Ascending chain condition

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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