---
tag: 8QPW65PZVW
kind: concept
name: Locally finite variety
msc: [08B05, 08B20]
specializes: [AJNZJD8442]
sources:
  - url: https://en.wikipedia.org/wiki/Locally_finite_variety
same_as:
  wikidata: Q6664684
  wikipedia: Locally finite variety
---

## Summary

A locally finite variety is a [[variety|AJNZJD8442]] of algebras in which every finitely generated member is finite.
Equivalently, its free algebras on finitely many generators are finite.
Many structural results on finite algebras, such as those of tame congruence theory, are stated for locally finite varieties.

## Explanations

### L1 human

In some kinds of algebra, a few elements can only produce finitely many new elements by the operations.
A variety is locally finite when this happens for every finite set of starting elements in every algebra of the variety.
Boolean algebras are an example: $n$ elements generate a Boolean algebra with at most $2^{2^n}$ elements.
Groups are not an example, since one element of infinite order already generates the infinite group $\Z$.