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  "body": "\n## Definition\n\nLet $\\mathcal{V}$ be a [[variety|AJNZJD8442]] of algebras of a fixed signature.\nThe variety $\\mathcal{V}$ is *finitely generated* if $\\mathcal{V} = \\mathbf{HSP}(\\mathcal{K})$ for some finite set $\\mathcal{K}$ of finite algebras.\nHere $\\mathbf{H}$, $\\mathbf{S}$ and $\\mathbf{P}$ are the [[class operators|S7MJVSP79N]] of homomorphic images, subalgebras and direct products.\n\n## Remarks\n\nIf $\\mathcal{K} = \\set{A_1, \\dots, A_k}$, then $\\mathbf{HSP}(\\mathcal{K}) = \\mathbf{HSP}(A_1 \\times \\cdots \\times A_k)$, so one finite algebra suffices.\nExample: the variety of abelian groups that satisfy the identity $6x = 0$ is generated by the finite group $\\Z/6\\Z$.\nEach such group is a direct sum of copies of $\\Z/2\\Z$ and $\\Z/3\\Z$, and both of these are subgroups of $\\Z/6\\Z$.\nA direct sum is a subgroup of the direct product, so each such group lies in $\\mathbf{SP}(\\Z/6\\Z)$.\nNon-example: the variety of all abelian groups is not finitely generated.\nIf $A$ is a finite abelian group with $m$ elements, then every group in $\\mathbf{HSP}(A)$ satisfies the identity $mx = 0$, while $\\Z$ does not.\nEvery finitely generated variety is a [[locally finite variety|8QPW65PZVW]].\n\n## Explanations\n\n### L1 human\n\nA variety is finitely generated when one finite algebra already produces the whole variety by taking subalgebras, products and images.\n",
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