where "sets" and "functions" are replaced by "posets" and "monotone mappings". Likewise, one can describe the completion process as a functor from the category of posets with monotone functions to some category of complete lattices with appropriate morphisms that is left adjoint to the forgetful functor in the converse direction. As long as one considers meet- or join-preserving functions as morphisms, this can easily be achieved through the so-called Dedekind–MacNeille completion. For this process, elements of the poset are mapped to (Dedekind-) cuts, which can then be mapped to the underlying posets of arbitrary complete lattices in much the same way as done for sets and free complete (semi-) lattices above. The aforementioned result that free complete lattices do not exist entails that an according free construction from a poset is not possible either. This is easily seen by considering posets with a discrete order, where every element only relates to itself. These are exactly the free posets on an underlying set. Would there be a free construction of complete lattices from posets, then both constructions could be composed, which contradicts the negative result above. Representation Already G. Birkhoff's Lattice Theory book contains a very useful representation method. It associates a complete lattice to any binary relation between two sets by constructing a Galois connection from the relation, which then leads to two dually isomorphic closure systems. Closure systems are intersection-closed families of sets. When ordered by the subset relation ⊆, they are complete lattices. A special instance of Birkhoff's construction starts from an arbitrary poset (P,≤) and constructs the Galois connection from the order relation ≤ between P and itself. The resulting complete lattice is the Dedekind-MacNeille completion. When this completion is applied to a poset that already is a complete lattice, then the result is isomorphic to the original one. Thus we immediately find that every complete lattice is represented by Birkhoff's method, up to isomorphism. The construction is utilized in formal concept analysis, where one represents real-word data by binary relations (called formal contexts) and uses the associated complete lattices (called concept lattices) for data analysis. The mathematics behind formal concept analysis therefore is the theory of complete lattices. Another representation is obtained as follows: A subset of a complete lattice is itself a complete lattice (when ordered with the induced order) if and only if it is the image of an increasing and idempotent (but not