---
tag: K7131908S7
kind: definition
name: Relatively free algebra
msc: [08B20]
defines: NYYV5V2FDD
uses: [GTS4DRRMZR, AJNZJD8442, 2TQD8NMG9P, M4ZHB3GXBP]
sources:
  - url: https://en.wikipedia.org/wiki/Free_object
---

## Definition

Let $A$ be an algebra, and let $V(A)$ be the [[variety|AJNZJD8442]] generated by $A$.
The algebra $A$ is *relatively free* if, for some subset $X \subseteq A$, it is a [[free algebra|GTS4DRRMZR]] of $V(A)$ on $X$ via the inclusion map.

## Remarks

Equivalently, $A$ has a [[generating set|2TQD8NMG9P]] $X$ such that every map $X \to A$ extends to an endomorphism of $A$.
This follows from the fact that a universal property for the class $\set{A}$ carries over to $V(A) = HSP(A)$, by [[Tarski's formula|M4ZHB3GXBP]].
Example: the free group $F_2$ on $\set{a, b}$ is relatively free, since every map $\set{a, b} \to F_2$ extends to an endomorphism.
Example: $\Z / n\Z$ with $X = \set{1}$ is relatively free; it is the free algebra on one generator of the variety of abelian groups satisfying $nx \approx 0$.
Non-example: the symmetric group $S_3$ is not relatively free.
Every generating set $X$ of $S_3$ contains a transposition $t$, since the even permutations form a proper subgroup.
A map $X \to S_3$ that sends $t$ to an element of order $3$ cannot extend to an endomorphism, since $t^2 = e$ would force the image $u$ to satisfy $u^2 = e$.