About: Birkhoff's representation theorem

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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.

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rdfs:label	Birkhoff's representation theorem (en)
rdfs:comment	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. (en)
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Link from a Wikipage to another Wikipage	Preorder Priestley space Prime power Proceedings of the London Mathematical Society Boolean lattice Algebra Universalis Antimatroid Bulletin of the London Mathematical Society Dedekind number Duke Mathematical Journal Independent set (graph theory) Compact space Mathematics Median graph Spectral space Garrett Birkhoff Greatest common divisor Mixed graph Equivalence of categories Hom-functor Closure (mathematics) Mathematical psychology Transitive closure Distributive lattice Divisor Lattice of stable matchings Stone duality Duality theory for distributive lattices Finite set Base (topology) Birkhoff's theorem (disambiguation) Discrete Mathematics (journal) Family of sets Hilary Priestley Knowledge space Free distributive lattice 2-satisfiability Intersection (set theory) Prime number Least common multiple Bijection Bitopological space Theorems in lattice theory Coherent space Boolean algebras canonically defined Bulletin of the AMS Connected component (graph theory) Open set Category theory Median algebra Union (set theory) Lattice homomorphism Implication graph Finite topological space Lattice theory Partial order Pairwise Stone space Topological space Subset Lower set Distributive law Composition of functions Birkhoff's HSP theorem Stone's representation theorem
Link from a Wikipage to an external page	http://projecteuclid.org/euclid.bams/1183510977
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has abstract	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. (en)
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is Link from a Wikipage to another Wikipage of	Antichain Antimatroid Ring of sets Dedekind number Dedekind–MacNeille completion Inclusion order

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