Editing Maximal and minimal elements (section)

== Directed sets ==
In a [[totally ordered set]], the terms maximal element and greatest element coincide, which is why both terms are used interchangeably in fields like [[Mathematical analysis|analysis]] where only total orders are considered. This observation applies not only to totally ordered subsets of any partially ordered set, but also to their order theoretic generalization via [[directed set]]s. In a directed set, every pair of elements (particularly pairs of incomparable elements) has a common upper bound within the set. If a directed set has a maximal element, it is also its greatest element,<ref group=proof>Let <math>m \in D</math> be maximal.  Let <math>x \in D</math> be arbitrary. Then the common upper bound <math>u</math> of <math>m</math> and <math>x</math> satisfies <math>u \ge m</math>, so <math>u=m</math> by maximality. Since <math>x\le u</math> holds by definition of <math>u</math>, we have <math>x\le m</math>. Hence <math>m</math> is the greatest element. <math>\blacksquare</math></ref> and hence its only maximal element. For a directed set without maximal or greatest elements, see examples 1 and 2 [[#Existence and uniqueness|above]].

Similar conclusions are true for minimal elements.

Further introductory information is found in the article on [[order theory]].