# Ascending chain condition

In

mathematics, a

poset *P* is said to satisfy the

**ascending chain condition** (ACC)
if every ascending chain

*a*_{1} ≤

*a*_{2} ≤ ... 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

*a*_{1} ≥

*a*_{2} ≥ ... 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.