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 the 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.
Read more about this topic: Chain Complete
Famous quotes containing the word properties:
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)