---
tag: GTS4DRRMZR
kind: concept
name: Free algebra in a variety
msc: [08B20]
sources:
  - url: https://en.wikipedia.org/wiki/Free_object
---

## Summary

The free algebra of a [[variety|AJNZJD8442]] $\mathcal{V}$ on a set $X$ is the algebra of $\mathcal{V}$ generated by $X$ with as few relations as possible.
The only relations among the generators are those forced by the identities of $\mathcal{V}$.
Its defining property is that every map from $X$ to an algebra of $\mathcal{V}$ extends to exactly one homomorphism.
Free groups, free abelian groups, polynomial rings and free monoids are free algebras of their varieties.
Every algebra of $\mathcal{V}$ is a quotient of a free algebra, which makes free algebras the starting point of presentations.

## Explanations

### L1 human

Take some letters $x, y, z$ and form all expressions in them that the operations allow.
Declare two expressions equal only when the laws of the variety force them to be equal.
In commutative rings with identity, this gives the polynomial ring $\Z[x, y, z]$: $xy$ equals $yx$, but $x^2$ does not equal $x$.
Because nothing extra was imposed, the letters can be sent anywhere, and the rest of the map is then determined.