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.[1]

Examples

Every complete lattice is chain-complete. Unlike complete lattices, chain-complete posets are relatively common. Examples include:

Properties

A poset is chain-complete if and only if it is a pointed dcpo.[1] However, this equivalence requires the axiom of choice.

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

References
  1. Markowsky, George (1976), "Chain-complete posets and directed sets with applications", Algebra Universalis, 6 (1): 53–68, doi:10.1007/bf02485815, MR 0398913, S2CID 16718857.
  2. Bourbaki, Nicolas (1949), "Sur le théorème de Zorn", Archiv der Mathematik, 2 (6): 434–437 (1951), doi:10.1007/bf02036949, MR 0047739, S2CID 117826806.
  3. Witt, Ernst (1951), "Beweisstudien zum Satz von M. Zorn", Mathematische Nachrichten, 4: 434–438, doi:10.1002/mana.3210040138, MR 0039776.
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