---
tag: QKGW3Z9BHW
kind: concept
name: Fully invariant congruence
msc: [08A30, 08B05, 08A35]
specializes: [DCSXXT4SER]
sources:
  - url: https://en.wikipedia.org/wiki/Variety_(universal_algebra)
---

## Summary

A fully invariant congruence of an [[algebra|QJGPT2W0FT]] is a [[congruence|DCSXXT4SER]] that every endomorphism of the algebra maps into itself.
On a [[term algebra|3YYKGWJJND]], the fully invariant congruences are exactly the [[equational theories|F1M51B4MDG]].
These are the sets of identities closed under the rules of equational logic.
This is how equational theories become lattice-theoretic objects in universal algebra.

## Explanations

### L1 human

A congruence says which elements count as equal.
It is fully invariant when this equality survives every way of mapping the algebra into itself by a homomorphism.
In a term algebra, an endomorphism is the same as a substitution of terms for variables.
So a fully invariant congruence there is a set of equations between terms that stays true after any substitution, which is exactly how identities like $x \cdot y = y \cdot x$ behave.