Editing Complete lattice (section)

== Examples ==
* Any non-empty finite lattice is trivially complete.
* The [[power set]] of a given set when ordered by [[subset|inclusion]]. The supremum is given by the [[union (set theory)|union]] and the infimum by the [[intersection (set theory)|intersection]] of subsets.
* The non-negative [[integer]]s ordered by [[divisibility]]. The least element of this lattice is the number 1 since it divides any other number. Perhaps surprisingly, the greatest element is 0, because it can be divided by any other number. The supremum of finite sets is given by the [[least common multiple]] and the infimum by the [[greatest common divisor]]. For infinite sets, the supremum will always be 0 while the infimum can well be greater than 1. For example, the set of all even numbers has 2 as the greatest common divisor.  If 0 is removed from this structure it remains a lattice but ceases to be complete.
* The subgroups of any given group under inclusion. (While the [[infimum]] here is the usual set-theoretic intersection, the [[supremum]] of a set of subgroups is the subgroup ''generated by'' the set-theoretic union of the subgroups, not the set-theoretic union itself.) If ''e'' is the identity of ''G'', then the [[trivial group]] {''e''} is the [[partial order|minimum]] subgroup of ''G'', while the [[partial order|maximum]] subgroup is the group ''G'' itself.
* The [[ideal (ring theory)|ideals]] of a [[ring (mathematics)|ring]], when ordered by inclusion. The supremum is given by the sum of ideals and the infimum by the intersection.
* The open sets of a [[topological space]], when ordered by inclusion. The supremum is given by the union of open sets and the infimum by the [[interior (topology)|interior]] of the intersection.

=== Non-examples ===
* The [[empty set]] is not a complete lattice. If it were a complete lattice, then in particular the empty set would have an infimum and supremum in the empty set, a contradiction.
* The [[rational numbers]] <math>\mathbb{Q}</math> with the usual order ≤ is not a complete lattice. It is a lattice with <math>\bigwedge \{x, y\} = \min\{x, y\}</math> and <math>\bigvee \{x, y\} = \max\{x, y\}</math>. However, <math>\mathbb{Q}</math> itself has no infimum or supremum, nor does <math>\{ x \in \mathbb{Q} | x^2 < 2 \}</math>.