Complete Lattice - Representation

Representation

There are various other mathematical concepts that can be used to represent complete lattices. One means of doing so is the Dedekind-MacNeille completion. When this completion is applied to a poset that already is a complete lattice, then the result is a complete lattice of sets which is isomorphic to the original one. Thus we immediately find that every complete lattice is isomorphic to a complete lattice of sets.

Another representation is obtained by noting that the image of any closure operator on a complete lattice is again a complete lattice (called its closure system). Since the identity function is a closure operator too, this shows that the complete lattices are exactly the images of closure operators on complete lattices. Now the Dedekind-MacNeille completion can also be cast into a closure operator: every set of elements is mapped to the least lower (or upper) Dedekind cut that contains this set. Such a least cut does indeed exist and one has a closure operator on the powerset lattice of all elements. In summary, one can say that every complete lattice is isomorphic to the image of a closure operator on a powerset lattice.

This in turn is utilized in formal concept analysis, where one uses binary relations (called formal contexts) to represent such closure operators.