Editing Maximal and minimal elements (section)

== Definition ==

Let <math>(P, \leq)</math> be a [[preordered set]] and let <math>S \subseteq P.</math> {{em|A '''maximal element''' of <math>S</math> with respect to <math>\,\leq\,</math>}} is an element <math>m \in S</math> such that 

:if <math>s \in S</math> satisfies <math>m \leq s,</math> then necessarily <math>s \leq m.</math> 

Similarly, {{em|a '''{{visible anchor|minimal element|Minimal element}}''' of <math>S</math> with respect to <math>\,\leq\,</math>}} is an element <math>m \in S</math> such that 

:if <math>s \in S</math> satisfies <math>s \leq m,</math> then necessarily <math>m \leq s.</math> 

Equivalently, <math>m \in S</math> is a minimal element of <math>S</math> with respect to <math>\,\leq\,</math> if and only if <math>m</math> is a maximal element of <math>S</math> with respect to <math>\,\geq,\,</math> where by definition, <math>q \geq p</math> if and only if <math>p \leq q</math> (for all <math>p, q \in P</math>).

If the subset <math>S</math> is not specified then it should be assumed that <math>S := P.</math> Explicitly, a {{em|{{visible anchor|maximal element|Maximal element}}}} (respectively, {{em|minimal element}}) {{em|of <math>(P, \leq)</math>}} is a maximal (resp. minimal) element of <math>S := P</math> with respect to <math>\,\leq.</math> 

If the preordered set <math>(P, \leq)</math> also happens to be a [[partially ordered set]] (or more generally, if the restriction <math>(S, \leq)</math> is a partially ordered set) then <math>m \in S</math> is a maximal element of <math>S</math> if and only if <math>S</math> contains no element strictly greater than <math>m;</math> explicitly, this means that there does not exist any element <math>s \in S</math> such that <math>m \leq s</math> and <math>m \neq s.</math> 
The characterization for minimal elements is obtained by using <math>\,\geq\,</math> in place of <math>\,\leq.</math>