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Encyclopedia > Descending chain condition

In mathematics, a poset P is said to satisfy the ascending chain condition (ACC) if every ascending chain a1 ≤ a2 ≤ ... of elements of P is eventually stationary, that is, there is some positive integer n such that am = an for all m > n. Similarly, P is said to satisfy the descending chain condition (DCC) if every descending chain a1 ≥ a2 ≥ ... of elements of P is eventually stationary (that is, there is no infinite descending chain).

The ascending chain condition on P is equivalent to the maximum condition: every nonempty subset of P has a maximal element. Similarly, the descending chain condition is equivalent to the minimum condition: every nonempty subset of P has a minimal element.

Every finite poset satisfies both ACC and DCC.

A totally ordered set that satisfies the descending chain condition is called a well-ordered set

Results from FactBites:

 Ascending chain condition - Wikipedia, the free encyclopedia (166 words) The ascending chain condition on P is equivalent to the maximum condition: every nonempty subset of P has a maximal element. Similarly, the descending chain condition is equivalent to the minimum condition: every nonempty subset of P has a minimal element. A totally ordered set that satisfies the descending chain condition is called a well-ordered set.
 Artinian - Wikipedia, the free encyclopedia (135 words) In mathematics, Artinian is an adjective that describes objects that satisfy particular cases of the descending chain condition. A ring is an Artinian ring if it satisfies the descending chain condition on ideals. The concept is named for Emil Artin, who classified all simple rings whose one-sided ideals satisfy the descending chain condition.
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