Infinite descending chain: Difference between revisions
Content deleted Content added
Tagging with {{settheory-stub}} using user scripts |
Merged content to Total order#Chains, redirecting; unopposed 2019 proposal; see Talk:Infinite descending chain (easy-merge) Tag: New redirect |
||
| (22 intermediate revisions by 17 users not shown) | |||
| Line 1: | Line 1: | ||
#REDIRECT [[Total order#Chains]] |
|||
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''. |
|||
{{R from merge}} |
|||
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. |
|||
{{R to section}} |
|||
If a partially ordered set does not contain any infinite descending chains, it is called [[Well-founded relation|well-founded]]. A totally ordered set without infinite descending chains is called [[well-order]]ed. |
|||
==See also== |
|||
* [[Descending chain condition]] |
|||
* [[Well-founded relation]] |
|||
[[Category:Order theory]] |
[[Category:Order theory]] |
||
[[Category:Wellfoundedness]] |
[[Category:Wellfoundedness]] |
||
[[zh:无穷降链]] |
|||
{{settheory-stub}} |
|||
Latest revision as of 20:15, 12 February 2020
Redirect to:
- From a merge: This is a redirect from a page that was merged into another page. This redirect was kept in order to preserve the edit history of this page after its content was merged into the content of the target page. Please do not remove the tag that generates this text (unless the need to recreate content on this page has been demonstrated) or delete this page.
- For redirects with substantive page histories that did not result from page merges use {{R with history}} instead.
- To a section: This is a redirect from a topic that does not have its own page to a section of a page on the subject. For redirects to embedded anchors on a page, use {{R to anchor}} instead.