Chain-complete partial order
In mathematics, specifically order theory, 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.
Every complete lattice is chain-complete. Unlike complete lattices, 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.
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.
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.
- Markowsky, George (1976), "Chain-complete posets and directed sets with applications", Algebra Universalis, 6 (1): 53–68, doi:10.1007/bf02485815, MR 0398913.
- Bourbaki, Nicolas (1949), "Sur le théorème de Zorn", Archiv der Mathematik, 2: 434–437 (1951), doi:10.1007/bf02036949, MR 0047739.
- Witt, Ernst (1951), "Beweisstudien zum Satz von M. Zorn", Mathematische Nachrichten, 4: 434–438, doi:10.1002/mana.3210040138, MR 0039776.