---
tag: 351WM99D5W
kind: definition
name: Free algebra in a variety
msc: [08B20]
defines: GTS4DRRMZR
uses: [AJNZJD8442, KBXSZC6BV1, 2TQD8NMG9P]
sources:
  - url: https://en.wikipedia.org/wiki/Free_object
---

## Definition

Let $\mathcal{V}$ be a [[variety|AJNZJD8442]] and let $X$ be a set.
A *free algebra of $\mathcal{V}$ on $X$* is an algebra $F \in \mathcal{V}$ together with a map $\eta \colon X \to F$ such that:

1. $F$ is [[generated|2TQD8NMG9P]] by $\eta(X)$.
2. For every $A \in \mathcal{V}$ and every map $f \colon X \to A$, there is a [[homomorphism|KBXSZC6BV1]] $h \colon F \to A$ with $h \circ \eta = f$.

It is written $F_{\mathcal{V}}(X)$, and $X$ is called a set of free generators.

## Remarks

Clause 1 makes the homomorphism $h$ in clause 2 unique, since two homomorphisms that agree on a generating set are equal.
A free algebra on $X$ is unique up to a unique isomorphism compatible with $\eta$.
If $\mathcal{V}$ contains an algebra with at least two elements, then $\eta$ is injective, and $X$ is usually regarded as a subset of $F$.
Example: in the variety of abelian groups, $F(X) = \Z^{(X)}$, the group of finitely supported functions $X \to \Z$, with $\eta(x)$ the indicator function of $x$.
A map $f \colon X \to A$ extends by $h(c) = \sum_{x} c(x) f(x)$, and this is a homomorphism.
Non-example: $\Z / 2\Z$ with $\eta(x) = 1$ is not free on $\set{x}$ in abelian groups, since the map $x \mapsto 1 \in \Z$ has no homomorphic extension $\Z / 2\Z \to \Z$.
When the signature has no constants and empty algebras are excluded, one takes $X$ nonempty.