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  "body": "\n## Summary\n\nThe free spectrum of a [[variety|AJNZJD8442]] $\\mathcal{V}$ is a function on the natural numbers.\nIt sends $n$ to the number of elements of the free algebra of $\\mathcal{V}$ on $n$ generators.\nIt is finite for all $n$ exactly when $\\mathcal{V}$ is locally finite.\nIts growth rate measures how much freedom the identities of $\\mathcal{V}$ leave, and it is used to separate and classify locally finite varieties.\n\n## Explanations\n\n### L1 human\n\nCount how many different expressions in $n$ letters a variety can tell apart.\nFor Boolean algebras on $n$ letters, every expression is a truth function of $n$ inputs, so there are $2^{2^n}$ of them.\nFor semilattices, an expression is determined by the set of letters it uses, so there are $2^n - 1$ of them.\nFor groups, already one letter gives infinitely many elements $x, x^2, x^3, \\ldots$, so the free spectrum is infinite.\n",
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