---
tag: 6RDM3069KC
kind: statement
role: theorem
name: Birkhoff's representation theorem
msc: [06D05, 06A06]
uses: [0HM0HQCK6N, ZHRQ0AT8PM, 0QV2JS9RJ3]
about: [0HM0HQCK6N]
sources:
  - url: https://en.wikipedia.org/wiki/Birkhoff%27s_representation_theorem
same_as:
  wikidata: Q17005116
  wikipedia: Birkhoff's representation theorem
---

## Statement

Let $L$ be a finite [[distributive lattice|0HM0HQCK6N]].
Let $J(L)$ be the set of [[join-irreducible elements|ZHRQ0AT8PM]] of $L$.
These are the elements $j$, other than the least element, such that $j = x \vee y$ implies $j = x$ or $j = y$.
Order $J(L)$ by the order of $L$, and let $\mathcal{O}(J(L))$ be the set of lower sets of the [[partially ordered set|0QV2JS9RJ3]] $J(L)$, ordered by inclusion.
Then the map
$$a \mapsto \set{j \in J(L) : j \le a}$$
is a lattice isomorphism from $L$ onto $\mathcal{O}(J(L))$, where meet and join in $\mathcal{O}(J(L))$ are intersection and union.

## Remarks

Every element $a$ of a finite lattice is the join of the join-irreducible elements below it, so the map is injective and order-preserving in both directions.
Distributivity is used for surjectivity: if $j \le x \vee y$ with $j$ join-irreducible, then $j = (j \wedge x) \vee (j \wedge y)$, so $j \le x$ or $j \le y$.
This shows that the map sends joins to unions, and so every lower set $D$ is the image of $\bigvee D$.

Example: the divisors of $12$ ordered by divisibility form a distributive lattice.
Its join-irreducible elements are the prime powers $2$, $4$ and $3$, with $2 \le 4$ and $3$ incomparable to both.
This poset has six lower sets, matching the six divisors $1, 2, 3, 4, 6, 12$; for example $6$ corresponds to $\set{2, 3}$.

Distributivity is necessary: in the [[diamond lattice M3|JQ6FJH2TP8]] all three middle elements are join-irreducible and pairwise incomparable, so the lower sets of $J(M_3)$ number $8$, while $M_3$ has $5$ elements.

## Explanations

### L1 human

Every finite distributive lattice looks like a family of sets closed under union and intersection.
The building blocks are the join-irreducible elements, which cannot be split as a join of smaller elements.
Each element is recorded by the set of building blocks below it.

### L2 human

The theorem gives a correspondence between finite posets and finite distributive lattices: a poset $P$ gives the lattice of its lower sets, and a lattice $L$ gives the poset $J(L)$.
This correspondence extends to a duality between the category of finite posets with order-preserving maps and the category of finite distributive lattices with bounded lattice homomorphisms.
For infinite distributive lattices, the analogue needs topology, as in [[Priestley duality|SA22SEC75K]].
In the infinite case, prime filters take the place of join-irreducible elements, and the [[prime ideal theorem for distributive lattices|ME8GM0VTWP]] guarantees that there are enough of them.