Editing Complete lattice (section)

== Representation ==
G. Birkhoff's book ''Lattice Theory'' 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 operator|closure systems]].<ref name="birkhoff">{{cite book |last=Birkhoff |first=Garrett |title=Lattice Theory |date=1967 |publisher=American Mathematical Society |publication-place=Providence, RI, USA |page=124 |edition=3rd |series=American Mathematical Society Colloquium Publications |volume=XXV |chapter=Complete Lattices |isbn=978-0821810255}}</ref> 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 [[order-isomorphism|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 [[closure operator|increasing and idempotent]] (but not necessarily extensive) self-map. 
The identity mapping has these two properties. Thus all complete lattices occur.