Editing Maximal and minimal elements (section)

== Existence and uniqueness ==
[[File:Zigzag poset.svg|thumb|x100px|A [[Fence (mathematics)|fence]] consists of minimal and maximal elements only (Example 3).]]

Maximal elements need not exist.

*'''Example 1:''' Let <math>S = [1, \infty) \subseteq \R</math> where <math>\R</math> denotes the [[real numbers]]. For all <math>m \in S,</math> <math>s = m + 1 \in S</math> but <math>m < s</math> (that is, <math>m \leq s</math> but not <math>m = s</math>). 
*'''Example 2:''' Let <math>S = \{ s \in \Q ~:~ 1 \leq s^2 \leq 2 \},</math> where <math>\Q</math> denotes the [[rational numbers]] and where [[Square_root_of_2#Proofs_of_irrationality|<math>\sqrt{2}</math>]] is irrational.

In general <math>\,\leq\,</math> is only a partial order on <math>S.</math> If <math>m</math> is a maximal element and <math>s \in S,</math> then it remains possible that neither <math>s \leq m</math> nor <math>m \leq s.</math> This leaves open the possibility that there exist more than one maximal elements.

*'''Example 3:''' In the [[Fence (mathematics)|fence]] <math>a_1 < b_1 > a_2 < b_2 > a_3 < b_3 > \ldots,</math> all the <math>a_i</math> are minimal and all <math>b_i</math> are maximal, as shown in the image.
*'''Example 4:''' Let ''A'' be a set with at least two elements and let <math>S = \{ \{ a \} ~:~ a \in A \}</math> be the subset of the [[power set]] <math>\wp(A)</math> consisting of [[Singleton (mathematics)|singleton subsets]], partially ordered by <math>\,\subseteq.</math> This is the discrete poset where no two elements are comparable and thus every element <math>\{ a \} \in S</math> is maximal (and minimal); moreover, for any distinct <math>a, b \in A,</math> neither <math>\{ a \} \subseteq \{ b \}</math> nor <math>\{ b \} \subseteq \{ a \}.</math>