---
tag: NVQJWDJG2T
kind: concept
name: Finitely generated variety
msc: [08B05, 08B26]
specializes: [AJNZJD8442]
sources:
  - url: https://en.wikipedia.org/wiki/Variety_(universal_algebra)
---

## Summary

A finitely generated variety is a [[variety|AJNZJD8442]] generated by finitely many finite algebras, or equivalently by one finite algebra.
It is the smallest variety that contains the given finite algebras.
Every finitely generated variety is a [[locally finite variety|8QPW65PZVW]], and such varieties are the main objects of the structure theory of finite algebras.

## Explanations

### L1 human

Start with one finite algebra, such as the two-element Boolean algebra.
Take all its subalgebras, all its direct powers, and all images of those under homomorphisms.
The class obtained is the variety generated by the algebra.
For the two-element Boolean algebra, this class is the variety of all Boolean algebras.