Ascending chain condition
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
am =
an 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.