---
tag: 9Q008GMZSW
kind: concept
name: Equational basis
msc: [08B05]
sources:
  - url: https://en.wikipedia.org/wiki/Variety_(universal_algebra)
---

## Summary

An equational basis of a class of algebras is a set of [[identities|PKD440G2XW]] from which all identities of the class follow.
It is a set of axioms for the [[equational theory|F1M51B4MDG]] of the class, and it defines the [[variety|AJNZJD8442]] generated by the class.
The usual axioms of groups, rings or lattices are equational bases, and a central question is whether a given algebra has a finite one.

## Explanations

### L1 human

An algebra satisfies infinitely many laws, and an equational basis is a list of laws from which all the others can be derived.
For groups, written with multiplication, inverse and identity element, three short laws suffice: associativity, $x e \approx x$ and $x x^{-1} \approx e$.
Every other law that holds in all groups, such as $(x y)^{-1} \approx y^{-1} x^{-1}$, can be derived from these.
A basis is useful because it is much shorter than the full list of laws.