Infinite descending chain: Difference between revisions
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Given a [[set]] ''S'' with a [[partial order]] ≤, an '''infinite descending chain''' is a [[chain (mathematics)|chain]] ''V'', that is, a subset of ''S'' upon which |
Given a [[set]] ''S'' with a [[partial order]] ≤, an '''infinite descending chain''' is a [[chain (mathematics)|chain]] ''V'', that is, a subset of ''S'' upon which ≤ defines a [[total order]], such that ''V'' has no ''minimal element'', that is, an element ''m'' such that for all elements ''n'' in ''V'' it holds that ''m'' ≤ ''n''. |
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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 chain on the [[natural number]]s, 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 chain on the [[natural number]]s, every chain of natural numbers has a minimal element. |
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Revision as of 23:20, 16 June 2005
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 minimal 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.