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  "body": "\n## Definition\n\nLet $\\mathcal{V}$ be a [[variety|AJNZJD8442]], and for each natural number $n$ let $F_{\\mathcal{V}}(n)$ be the [[free algebra|GTS4DRRMZR]] of $\\mathcal{V}$ on a set of $n$ generators.\nThe *free spectrum* of $\\mathcal{V}$ is the function\n$$f_{\\mathcal{V}}(n) = \\abs{F_{\\mathcal{V}}(n)}, \\qquad n \\in \\N,$$\nwith values in $\\N \\cup \\set{\\infty}$.\n\n## Remarks\n\nThe values are all finite exactly when $\\mathcal{V}$ is a [[locally finite variety|8QPW65PZVW]].\nWhen $\\mathcal{V}$ is generated by a finite algebra $A$, the free algebra $F_{\\mathcal{V}}(n)$ embeds in $A^{A^n}$, so $f_{\\mathcal{V}}(n) \\le \\abs{A}^{\\abs{A}^n}$.\nExample: for Boolean algebras, $f(n) = 2^{2^n}$, since $F(n)$ is the algebra of all Boolean functions $\\set{0, 1}^n \\to \\set{0, 1}$.\nExample: for semilattices, $f(n) = 2^n - 1$, since $F(n)$ is the set of nonempty subsets of the generators under union.\nNon-example: the variety of all groups has no finite spectrum value for $n \\ge 1$, since the free group on one generator is $\\Z$.\nSome authors start the spectrum at $n = 1$, or use the related function $\\log_2 f_{\\mathcal{V}}(n)$.\n",
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