Infinite descending chain: Difference between revisions
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#REDIRECT [[Total order#Chains]] |
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{{mergeto|totally ordered set|date=January 2019}} |
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Given a [[Set (mathematics)|set]] ''S'' with a [[partial order]] ≤, an '''infinite descending chain''' is an infinite, strictly decreasing sequence of elements ''x<sub>1</sub> > x<sub>2</sub> > ... > x<sub>n</sub> > ...'' |
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As an example, in the set of [[integer]]s, the [[Chain (order theory)|chain]] −1, −2, −3, ... is an infinite descending chain, but there exists no infinite descending chain on the [[natural number]]s, as every [[Chain (order theory)|chain]] of natural numbers has a minimal element. |
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If a partially ordered set does not possess any infinite descending chains, it is said then, that it satisfies the [[descending chain condition]]. Assuming the [[axiom of choice]], the descending chain condition on a partially ordered set is equivalent to requiring that the corresponding [[strict order]] is [[well-founded]]. A stronger condition, that there be no infinite descending chains and no infinite [[Antichain|antichains]], defines the [[well-quasi-ordering]]s. A totally ordered set without infinite descending chains is called [[well-order]]ed. |
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==See also== |
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* [[Artinian ring]] |
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==References== |
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{{Reflist}} |
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* [[Yiannis N. Moschovakis]] (2006) ''Notes on set theory'', [[Undergraduate Texts in Mathematics]] (Birkhäuser) {{ISBN|0-387-28723-X}}, p.116 |
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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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{{settheory-stub}} |
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Latest revision as of 20:15, 12 February 2020
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