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Concept
Birkhoff's representation theorem
Summary
This is about lattice theory. For other similarly named results, see Birkhoff's theorem (disambiguation).
In mathematics, Birkhoff's representation theorem for distributive lattices states that the elements of any finite distributive lattice can be represented as finite sets, in such a way that the lattice operations correspond to unions and intersections of sets. The theorem can be interpreted as providing a one-to-one correspondence between distributive lattices and partial orders, between quasi-ordinal knowledge spaces and preorders, or between finite topological spaces and preorders. It is named after Garrett Birkhoff, who published a proof of it in 1937.
The name “Birkhoff's representation theorem” has also been applied to two other results of Birkhoff, one from 1935 on the representation of Boolean algebras as families of sets closed under union, intersection, and complement (so-called fields of sets, closely related to the rings of sets used by Birkhoff to represent distributive lattices), and Birkhoff's HSP theorem representing algebras as products of irreducible algebras. Birkhoff's representation theorem has also been called the fundamental theorem for finite distributive lattices.
Many lattices can be defined in such a way that the elements of the lattice are represented by sets, the join operation of the lattice is represented by set union, and the meet operation of the lattice is represented by set intersection. For instance, the Boolean lattice defined from the family of all subsets of a finite set has this property. More generally any finite topological space has a lattice of sets as its family of open sets. Because set unions and intersections obey the distributive law, any lattice defined in this way is a distributive lattice. Birkhoff's theorem states that in fact all finite distributive lattices can be obtained this way, and later generalizations of Birkhoff's theorem state a similar thing for infinite distributive lattices.
Consider the divisors of some composite number, such as (in the figure) 120, partially ordered by divisibility.
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