---
tag: QP06QH2Q7C
kind: concept
name: Congruence generated by a relation
msc: [08A30]
sources:
  - url: https://en.wikipedia.org/wiki/Congruence_relation
---

## Summary

The congruence generated by a relation $R$ on an [[algebra|QJGPT2W0FT]] is the smallest [[congruence|DCSXXT4SER]] that contains $R$.
It exists because any intersection of congruences is a congruence.
It is the tool for forcing given pairs of elements to become equal while keeping the operations meaningful, as in presentations by generators and relations.

## Explanations

### L1 human

Suppose you want some pairs of elements of an algebra to count as equal.
Once $a$ counts as equal to $b$, compatibility forces many other pairs to be equal too, such as $a + c$ and $b + c$.
The congruence generated by your pairs collects exactly what is forced and nothing more.
In the group of integers under addition, asking for $0$ to equal $6$ forces exactly congruence modulo $6$.