missing). This discussion is also found in the article on semilattices. Complete sublattices A sublattice M of a complete lattice L is called a complete sublattice of L if for every subset A of M the elements and , as defined in L, are actually in M. If the above requirement is lessened to require only non-empty meet and joins to be in L, the sublattice M is called a closed sublattice of M. Conditionally Complete Lattices A lattice is said to be "conditionally complete" if it satisfies either or both of the following properties: Any subset bounded above has a least upper bound Any subset bounded below has a greatest lower bound Examples Any non-empty finite lattice is trivially complete. 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 [0,1] 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. 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 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