Relatively free algebra

Remarks

Equivalently, $A$ has a generating set $X$ 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. 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 $u^2 = e$.