Infinite descending chain: Difference between revisions
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Given a [[Set (mathematics)|set]] ''S'' with a [[partial order]] |
Given a [[Set (mathematics)|set]] ''S'' with a [[partial order]] ≤, an '''infinite descending chain''' is a [[Chain (order theory)|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''. |
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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. |
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* [[Well-founded relation]] |
* [[Well-founded relation]] |
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{{DEFAULTSORT:Infinite Descending Chain}} |
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[[Category:Order theory]] |
[[Category:Order theory]] |
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[[Category:Wellfoundedness]] |
[[Category:Wellfoundedness]] |
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[[Category:Articles lacking sources (Erik9bot)]] |
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[[zh:无穷降链]] |
[[zh:无穷降链]] |
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Revision as of 11:09, 16 December 2009
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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. A totally ordered set without infinite descending chains is called well-ordered.