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  "body": "\n## Definition\n\nLet $\\mathcal{V}$ be a [[variety|AJNZJD8442]] of algebras.\nThe variety $\\mathcal{V}$ is *locally finite* if every [[finitely generated algebra|WR4WF1K2KW]] in $\\mathcal{V}$ is finite.\n\n## Remarks\n\nAn equivalent condition is that the [[free algebra|GTS4DRRMZR]] $F_{\\mathcal{V}}(n)$ of $\\mathcal{V}$ on $n$ generators is finite for every natural number $n$.\nThis holds because every algebra of $\\mathcal{V}$ generated by $n$ elements is a homomorphic image of $F_{\\mathcal{V}}(n)$.\nExample: the variety of Boolean algebras is locally finite.\nA Boolean algebra generated by $n$ elements is an image of the free Boolean algebra on $n$ generators, which has $2^{2^n}$ elements.\nNon-example: the variety of groups is not locally finite.\nThe group $(\\Z, +)$ is generated by the single element $1$ and is infinite.\n\n## Explanations\n\n### L1 human\n\nIn a locally finite variety, a finite set of elements can never generate an infinite algebra.\n",
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