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  "body": "\n## Definition\n\nLet $\\mathcal{V}$ be a [[variety|AJNZJD8442]] of algebras of signature $\\sigma$.\nA *subvariety* of $\\mathcal{V}$ is a variety $\\mathcal{W}$ of signature $\\sigma$ such that $\\mathcal{W} \\subseteq \\mathcal{V}$.\n\n## Remarks\n\nIf $\\mathcal{V} = \\operatorname{Mod}(\\Sigma)$, the subvarieties of $\\mathcal{V}$ are exactly the classes $\\operatorname{Mod}(\\Sigma \\cup \\Gamma)$ for sets $\\Gamma$ of [[identities|PKD440G2XW]] of signature $\\sigma$.\nExample: abelian groups form a subvariety of the variety of groups, since they are the groups that satisfy $xy \\approx yx$.\nNon-example: the class of finite groups is contained in the variety of groups but is not a subvariety.\nIt is not closed under products: $\\Z$ is a subgroup of $\\prod_{n \\ge 1} \\Z / n\\Z$, generated by the element $(1, 1, \\ldots)$.\nEvery variety has itself and the trivial variety, of one-element algebras, as subvarieties.\n",
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