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Given a set S with a partial order ≤, an infinite descending chain is a chain V that is a subset of S upon which ≤ defines a total order such that V has no least element, that is, an element m such that for all elements n in V it holds that m ≤ n.

As an example, in the set of integers, the chain −1, −2, −3, ... is an infinite descending chain, but there exists no infinite descending chain on the natural numbers, as every chain of natural numbers has a minimal element.

If a partially ordered set does not contain any infinite descending chains, it is called well-founded or, in some case, Artinian; it is then said to satisfy the descending chain condition. A stronger condition, that there be no infinite descending chains and no infinite antichains, defines the well-quasi-orderings. A totally ordered set without infinite descending chains is called well-ordered.

References

Yiannis N. Moschovakis (2006) Notes on set theory, Undergraduate texts in mathematics (Birkhäuser) ISBN 038728723X, p.116

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	As an example, in the set of [[integer]]s, the chain −1, −2, −3, ... is an infinite descending chain, but there exists no infinite descending chain on the [[natural number]]s, as every chain of natural numbers has a minimal element.		As an example, in the set of [[integer]]s, the chain −1, −2, −3, ... is an infinite descending chain, but there exists no infinite descending chain on the [[natural number]]s, as every chain of natural numbers has a minimal element.

	If a partially ordered set does not contain any infinite descending chains, it is called [[Well-founded relation\|well-founded]] ~~and~~ said to satisfy the [[descending chain condition]]. A stronger condition, that there be no infinite descending chains and no infinite antichains, defines the [[well-quasi-ordering]]s. A totally ordered set without infinite descending chains is called [[well-order]]ed.		If a partially ordered set does not contain any infinite descending chains, it is called [[Well-founded relation\|well-founded]] or, in some case, [[Artinian]]; it is then said to satisfy the [[descending chain condition]]. A stronger condition, that there be no infinite descending chains and no infinite antichains, defines the [[well-quasi-ordering]]s. A totally ordered set without infinite descending chains is called [[well-order]]ed.