---
tag: DZGNHKE3X1
kind: statement
role: theorem
name: Birkhoff–Frink theorem
msc: [06B23, 08A30, 06A15]
uses: [D7A0ZGZT5S, 9F7TYES4KE, 3J0BTXSCVN, 2HJGRABSPZ]
about: [3J0BTXSCVN]
sources:
  - url: https://en.wikipedia.org/wiki/Compact_element
---

## Statement

Let $L$ be a [[complete lattice|D7A0ZGZT5S]].
The following are equivalent.

1. $L$ is an [[algebraic lattice|3J0BTXSCVN]].
2. $L$ is isomorphic to the lattice of subalgebras of some algebra $A$, a set with a family of finitary operations, ordered by inclusion.
3. $L$ is isomorphic to the lattice of closed sets of an algebraic [[closure operator|2HJGRABSPZ]] $c$ on some set $X$.
   Algebraic means that $c(S) = \bigcup \set{c(F) : F \subseteq S \text{ finite}}$ for every $S \subseteq X$.

## Remarks

From 2 to 3: the subalgebra generated by a set is an algebraic closure operator, since each operation has finitely many arguments.
From 3 to 1: in the lattice of closed sets, the [[compact elements|9F7TYES4KE]] are the closures of finite sets, and every closed set is the union of these.
From 1 to 2: one equips the set $K$ of compact elements of $L$ with operations whose subalgebras are exactly the ideals of the join-semilattice $K$.
The map $a \mapsto \set{k \in K : k \le a}$ identifies $L$ with these ideals.
For example, the lattice of subgroups of a group and the lattice of ideals of a ring are algebraic.
The lattice of closed subsets of $[0, 1]$ is complete but not algebraic, so it is not a subalgebra lattice.

## Explanations

### L1 human

The subgroups of a group, the subspaces of a vector space and the ideals of a ring all form complete lattices of a special kind, called algebraic.
The theorem says that the lattices of "sub-things" of algebraic structures are exactly the algebraic lattices.
Every algebraic lattice is the lattice of substructures of some suitable algebra.

### L2 human

The theorem explains why compactness is the right abstraction of "finitely generated": the compact elements of a subalgebra lattice are the finitely generated subalgebras.
It is the first of two representation results: the [[Grätzer–Schmidt theorem|2BZ2ZXH03B]] shows that the same lattices are also exactly the congruence lattices of algebras.
Together with the description of algebraic lattices as ideal lattices of join-semilattices with $0$, it gives several equivalent pictures of the same class.