Difference between revisions of "Ascending chain condition"
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Similarly, if every descending chain | Similarly, if every descending chain | ||
<cmath> x_0 \geqslant x_1 \geqslant x_2 \geqslant \dotsc </cmath> | <cmath> x_0 \geqslant x_1 \geqslant x_2 \geqslant \dotsc </cmath> | ||
− | stabilizes, we say that <math>S</math> satisfies the '''descending chain condition''' ('''DCC'''). A set <math>S</math> with an ordering <math>\leqslant</math> | + | stabilizes, we say that <math>S</math> satisfies the '''descending chain condition''' ('''DCC'''). A set <math>S</math> with an ordering <math>\leqslant</math> satisfies ACC if and only if |
its opposite ordering satisfies DCC. | its opposite ordering satisfies DCC. | ||
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say that <math>M</math> is [[Noetherian]]. If this set satisfies DCC, we say | say that <math>M</math> is [[Noetherian]]. If this set satisfies DCC, we say | ||
that <math>M</math> is [[Artinian]]. | that <math>M</math> is [[Artinian]]. | ||
+ | |||
+ | '''Theorem.''' A partially ordered set <math>S</math> satisfies the ascending | ||
+ | chain condition if and only if every subset of <math>S</math> has a | ||
+ | [[maximal element]]. | ||
+ | |||
+ | ''Proof.'' First, suppose that every subset of <math>S</math> has a maximal | ||
+ | element. Then every ascending chain in <math>S</math> has a maximal element, | ||
+ | so <math>S</math> satisfies ACC. | ||
+ | |||
+ | Now, suppose that some subset of <math>S</math> has no maximal element. Then | ||
+ | we can recursively define elements <math>x_0, x_1, \dotsc</math> such that | ||
+ | <math>x_{n+1} > x_n</math>, for all <math>n\ge 0</math>. This sequence constitutes | ||
+ | an ascending chain that does not stabilize, so <math>S</math> does not | ||
+ | satisfy ACC. <math>\blacksquare</math> | ||
+ | |||
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Latest revision as of 18:00, 15 December 2018
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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