In
set theory, a statement (equivalent to the
Axiom of Choice) which asserts that: If S is any
non-empty partially ordered set in which every
chain has an
upper bound, then S has a
maximal element.
It should be noted that
Zorn's lemma states that under the given conditions S will have a
maximal element, it does not say how many
maximal elements S may have. A
maximal element in a
poset is an
element such that if any other
element is greater than or equal to it, it must in fact be equal to it. A
chain in a
poset consists of a
sub-poset in which every
element is
comparable.
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