---
tag: REE07CSXHR
kind: concept
name: Congruence lattice
msc: [08A30, 06B15]
sources:
  - url: https://en.wikipedia.org/wiki/Congruence_lattice_problem
---

## Summary

The congruence lattice of an [[algebra|QJGPT2W0FT]] is the set of all its [[congruences|DCSXXT4SER]], ordered by inclusion.
It is an [[algebraic lattice|3J0BTXSCVN]], and it records all the [[quotient algebras|N5KE7FH6CX]] of the algebra and how they relate.
Many properties of algebras and varieties, such as simplicity, subdirect irreducibility and congruence modularity, are properties of this lattice.
By a theorem of Grätzer and Schmidt, every algebraic lattice is isomorphic to the congruence lattice of some algebra.

## Explanations

### L1 human

Each congruence is one way of gluing elements of an algebra together.
Some gluings are finer than others, and comparing them by inclusion gives a lattice.
At the bottom is "glue nothing", and at the top is "glue everything".
For a group, this lattice is the same as the lattice of normal subgroups.
For the group $\Z/6\Z$ it has four elements, matching the subgroups $\set{0}$, $\set{0, 3}$, $\set{0, 2, 4}$ and the whole group.