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

$f \leq g$

if and only if when

$f : A \to X$ and $g : B \to X$

we have

$A \subseteq B$ 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.