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. 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. The lattice of self-adjoint projections (also known as orthogonal projections) of a von Neumann algebra. Locally finite complete lattices A complete lattice L is said to be locally finite if the supremum of any infinite subset is equal to 1, or equivalently, the set is finite for any . The lattice (N, |) is locally finite. Note that in this lattice, the element generally denoted "0" is actually 1 and vice versa. Morphisms of complete lattices The traditional morphisms between complete lattices are the complete homomorphisms (or complete lattice homomorphisms). These are characterized as functions that preserve all joins and all meets. Explicitly, this means that a function f: L→M between two complete lattices L and M is a complete homomorphism if and , for all subsets A of L. Such functions are automatically monotonic, but the condition of being a complete homomorphism is in fact much more specific. For this reason, it can be useful to consider weaker notions of morphisms, that are only required to preserve all joins (giving a category Sup) or all meets (giving a category Inf), which are indeed inequivalent conditions. This notion may be considered as a homomorphism of complete meet-semilattices or complete join-semilattices, respectively. Galois connections and adjoints Furthermore, morphisms that preserve all joins are equivalently characterized as the lower adjoint part of a unique Galois connection. For any pair of preorders P and Q, these are given by pairs of monotone functions f and g such that where f is called the lower adjoint and g is called