Editing Maximal and minimal elements (section)

{{Short description|Element that is not ≤ (or ≥) any other element}}
[[File:Lattice of the divisibility of 60 narrow 1,2,3,4.svg|thumb|[[Hasse diagram]] of the set ''P'' of [[divisor]]s of 60, partially ordered by the relation "''x'' divides ''y''". The red subset <math>S</math> = {1,2,3,4} has two maximal elements, viz. 3 and 4, and one minimal element, viz. 1, which is also its least element.]]
In [[mathematics]], especially in [[order theory]], a '''maximal element''' of a [[subset]] <math>S</math> of some [[preordered set]] is an element of <math>S</math> that is not smaller than any other element in <math>S</math>. A '''minimal element''' of a subset <math>S</math> of some preordered set is defined [[Duality (order theory)|dually]] as an element of <math>S</math> that is not greater than any other element in <math>S</math>.

The notions of maximal and minimal elements are weaker than those of [[greatest element and least element]] which are also known, respectively, as maximum and minimum. The maximum of a subset <math>S</math> of a preordered set is an element of <math>S</math> which is greater than or equal to any other element of <math>S,</math> and the minimum of <math>S</math> is again defined dually. In the particular case of a [[partially ordered set]], while there can be at most one maximum and at most one minimum there may be multiple maximal or minimal elements.<ref>{{citation|title=A Discrete Transition to Advanced Mathematics|first1=Bettina|last1=Richmond|author1-link=Bettina Richmond|first2=Thomas|last2=Richmond|publisher=American Mathematical Society|year=2009|isbn=978-0-8218-4789-3|page=181|url=https://books.google.com/books?id=HucyKYx0_WwC&pg=PA181}}.</ref><ref>{{citation|title=Group Theory|first=William Raymond|last=Scott|edition=2nd|publisher=Dover|year=1987|isbn=978-0-486-65377-8|url=https://books.google.com/books?id=kt4o5ZTwH4wC&pg=PA22|page=22}}</ref> Specializing further to [[totally ordered set]]s, the notions of maximal element and maximum coincide, and the notions of minimal element and minimum coincide.

As an example, in the collection
<math display="block">S := \left\{ \{d, o\}, \{d, o, g\}, \{g, o, a, d\}, \{o, a, f\} \right\}</math>
ordered by [[Inclusion (set theory)|containment]], the element {''d'', ''o''} is minimal as it contains no sets in the collection, the element {''g'', ''o'', ''a'', ''d''} is maximal as there are no sets in the collection which contain it, the element {''d'', ''o'', ''g''} is neither, and the element {''o'', ''a'', ''f''} is both minimal and maximal. By contrast, neither a maximum nor a minimum exists for <math>S.</math>

[[Zorn's lemma]] states that every partially ordered set for which every totally ordered subset has an [[upper bound]] contains at least one maximal element. This lemma is equivalent to the [[well-ordering theorem]] and the [[axiom of choice]]<ref>{{cite book
|last=Jech
|first=Thomas
|author-link=Thomas Jech
|title=The Axiom of Choice
|year=2008
|orig-year=originally published in 1973
|publisher=[[Dover Publications]]
|isbn=978-0-486-46624-8
}}</ref> and implies major results in other mathematical areas like the [[Hahn–Banach theorem]], the [[Kirszbraun theorem]], [[Tychonoff's theorem]], the existence of a [[Hamel basis]] for every vector space, and the existence of an [[algebraic closure]] for every [[Field (mathematics)|field]].