of elements of P is eventually stationary, that is, there is some positive integer n such that a_m = a_n for all m > n. Similarly, P is said to satisfy the descending chain condition (DCC) if every descending chain a₁ ≥ a₂ ≥ ... of elements of P is eventually stationary (that is, there is no infinite descending chain).

The ascending chain condition on P is equivalent to the maximum condition: every nonempty subset of P has a maximal element. Similarly, the descending chain condition is equivalent to the minimum condition: every nonempty subset of P has a minimal element.

Every finite poset satisfies both ACC and DCC.

A totally ordered set that satisfies the descending chain condition is called a well-ordered set

See also Noetherian and Artinian.