Editing Ascending chain condition (section)

== Definition ==
A [[partially ordered set]] (poset) ''P'' is said to satisfy the '''ascending chain condition''' (ACC) if no infinite strictly ascending sequence 
: <math>a_1 < a_2 < a_3 < \cdots</math>
of elements of ''P'' exists.{{sfn|Hazewinkel|p=580|ps=none}} 
Equivalently,{{efn|Proof: first, a strictly increasing sequence cannot stabilize, obviously. Conversely, suppose there is an ascending sequence that does not stabilize; then clearly it contains a strictly increasing (necessarily infinite) subsequence.}} every weakly ascending sequence
: <math>a_1 \leq a_2 \leq a_3 \leq \cdots,</math>
of elements of ''P'' eventually stabilizes, meaning that there exists a positive integer ''n'' such that
: <math>a_n = a_{n+1} = a_{n+2} = \cdots.</math>
Similarly, ''P'' is said to satisfy the '''descending chain condition''' (DCC) if there is no infinite strictly descending chain of elements of ''P''.{{sfn|Hazewinkel|p=580|ps=none}} Equivalently, every weakly descending sequence
: <math>a_1 \geq a_2 \geq a_3 \geq \cdots</math>
of elements of ''P'' eventually stabilizes.

=== Comments ===
* Assuming the [[axiom of dependent choice]], the descending chain condition on (possibly infinite) poset ''P'' is equivalent to ''P'' being [[well-founded]]: every nonempty subset of ''P'' has a minimal element (also called the '''minimal condition''' or '''minimum condition'''). A [[total order|totally ordered set]] that is well-founded is a [[well-order|well-ordered set]].
* Similarly, the ascending chain condition is equivalent to ''P'' being converse well-founded (again, assuming dependent choice): every nonempty subset of ''P'' has a maximal element (the '''maximal condition''' or '''maximum condition''').
* Every finite poset satisfies both the ascending and descending chain conditions, and thus is both well-founded and converse well-founded.