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Difference between revisions of "Ascending chain condition"

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(Undo revision 99456 by Erich2007 (talk))
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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>
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stabilizes, we say that <math>S</math> satisfies the '''descending chain condition''' ('''DCC''').  A set <math>S</math> with an ordering <math>\leqslant</math> satisifes ACC if and only if
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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]].
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 +
'''Theorem.''' A partially ordered set <math>S</math> satisfies the ascending
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chain condition if and only if every subset of <math>S</math> has a
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[[maximal element]].
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''Proof.''  First, suppose that every subset of <math>S</math> has a maximal
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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
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<math>x_{n+1} > x_n</math>, for all <math>n\ge 0</math>.  This sequence constitutes
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an ascending chain that does not stabilize, so <math>S</math> does not
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satisfy ACC.  <math>\blacksquare</math>
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Latest revision as of 17:00, 15 December 2018

Let $S$ be a partially ordered set. We say that $S$ satisfies the ascending chain condition (ACC) if every ascending chain \[x_0 \leqslant x_1 \leqslant x_2 \leqslant \dotsc\] eventually stabilizes; that is, there is some $N\ge 0$ such that $x_n = x_N$ for all $n\ge N$.

Similarly, if every descending chain \[x_0 \geqslant x_1 \geqslant x_2 \geqslant \dotsc\] stabilizes, we say that $S$ satisfies the descending chain condition (DCC). A set $S$ with an ordering $\leqslant$ satisfies ACC if and only if its opposite ordering satisfies DCC.

Every finite ordered set necessarily satisfies both ACC and DCC.

Let $A$ be a ring, and let $M$ be an $A$-module. If the set of sub-modules of $M$ with the ordering of $M$ satifies ACC, we say that $M$ is Noetherian. If this set satisfies DCC, we say that $M$ is Artinian.

Theorem. A partially ordered set $S$ satisfies the ascending chain condition if and only if every subset of $S$ has a maximal element.

Proof. First, suppose that every subset of $S$ has a maximal element. Then every ascending chain in $S$ has a maximal element, so $S$ satisfies ACC.

Now, suppose that some subset of $S$ has no maximal element. Then we can recursively define elements $x_0, x_1, \dotsc$ such that $x_{n+1} > x_n$, for all $n\ge 0$. This sequence constitutes an ascending chain that does not stabilize, so $S$ does not satisfy ACC. $\blacksquare$


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