Complete Lattice - Examples

Examples

• The power set of a given set, ordered by inclusion. The supremum is given by the union and the infimum by the intersection of subsets.
• The unit interval and the extended real number line, with the familiar total order and the ordinary suprema and infima. Indeed, a totally ordered set (with its order topology) is compact as a topological space if it is complete as a lattice.
• The non-negative integers, ordered by divisibility. The least element of this lattice is the number 1, since it divides any other number. Maybe 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 minimum subgroup of G, while the maximum subgroup is the group G itself.
• The submodules of a module, ordered by inclusion. The supremum is given by the sum of submodules and the infimum by the intersection.
• The ideals of a ring, 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, ordered by inclusion. The supremum is given by the union of open sets and the infimum by the interior of the intersection. On the other hand, if we define infimum to be set intersection, the open sets form a bounded but not complete lattice; in general, arbitrary intersections of open sets are not open.
• The convex subsets of a real or complex vector space, ordered by inclusion. The infimum is given by the intersection of convex sets and the supremum by the convex hull of the union.
• The topologies on a set, ordered by inclusion. The infimum is given by the intersection of topologies, and the supremum by the topology generated by the union of topologies.
• The lattice of all transitive relations on a set.
• The lattice of all sub-multisets of a multiset.
• The lattice of all equivalence relations on a set; the equivalence relation ~ is considered to be smaller (or "finer") than ≈ if x~y always implies x≈y.
• Any non-empty finite lattice is trivially a complete lattice.