Ascending chain condition
Let be a partially ordered set. We say that
satisfies the ascending chain condition (ACC) if every ascending chain
eventually stabilizes; that is, there is some
such that
for all
.
Similarly, if every descending chain
stabilizes, we say that
satisfies the descending chain condition (DCC). A set
with an ordering
satisfies ACC if and only if
its opposite ordering satisfies DCC.
Every finite ordered set necessarily satisfies both ACC and DCC.
Let be a ring, and let
be an
-module. If the set
of sub-modules of
with the ordering of
satifies ACC, we
say that
is Noetherian. If this set satisfies DCC, we say
that
is Artinian.
Theorem. A partially ordered set satisfies the ascending
chain condition if and only if every subset of
has a
maximal element.
Proof. First, suppose that every subset of has a maximal
element. Then every ascending chain in
has a maximal element,
so
satisfies ACC.
Now, suppose that some subset of has no maximal element. Then
we can recursively define elements
such that
, for all
. This sequence constitutes
an ascending chain that does not stabilize, so
does not
satisfy ACC.
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