Editing Greatest element and least element (section)

== Examples ==

[[File:KeinVerband.svg|thumb|upright=0.5|[[Hasse diagram]] of example 2]]
* The subset of [[integer]]s has no upper bound in the set <math>\mathbb{R}</math> of [[real number]]s.
* Let the relation <math>\,\leq\,</math> on <math>\{ a, b, c, d \}</math> be given by <math>a \leq c,</math> <math>a \leq d,</math> <math>b \leq c,</math> <math>b \leq d.</math> The set <math>\{ a, b \}</math> has upper bounds <math>c</math> and <math>d,</math> but no least upper bound, and no greatest element (cf. picture).
* In the [[rational number]]s, the set of numbers with their square less than 2 has upper bounds but no greatest element and no least upper bound.
* In <math>\mathbb{R},</math> the set of numbers less than 1 has a least upper bound, viz. 1, but no greatest element.
* In <math>\mathbb{R},</math> the set of numbers less than or equal to 1 has a greatest element, viz. 1, which is also its least upper bound.
* In <math>\mathbb{R}^2</math> with the [[product order]], the set of pairs <math>(x, y)</math> with <math>0 < x < 1</math> has no upper bound.
* In <math>\mathbb{R}^2</math> with the [[lexicographical order]], this set has upper bounds, e.g. <math>(1, 0).</math> It has no least upper bound.