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  "body": "\n## Statement\n\nLet $K$ be a class of algebras of a fixed signature $\\sigma$.\nThen $K$ is a [[variety|AJNZJD8442]] if and only if $K$ is closed under the [[class operators|S7MJVSP79N]] $H$, $S$ and $P$.\nHere a variety is the class of all models of some set of [[identities|PKD440G2XW]].\n\n## Remarks\n\nOne direction is a direct check: identities are preserved by homomorphic images, by subalgebras and by direct products.\nFor the converse, let $\\Sigma$ be the set of identities that hold in every member of $K$, and let $A$ satisfy $\\Sigma$.\nThe free algebra of $HSP(K)$ on a set of the size of $A$ is a subalgebra of a product of members of $K$.\nSince $A$ satisfies $\\Sigma$, the algebra $A$ is a homomorphic image of it.\nSo $A \\in HSP(K) = K$.\nExample: groups form a variety, and the theorem confirms that subgroups, quotient groups and products of groups are groups.\nClosure under products cannot be dropped.\nThe class of groups with at most two elements is closed under $H$ and $S$, but it does not contain $\\Z / 2\\Z \\times \\Z / 2\\Z$.\n\n## Explanations\n\n### L1 human\n\nThere are two ways to describe a class of algebras.\nOne is by the equations its members obey.\nThe other is by the constructions it is closed under.\nBirkhoff's theorem says that being described by equations is the same as being closed under taking images, subalgebras and products.\n\n### L2 human\n\nThe theorem lets one decide whether a class is a variety without writing identities.\nTo show that a class is not equationally definable, one finds a product, subalgebra or image that leaves the class.\nFor example, torsion-free abelian groups are not a variety, because their quotient $\\Z / 2\\Z$ has torsion.\nThe proof introduces free algebras of $HSP(K)$, which are the main tool of the theory.\nA finer form, Tarski's formula $V(K) = HSP(K)$, describes the variety generated by any class.\n",
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  "name": "Birkhoff's HSP theorem",
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