---
tag: YGYZAQ12MZ
kind: concept
name: Finitely based algebra
msc: [08B05]
sources:
  - url: https://en.wikipedia.org/wiki/Variety_(universal_algebra)
---

## Summary

An algebra is finitely based when all of its [[identities|PKD440G2XW]] follow from finitely many of them, that is, when it has a finite [[equational basis|9Q008GMZSW]].
Every finite group, every finite ring and every finite lattice is finitely based, but some finite algebras are not.
Deciding which finite algebras are finitely based is the subject of Tarski's finite basis problem.

## Explanations

### L1 human

Any algebra satisfies infinitely many laws.
It is finitely based if a finite list of these laws is enough to derive all the others.
The integers under addition are finitely based: the laws of abelian groups give every law they satisfy.
Surprisingly, there are algebras with only a few elements whose laws cannot be captured by any finite list.