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  "body": "\n## Definition\n\nLet $A$ be an algebra, and let $V(A)$ be the [[variety|AJNZJD8442]] generated by $A$.\nThe 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.\n\n## Remarks\n\nEquivalently, $A$ has a [[generating set|2TQD8NMG9P]] $X$ such that every map $X \\to A$ extends to an endomorphism of $A$.\nThis 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]].\nExample: 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.\nExample: $\\Z / n\\Z$ with $X = \\set{1}$ is relatively free; it is the free algebra on one generator of the variety of abelian groups satisfying $nx \\approx 0$.\nNon-example: the symmetric group $S_3$ is not relatively free.\nEvery generating set $X$ of $S_3$ contains a transposition $t$, since the even permutations form a proper subgroup.\nA 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 $u^2 = e$.\n",
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