Birkhoff's representation theorem

This is about <a href="/facts/Lattice_theory/5ak6ufky">lattice theory</a>. For other similarly named results, see <a href="/facts/Birkhoff%2527s_theorem_(disambiguation)/q60TEf0q">Birkhoff's theorem (disambiguation)</a>.
In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, Birkhoff's representation theorem for distributive lattices states that the elements of any <a href="/facts/Finite_set/za1newlP">finite</a> <a href="/facts/Distributive_lattice/97DFz1cu">distributive lattice</a> can be represented as <a href="/facts/Finite_set/za1newlP">finite sets</a>, in such a way that the lattice operations correspond to <a href="/facts/Union_(set_theory)/KVVXKfWl">unions</a> and <a href="/facts/Intersection_(set_theory)/RPfUNJh9">intersections</a> of sets. Here, a lattice is an abstract structure with two <a href="/facts/Binary_operation/CYtow0bz">binary operations</a>, the "meet" and "join" operations, which must obey certain axioms; it is distributive if these two operations obey the <a href="/facts/Distributive_law/0LNBrVJl">distributive law</a>. The union and intersection operations, in a family of sets that is closed under these operations, automatically form a distributive lattice, and Birkhoff's representation theorem states that (up to isomorphism) every finite distributive lattice can be formed in this way. It is named after <a href="/facts/Garrett_Birkhoff/5jeI7Wmr">Garrett Birkhoff</a>, who published a proof of it in 1937.
The theorem can be interpreted as providing a <a href="/facts/Bijection/ayC7YE3z">one-to-one correspondence</a> between distributive lattices and <a href="/facts/Partial_order/qb4a3FAX">partial orders</a>, between <a href="/facts/Knowledge_space/87nUIIv8">quasi-ordinal knowledge spaces</a> and <a href="/facts/Preorder/uubzHSWm">preorders</a>, or between <a href="/facts/Finite_topological_space/HKD3e0qp">finite topological spaces</a> and preorders.
The name “Birkhoff's representation theorem” has also been applied to two other results of Birkhoff, one from 1935 on the <a href="/facts/Boolean_algebras_canonically_defined/ztEsAwX0">representation of Boolean algebras</a> as families of sets closed under union, intersection, and <a href="/facts/Complement_(set_theory)/yWUePIYY">complement</a> (so-called fields of sets, closely related to the rings of sets used by Birkhoff to represent distributive lattices), and <a href="/facts/Birkhoff%2527s_HSP_theorem/vXWYFyDI">Birkhoff's HSP theorem</a> representing algebras as products of irreducible algebras. Birkhoff's representation theorem has also been called the fundamental theorem for finite distributive lattices.

Birkhoff's representation theorem open-in-new

Birkhoff's representation theorem