Editing Zorn's lemma (section)

==Statement of the lemma==
Preliminary notions:
* A [[set (mathematics)|set]] ''P'' equipped with a [[binary relation]] ≤ that is [[Reflexive relation|reflexive]] ({{nowrap|''x'' ≤ ''x''}} for every ''x''), [[Antisymmetric relation|antisymmetric]] (if both {{nowrap|''x'' ≤ ''y''}} and {{nowrap|''y'' ≤ ''x''}} hold, then {{nowrap|1=''x'' = ''y''}}), and [[Transitive relation|transitive]] (if {{nowrap|''x'' ≤ ''y''}} and {{nowrap|''y'' ≤ ''z''}} then {{nowrap|''x'' ≤ ''z''}}) is said to be [[partially ordered set|(partially) ordered]] by ≤. Given two elements ''x'' and ''y'' of ''P'' with ''x'' ≤ ''y'', ''y'' is said to be [[greater than or equal to]] ''x''. The word "partial" is meant to indicate that not every pair of elements of a partially ordered set is required to be [[Comparability|comparable]] under the order relation, that is, in a partially ordered set ''P'' with order relation ≤ there may be elements ''x'' and ''y'' with neither ''x'' ≤ ''y'' nor ''y'' ≤ ''x''. An ordered set in which every pair of elements is comparable is called [[totally ordered set|totally ordered]].
* Every subset ''S'' of a partially ordered set ''P'' can itself be seen as partially ordered by [[Binary_relation#Restriction|restricting]] the order relation inherited from ''P'' to ''S''. A subset ''S'' of a partially ordered set ''P'' is called a [[chain (order theory)|chain]] (in ''P'') if it is totally ordered in the inherited order.
* An element ''m'' of a partially ordered set ''P'' with order relation ≤ is [[maximal element|maximal]] (with respect to ≤) if there is no other element of ''P'' greater than ''m'', that is, there is no ''s'' in ''P'' with ''s'' ≠ ''m'' and ''m'' ≤ ''s''. Depending on the order relation, a partially ordered set may have any number of maximal elements. However, a totally ordered set can have at most one maximal element.
* Given a subset ''S'' of a partially ordered set ''P'', an element ''u'' of ''P'' is an [[upper bound]] of ''S'' if it is greater than or equal to every element of ''S''. Here, ''S'' is not required to be a chain, and ''u'' is required to be comparable to every element of ''S'' but need not itself be an element of ''S''.

Zorn's lemma can then be stated as:

{{math_theorem|name=Zorn's lemma|math_statement=<ref>{{harvnb|Halmos|1960|loc=§ 16.}}</ref><ref>{{cite book|last1=Lang|first1=Serge|author-link1=Serge Lang|title=Algebra|edition=Revised 3rd|publisher=Springer-Verlag|series=Graduate Texts in Mathematics|volume=211|date=2002|isbn=978-0-387-95385-4|page=880}}, {{cite book|last1=Dummit|first1=David S.|last2=Foote|first2=Richard M.|title=Abstract Algebra|edition=2nd|publisher=Prentice Hall|date=1998|isbn=978-0-13-569302-5|page=875}}, and {{cite book|last1=Bergman|first1=George M|author-link1=George Bergman|title=An Invitation to General Algebra and Universal Constructions|edition=2nd|publisher=Springer-Verlag|series=Universitext|date=2015|isbn=978-3-319-11477-4|page=162|url=https://math.berkeley.edu/~gbergman/245/}}.</ref> Let <math>P</math> be a [[partially ordered set]] that satisfies the following two properties:
# <math>P</math> is nonempty; 
# Every [[chain (order theory)|chain]] in ''P'' has an [[upper bound]] in ''P''.
Then <math>P</math> has at least one [[maximal element]].}}

In fact, property (1) is redundant, since property (2) says, in particular, that the empty chain has an upper bound in <math>P</math>, implying <math>P</math> is nonempty. However, in practice, one often checks (1) and then verifies (2) only for nonempty chains, since the case of the empty chain is taken care by (1).

In the terminology of Bourbaki, a partially ordered set is called '''inductive''' if each chain has an upper bound in the set (in particular, the set is then nonempty).<ref>{{harvnb|Bourbaki|1970|loc=Ch. III., §2., no. 4., Definition 3.}}</ref> Then the lemma can be stated as:

{{Math theorem|name=Zorn's lemma|math_statement=<ref>{{harvnb|Bourbaki|1970|loc=Ch. III., §2., no. 4., Théorème 2.}}</ref> Each inductive set has a maximal element.}}

For some applications, the following variant may be useful.

{{Math theorem|name=Corollary|math_statement=<ref>{{harvnb|Bourbaki|1970|loc=Ch. III., §2., no. 4., Corollaire 1.}}</ref> Let <math>P</math> be a partially ordered set in which every chain has an upper bound and <math>a</math> an element in <math>P</math>. Then there exists a maximal element <math>b</math> in <math>P</math> such that <math>b \ge a</math>.}}

Indeed, let <math>Q = \{ x \in P \mid x \ge a \}</math> with the partial ordering from <math>P</math>. Then, for a chain in <math>Q</math>, an upper bound in <math>P</math> is in <math>Q</math> and so <math>Q</math> satisfies the hypothesis of Zorn's lemma and a maximal element in <math>Q</math> is a maximal element in <math>P</math> as well.