---
tag: WZ157RXTNV
kind: definition
name: Locally finite variety
msc: [08B05, 08B20]
defines: 8QPW65PZVW
uses: [AJNZJD8442, WR4WF1K2KW, GTS4DRRMZR]
sources:
  - url: https://en.wikipedia.org/wiki/Locally_finite_variety
---

## Definition

Let $\mathcal{V}$ be a [[variety|AJNZJD8442]] of algebras.
The variety $\mathcal{V}$ is *locally finite* if every [[finitely generated algebra|WR4WF1K2KW]] in $\mathcal{V}$ is finite.

## Remarks

An equivalent condition is that the [[free algebra|GTS4DRRMZR]] $F_{\mathcal{V}}(n)$ of $\mathcal{V}$ on $n$ generators is finite for every natural number $n$.
This holds because every algebra of $\mathcal{V}$ generated by $n$ elements is a homomorphic image of $F_{\mathcal{V}}(n)$.
Example: the variety of Boolean algebras is locally finite.
A Boolean algebra generated by $n$ elements is an image of the free Boolean algebra on $n$ generators, which has $2^{2^n}$ elements.
Non-example: the variety of groups is not locally finite.
The group $(\Z, +)$ is generated by the single element $1$ and is infinite.

## Explanations

### L1 human

In a locally finite variety, a finite set of elements can never generate an infinite algebra.