Chain complete

In order-theoretic mathematics, a partially ordered set is chain complete if every chain in it has a least upper bound. It is ω-complete when every increasing sequence of elements (a type of countable chain) has a least upper bound; the same notion can be extended to other cardinalities of chains.^[1]

Examples

Every complete poset is chain complete. Unlike complete posets, chain complete posets are relatively common. Examples include:

• 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 restriction.
• The set of all partial choice functions on a collection of non-empty sets, ordered by restriction.
• The set of all prime ideals of a ring, ordered by inclusion.
• The set of all consistent theories of a first-order language.

Properties

Zorn's lemma states that, if a poset has an upper bound for every chain, then it has a maximal element. Thus, it applies to chain complete posets, but is more general in that it allows chains that have upper bounds but do not have least upper bounds.

Chain complete posets also obey the Bourbaki–Witt theorem, a fixed point theorem stating that, if f is a function from a chain complete poset to itself with the property that, for all x, f(x) ≥ x, then f has a fixed point. This theorem, in turn, can be used to prove that Zorn's lemma is a consequence of the axiom of choice.^[2][3]

By analogy with the Dedekind–MacNeille completion of a partially ordered set, every partially ordered set can be extended uniquely to a minimal chain-complete poset.^[1]

See also

• Completeness (order theory)

References