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  "body": "\n## Summary\n\nA variety of algebras is the class of all algebras of a fixed signature that satisfy a given set of identities.\nGroups, rings, lattices and modules over a fixed ring are the standard examples.\nVarieties are the basic objects of [[universal algebra|W82R6N0GM9]], because many constructions, such as free algebras and presentations, work in every variety at once.\nBy Birkhoff's theorem, a class of algebras is a variety exactly when it is closed under homomorphic images, subalgebras and direct products.\n\n## Explanations\n\n### L1 human\n\nPick some operations and some equations that the operations must obey.\nThe variety is the collection of every algebra, of any size, in which these equations hold.\nFor example, the equations $x(yz) = (xy)z$, $ex = xe = x$ and $x x^{-1} = x^{-1} x = e$ in the operations $\\cdot$, $e$ and ${}^{-1}$ define the variety of groups.\nA field is not described this way, because the inverse of $0$ is undefined, and indeed fields do not form a variety.\n",
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