---
tag: 3H2W1MQFGM
kind: definition
name: Absolutely free algebra
msc: [08B20, 08A40]
defines: TTTEV3NWWM
uses: [KBXSZC6BV1, 2TQD8NMG9P, 3YYKGWJJND]
sources:
  - url: https://en.wikipedia.org/wiki/Term_algebra
---

## Definition

Let $\sigma$ be a signature of function symbols and let $X$ be a set.
An *absolutely free algebra* of signature $\sigma$ on $X$ is an algebra $F$ of signature $\sigma$ with a map $\eta \colon X \to F$ such that:

1. $F$ is [[generated|2TQD8NMG9P]] by $\eta(X)$.
2. For every algebra $A$ of signature $\sigma$ and every map $f \colon X \to A$, there is a [[homomorphism|KBXSZC6BV1]] $h \colon F \to A$ with $h \circ \eta = f$.

## Remarks

This is a free algebra in the variety of all algebras of signature $\sigma$, the variety defined by the empty set of identities.
The [[term algebra|3YYKGWJJND]] $T(X)$, whose elements are terms over $X$, is an absolutely free algebra on $X$.
Every absolutely free algebra on $X$ is isomorphic to it.
Let $F$ be generated by $\eta(X)$.
Then $F$ is absolutely free on $X$ exactly when $\eta$ and each basic operation are injective, and the images of different operations are disjoint and do not meet $\eta(X)$.
Example: for one unary symbol $s$ and $X = \set{x}$, the algebra $\N$ with $s(n) = n + 1$ and $\eta(x) = 0$ is absolutely free; the extension of $f$ sends $n$ to $s^n(f(x))$.
Non-example: $\Z$ with $s(n) = n + 1$ and $\eta(x) = 0$ is not absolutely free on $\set{x}$, since it is not generated by $0$.