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'' ''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'' <= ''v''. |
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'' <= ''v''. |
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If a partially ordered set does not contain any infinite descending chains, it is called [[well-founded set|well-founded]]. A total ordered set without infinite descending chains is called [[well-order|well-ordered]]. |
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Revision as of 15:51, 25 February 2002
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 <= v.
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.