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  "body": "\n## Definition\n\nLet $\\sigma$ be a signature of function symbols, and let $\\Sigma$ be a set of [[identities|PKD440G2XW]] $s \\approx t$ in the language of $\\sigma$.\nAn algebra $A$ of signature $\\sigma$ *satisfies* $s \\approx t$ if $s$ and $t$ take the same value in $A$ for every assignment of the variables.\nThe class $\\operatorname{Mod}(\\Sigma)$ is the class of all algebras of signature $\\sigma$ that satisfy every identity in $\\Sigma$.\nA *variety* (or *equational class*) of signature $\\sigma$ is a class $\\mathcal{V}$ of algebras of signature $\\sigma$ with $\\mathcal{V} = \\operatorname{Mod}(\\Sigma)$ for some $\\Sigma$.\n\n## Remarks\n\nExample: abelian groups, in the signature $(+, -, 0)$, form a variety.\nIt is $\\operatorname{Mod}(\\Sigma)$ for $\\Sigma$ the identities $(x + y) + z \\approx x + (y + z)$, $x + 0 \\approx x$, $x + (-x) \\approx 0$ and $x + y \\approx y + x$.\nNon-example: fields do not form a variety of signature $(+, \\cdot, -, 0, 1)$.\nEvery variety is closed under direct products, but the product $\\F_2 \\times \\F_2$ of two fields is not a field, since $(1, 0) \\cdot (0, 1) = (0, 0)$.\nBy Birkhoff's theorem, a class is a variety if and only if it is closed under homomorphic images, subalgebras and direct products.\nSome authors write $\\operatorname{Mod}(\\Sigma)$ as $V(\\Sigma)$.\n",
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