---
tag: Z937W8RW4N
kind: definition
name: Congruence lattice
msc: [08A30, 06B15]
defines: REE07CSXHR
uses: [QJGPT2W0FT, DCSXXT4SER, QP06QH2Q7C, D7A0ZGZT5S, 3J0BTXSCVN, 9F7TYES4KE, 05J9NWX07X]
sources:
  - url: https://en.wikipedia.org/wiki/Congruence_lattice_problem
---

## Definition

Let $\mathbf{A}$ be an [[algebra|QJGPT2W0FT]] with universe $A$.
Let $\operatorname{Con} \mathbf{A}$ be the set of all [[congruences|DCSXXT4SER]] on $\mathbf{A}$.
The *congruence lattice* of $\mathbf{A}$ is the partially ordered set $(\operatorname{Con} \mathbf{A}, \subseteq)$.
Its meet and join are
$$\theta \wedge \psi = \theta \cap \psi, \qquad \theta \vee \psi = \operatorname{Cg}(\theta \cup \psi),$$
where $\operatorname{Cg}$ is the [[congruence generated by a relation|QP06QH2Q7C]].

## Remarks

The join $\theta \vee \psi$ is also the [[transitive closure|05J9NWX07X]] of $\theta \cup \psi$, that is, the union of the relations $\theta \circ \psi$, $\theta \circ \psi \circ \theta$, and so on.
$\operatorname{Con} \mathbf{A}$ is a [[complete lattice|D7A0ZGZT5S]] with least element $\Delta_A$ and greatest element $\nabla_A = A \times A$.
It is an [[algebraic lattice|3J0BTXSCVN]], and its [[compact elements|9F7TYES4KE]] are the finitely generated congruences $\operatorname{Cg}(R)$ with $R$ finite.

Example: for the group $\Z/6\Z$, the congruences correspond to the subgroups $\set{0}$, $\set{0, 3}$, $\set{0, 2, 4}$ and $\Z/6\Z$.
So $\operatorname{Con}(\Z/6\Z)$ has four elements, with two incomparable elements in the middle.

Non-example: the union of two congruences need not be their join.
On the group $\Z$, the union of congruence modulo $2$ and modulo $3$ contains $(0, 2)$ and $(2, 5)$ but not $(0, 5)$, so it is not transitive.