---
tag: W82R6N0GM9
kind: concept
name: Universal algebra
msc: [08A05, 03C05]
sources:
  - url: https://en.wikipedia.org/wiki/Universal_algebra
same_as:
  wikidata: Q1636734
  wikipedia: Universal algebra
---

## Summary

Universal algebra is the part of algebra that studies [[algebraic structures|QJGPT2W0FT]] in general, rather than groups, rings or lattices one at a time.
Its objects are sets with a family of [[finitary operations|EJ8YJVM9BW]], described by a [[signature|WDVVEYD6AT]].
It treats [[homomorphisms|KBXSZC6BV1]], [[subalgebras|676H7EGTKJ]], [[congruences|DCSXXT4SER]] and products for all such structures at once.
Its central results, such as [[Birkhoff's HSP theorem|5083DHSK82]], describe the classes of algebras defined by identities.

## Explanations

### L1 human

Groups, rings, vector spaces and lattices are all sets with some operations that obey some rules.
Many facts about them have the same proof: for example, the image of a homomorphism is always closed under the operations.
Universal algebra proves such facts once, for every kind of structure.
For example, a group is a set with a binary operation $\cdot$, a unary operation ${}^{-1}$ and a constant $e$.
A ring is a set with two binary operations, one unary operation and two constants.
The general theory then applies to both.