Infinite descending chain

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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, 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 total ordered set without infinite descending chains is called well-ordered.

 

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