---
tag: KBXSZC6BV1
kind: concept
name: Homomorphism
msc: [08A05, 03C07]
sources:
  - url: https://en.wikipedia.org/wiki/Homomorphism
same_as:
  wikidata: Q215111
  wikipedia: Homomorphism
---

## Summary

A homomorphism is a map between two [[structures|983GZBQYT4]] of the same [[signature|WDVVEYD6AT]] that preserves the operations, the constants and the relations.
[[Group homomorphisms|0JYTG5K9Y4]], [[ring homomorphisms|0VTH8KPCG2]] and [[linear maps|0NRYHJPQG2]] are all homomorphisms for suitable signatures.
Homomorphisms are the maps used to compare structures: [[isomorphisms|PZMGQMDN25]], [[embeddings|SCRB74SWQN]] and [[quotient maps|N5KE7FH6CX]] are special homomorphisms.

## Explanations

### L1 human

A homomorphism translates one structure into another without breaking its rules.
If $h$ is a homomorphism for addition, then computing $a + b$ and then applying $h$ gives the same as applying $h$ to $a$ and $b$ and then adding.
[[Reduction of integers modulo a fixed number|BV08DGBD7C]] is the standard example: the remainder of a sum is determined by the remainders of the summands.
The map $x \mapsto x + 1$ on the integers is [[not a homomorphism|3A3JJPJHQR]] for addition, since it does not send $0$ to $0$.