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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 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. A totally ordered set without infinite descending chains is called well-ordered.

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Last updated on Friday September 19, 2008 at 13:27:10 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Friday September 19, 2008 at 13:27:10 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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