Editing Maximal and minimal elements (section)

== Greatest and least elements ==
{{main article|Greatest and least elements}}
For a partially ordered set <math>(P, \leq),</math> the [[irreflexive kernel]] of <math>\,\leq\,</math> is denoted as <math>\,<\,</math> and is defined by <math>x < y</math> if <math>x \leq y</math> and <math>x \neq y.</math>
For arbitrary members <math>x, y \in P,</math> exactly one of the following cases applies:
#<math>x < y</math>;
#<math>x = y</math>;
#<math>y < x</math>;
#<math>x</math> and <math>y</math> are incomparable.
Given a subset <math>S \subseteq P</math> and some <math>x \in S,</math>
* if case 1 never applies for any <math>y \in S,</math> then <math>x</math> is a maximal element of <math>S,</math> as defined above; 
* if case 1 and 4 never applies for any <math>y \in S,</math> then <math>x</math> is called a {{em|[[greatest element]]}} of <math>S.</math>
Thus the definition of a greatest element is stronger than that of a maximal element.

Equivalently, a greatest element of a subset <math>S</math> can be defined as an element of <math>S</math> that is greater than every other element of <math>S.</math> 
A subset may have at most one greatest element.<ref group=proof>If <math>g_1</math> and <math>g_2</math> are both greatest, then <math>g_1 \leq g_2</math> and <math>g_2 \leq g_1,</math> and hence <math>g_1 = g_2</math> by [[antisymmetry]]. <math>\blacksquare</math></ref>

The greatest element of <math>S,</math> if it exists, is also a maximal element of <math>S,</math><ref group=proof>If <math>g</math> is the greatest element of <math>S</math> and <math>s \in S,</math> then <math>s \leq g.</math> By [[antisymmetry]], this renders (<math>g \leq s</math> and <math>g \neq s</math>) impossible. <math>\blacksquare</math></ref> and the only one.<ref group=proof>If <math>m</math> is a maximal element then <math>m \leq g</math> (because <math>g</math> is greatest) and thus <math>m = g</math> since <math>m</math> is maximal. <math>\blacksquare</math></ref>
By [[contraposition]], if <math>S</math> has several maximal elements, it cannot have a greatest element; see example 3.
If <math>P</math> satisfies the [[ascending chain condition]], a subset <math>S</math> of <math>P</math> has a greatest element [[if, and only if]], it has one maximal element.<ref group=proof>{{em|Only if}}: see above. &mdash; {{em|If}}: Assume for contradiction that <math>S</math> has just one maximal element, <math>m,</math> but no greatest element. Since <math>m</math> is not greatest, some <math>s_1 \in S</math> must exist that is incomparable to <math>m.</math> Hence <math>s_1 \in S</math> cannot be maximal, that is, <math>s_1 < s_2</math> must hold for some <math>s_2 \in S.</math> The latter must be incomparable to <math>m,</math> too, since <math>m < s_2</math> contradicts <math>m</math>'s maximality while <math>s_2 \leq m</math> contradicts the incomparability of <math>m</math> and <math>s_1.</math> Repeating this argument, an infinite ascending chain <math>s_1 < s_2 < \ldots < s_n < \cdots</math> can be found (such that each <math>s_i</math> is incomparable to <math>m</math> and not maximal). This contradicts the ascending chain condition. <math>\blacksquare</math></ref>

When the restriction of <math>\,\leq\,</math> to <math>S</math> is a [[total order]] (<math>S = \{ 1, 2, 4 \}</math> in the topmost picture is an example), then the notions of maximal element and greatest element coincide.<ref group=proof>Let <math>m \in S</math> be a maximal element, for any <math>s \in S</math> either <math>s \leq m</math> or <math>m \leq s.</math> In the second case, the definition of maximal element requires that <math>s = m,</math> so it follows that <math>s \leq m.</math> In other words, <math>m</math> is a greatest element. <math>\blacksquare</math></ref> 
This is not a necessary condition: whenever <math>S</math> has a greatest element, the notions coincide, too, as stated above.
If the notions of maximal element and greatest element coincide on every two-element subset <math>S</math> of <math>P.</math> then <math>\,\leq\,</math> is a total order on <math>P.</math><ref group=proof>If <math>a, b \in P</math> were incomparable, then <math>S = \{ a, b \}</math> would have two maximal, but no greatest element, contradicting the coincidence. <math>\blacksquare</math></ref>

Dual to ''greatest'' is the notion of ''least element'' that relates to ''minimal'' in the same way as ''greatest'' to ''maximal''.