Ascending chain condition: Difference between revisions

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. $\blacksquare$

This article is a stub. Help us out by expanding it.

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+Let <math>S</math> be a [[partially ordered set]].  We say that <math>S</math> satisfies the '''ascending chain condition''' ('''ACC''') if every ascending chain
+<cmath> x_0 \leqslant x_1 \leqslant x_2 \leqslant \dotsc </cmath>
+eventually stabilizes; that is, there is some <math>N\ge 0</math> such that
+<math>x_n = x_N</math> for all <math>n\ge N</math>.
+Similarly, if every descending chain
+<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> satisfies ACC if and only if
+its opposite ordering satisfies DCC.
+Every [[finite]] ordered set necessarily satisfies both ACC and
+DCC.
+Let <math>A</math> be a [[ring]], and let <math>M</math> be an <math>A</math>-module.  If the set
+of sub-modules of <math>M</math> with the ordering of <math>M</math> satifies ACC, we
+say that <math>M</math> is [[Noetherian]].  If this set satisfies DCC, we say
+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>
+{{stub}}
+== See also ==
+* [[Noetherian]]
+* [[Artinian]]
+* [[Partially ordered set]]
+* [[Zorn's Lemma]]<!-- haha-->
+[[Category:Set theory]]
+[[Category:Ring theory]]

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Ascending chain condition: Difference between revisions

Latest revision as of 17:00, 15 December 2018

See also