---
tag: QBHNB4CP22
kind: definition
name: Finitely generated variety
msc: [08B05, 08B26]
defines: NVQJWDJG2T
uses: [AJNZJD8442, S7MJVSP79N]
sources:
  - url: https://en.wikipedia.org/wiki/Variety_(universal_algebra)
---

## Definition

Let $\mathcal{V}$ be a [[variety|AJNZJD8442]] of algebras of a fixed signature.
The variety $\mathcal{V}$ is *finitely generated* if $\mathcal{V} = \mathbf{HSP}(\mathcal{K})$ for some finite set $\mathcal{K}$ of finite algebras.
Here $\mathbf{H}$, $\mathbf{S}$ and $\mathbf{P}$ are the [[class operators|S7MJVSP79N]] of homomorphic images, subalgebras and direct products.

## Remarks

If $\mathcal{K} = \set{A_1, \dots, A_k}$, then $\mathbf{HSP}(\mathcal{K}) = \mathbf{HSP}(A_1 \times \cdots \times A_k)$, so one finite algebra suffices.
Example: the variety of abelian groups that satisfy the identity $6x = 0$ is generated by the finite group $\Z/6\Z$.
Each such group is a direct sum of copies of $\Z/2\Z$ and $\Z/3\Z$, and both of these are subgroups of $\Z/6\Z$.
A direct sum is a subgroup of the direct product, so each such group lies in $\mathbf{SP}(\Z/6\Z)$.
Non-example: the variety of all abelian groups is not finitely generated.
If $A$ is a finite abelian group with $m$ elements, then every group in $\mathbf{HSP}(A)$ satisfies the identity $mx = 0$, while $\Z$ does not.
Every finitely generated variety is a [[locally finite variety|8QPW65PZVW]].

## Explanations

### L1 human

A variety is finitely generated when one finite algebra already produces the whole variety by taking subalgebras, products and images.