---
tag: TTTEV3NWWM
kind: concept
name: Absolutely free algebra
msc: [08B20, 08A40]
specializes: [GTS4DRRMZR]
sources:
  - url: https://en.wikipedia.org/wiki/Term_algebra
---

## Summary

An absolutely free algebra is an algebra that is free in the class of all algebras of its signature, with no identities imposed at all.
Up to isomorphism, the absolutely free algebra on a set $X$ is the [[term algebra|3YYKGWJJND]] over $X$, whose elements are the formal terms.
Every free algebra of a variety is a quotient of an absolutely free algebra, so terms are the common raw material of all free algebras.

## Explanations

### L1 human

Take generators $x, y$ and one binary operation $*$.
Build every expression, such as $x * y$, $(x * y) * x$ and $x * (y * x)$, and treat two expressions as equal only when they are written the same way.
This algebra satisfies no law, not even $x * y = y * x$ or associativity.
It is "absolutely" free because no rule restricts it.