Let M be a set and A a subset of M with "=>" a preorder on M and "=>" on A a weak order=> A is a chain on M.

A chain is maximal if there does not exist C' superset C s.t. C' is a chain on M.

An upperbound of A subset of M <=> There exists m in M s.t. m"=>"a forall a in A.

HausDorff's Maximal principle:

axiom of choice => every chain in a preordered set is contained in a maximal chain.

Zorn's Lemma: all chains in a preordered set have an upper bound => M has a maximal element.