---
tag: AJNZJD8442
kind: concept
name: Variety of algebras
msc: [08B99, 03C05]
sources:
  - url: https://en.wikipedia.org/wiki/Variety_(universal_algebra)
same_as:
  wikidata: Q3502010
  wikipedia: Variety (universal algebra)
---

## Summary

A variety of algebras is the class of all algebras of a fixed signature that satisfy a given set of identities.
Groups, rings, lattices and modules over a fixed ring are the standard examples.
Varieties are the basic objects of [[universal algebra|W82R6N0GM9]], because many constructions, such as free algebras and presentations, work in every variety at once.
By Birkhoff's theorem, a class of algebras is a variety exactly when it is closed under homomorphic images, subalgebras and direct products.

## Explanations

### L1 human

Pick some operations and some equations that the operations must obey.
The variety is the collection of every algebra, of any size, in which these equations hold.
For example, the equations $x(yz) = (xy)z$, $ex = xe = x$ and $x x^{-1} = x^{-1} x = e$ in the operations $\cdot$, $e$ and ${}^{-1}$ define the variety of groups.
A field is not described this way, because the inverse of $0$ is undefined, and indeed fields do not form a variety.