Galois connection
Every Galois connection (or residuated mapping) gives rise to a closure operator (as is explained in that article). In fact, "every" closure operator arises in this way from a suitable Galois connection. The Galois connection is not uniquely determined by the closure operator. One Galois connection that gives rise to the closure operator cl can be described as follows: if "A" is the set of closed elements with respect to cl, then cl: "P" → "A" is the lower adjoint of a Galois connection between "P" and "A", with the upper adjoint being the embedding of "A" into "P". Furthermore, every lower adjoint of an embedding of some subset into "P" is a closure operator. "Closure operators are lower adjoints of embeddings." Note however that not every embedding has a lower adjoint.