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**.

Read more about Birkhoff's Representation Theorem: Understanding The Theorem, Examples, The Partial Order of Join-irreducibles, Birkhoff's Theorem, Rings of Sets and Preorders, Functoriality, Generalizations

### Other articles related to "representation, representations":

... The

**Representation**of the People Act 1832 The

**Representation**of the People (Scotland) Act 1832 The

**Representation**of the People (Ireland) Act 1832 The

**Representation**of the. 2) Act 1920 The

**Representation**of the People Act 1921 The

**Representation**of the People Act 1922 The

**Representation**of the People (No. 2) Act 1922 The

**Representation**of the People (Economy Provisions) Act 1926 — an alternative citation for Part III of the Economy (Miscellaneous Provisions ...

... Today, each Representative represents about 40,000 residents ... Their districts are named for the counties they are in and tend to stay within one county, although districts occasionally cross county lines ...

...

**Representation**(mathematics) and group

**representation**Multiple

**representations**(mathematics education) Knowledge

**representation**, the study of formal ways to describe knowledge ...

... A

**representation**V of a finite group G over an algebraically closed field K of characteristic zero is faithful (as a

**representation**) if and only if every irreducible

**representation**of G occurs as a ... Also, V is faithful (as a

**representation**) if and only if every irreducible

**representation**of G occurs as a subrepresentation of (the n-th tensor power of the ...

... The chief concern of the article is the

**representation**of slaves in relation to taxation and

**representation**... are property as well as people, therefore requiring some

**representation**... This

**representation**is decided to be every 3 out of 5 slaves are to be counted or 3/5 of the total number of slaves ...

### Famous quotes containing the word theorem:

“To insure the adoration of a *theorem* for any length of time, faith is not enough, a police force is needed as well.”

—Albert Camus (1913–1960)