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  "body": "\n## Summary\n\nThe equational theory of a class of algebras is the set of all [[identities|PKD440G2XW]] that hold in every member of the class.\nA set of identities is called an equational theory when it arises in this way.\nEquational theories correspond one to one to varieties of algebras, with larger theories giving smaller varieties.\n\n## Explanations\n\n### L1 human\n\nCollect every law that holds in all groups: associativity, $x \\cdot e \\approx x$, $(x y)^{-1} \\approx y^{-1} x^{-1}$, and infinitely many more.\nThis collection is the equational theory of groups.\nIt does not contain $x y \\approx y x$, because some groups are not commutative.\nThe equational theory of abelian groups is bigger: it contains $x y \\approx y x$ and everything that follows from it.\n",
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