Science Fair Project Encyclopedia
Chain complete
In mathematics, a partially ordered set in order theory is chain complete if every chain in it has an upper bound.
Unlike complete posets, chain complete posets are relatively common. Examples include:
- Any complete poset
- The set of all linearly independent subsets of a vector space V, ordered by inclusion.
- The set of all partial functions on a set, ordered by
- iff when
and 
- we have
and g | A = f.
- If A is a collection of non-empty sets, the set of all partial choice functions on A, ordered as above.
- The set of all ideals of a ring, ordered by inclusion.
- The set of all consistent theories of a first order language.
Chain complete posets are interesting because of the Bourbaki-Witt theorem, and their connection with Zorn's Lemma.
Last updated: 08-29-2005 01:46:20
10-26-2009 08:16:03
The contents of this article is licensed from www.wikipedia.org under the GNU Free Documentation License. Click here to see the transparent copy and copyright details
The contents of this article is licensed from www.wikipedia.org under the GNU Free Documentation License. Click here to see the transparent copy and copyright details