---
tag: 2BZ2ZXH03B
kind: statement
role: theorem
name: Grätzer–Schmidt theorem
msc: [08A30, 06B15, 06B23]
uses: [D7A0ZGZT5S, 9F7TYES4KE, 3J0BTXSCVN]
about: [3J0BTXSCVN]
sources:
  - url: https://en.wikipedia.org/wiki/Congruence_lattice_problem
---

## Statement

Let $L$ be an [[algebraic lattice|3J0BTXSCVN]].
Then there is an algebra $A$, a set with a family of finitary operations, with the following property.
The lattice $\operatorname{Con} A$ of congruences of $A$, ordered by inclusion, is isomorphic to $L$.

## Remarks

A congruence of an algebra $A$ is an equivalence relation on $A$ that is compatible with every operation of $A$.
For every algebra $A$, the lattice $\operatorname{Con} A$ is algebraic: its [[compact elements|9F7TYES4KE]] are the finitely generated congruences, and every congruence is the join of the congruences generated by single pairs.
So the theorem says that the algebraic lattices are exactly the congruence lattices of algebras, up to isomorphism.
The proof is an explicit construction: the operations of $A$ are chosen so that the finitely generated congruences of $A$ behave like the compact elements of $L$.
The algebra produced may have infinitely many operations; the representation of finite lattices as congruence lattices of finite algebras is a different and harder question.

## Explanations

### L1 human

Every algebra, such as a group, a ring or a lattice, has congruences, which are the ways to glue its elements together compatibly with the operations.
The congruences of one algebra form a lattice with a special property called algebraic.
The theorem says that this is the only restriction: every algebraic lattice arises in this way from some algebra.

### L2 human

In a course on universal algebra this theorem closes the description of congruence lattices.
It sits next to the [[Birkhoff–Frink theorem|DZGNHKE3X1]], which says the same thing for lattices of subalgebras.
For special classes of algebras the question changes: the congruence lattice of a lattice is always distributive, and which distributive algebraic lattices occur there is the congruence lattice problem.