User:Bernhard Ganter/sandbox

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 &sube;, they are complete lattices.

A special instance of Birkhoff's construction starts from an arbitrary poset (P,&le;) and constructs the Galois connection from the order relation &le; 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.

This 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: Let (L,&le;) be a complete lattice and φ:L&rarr;L be a mapping. The image of φ with the induced order is a complete lattice if and only if φ is increasing and idempotent (but not necessarily extensive). The identity mapping obviously has these two properties. Thus all complete lattices occur.