- Chain complete
-
In mathematics, a partially ordered set in order theory is chain complete if every chain in it has a least 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
- if and only if 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 prime 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.
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