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  "body": "\n## Definition\n\nLet $\\sigma$ be a signature with only function symbols and let $X$ be a countably infinite set of variables.\nFor a class $\\mathcal{K}$ of algebras of signature $\\sigma$, the *equational theory* of $\\mathcal{K}$ is the set\n$$\\operatorname{Id}(\\mathcal{K}) = \\set{s \\approx t : s, t \\text{ terms over } \\sigma \\text{ and } X, \\ \\mathcal{K} \\models s \\approx t}$$\nof all [[identities|PKD440G2XW]] satisfied by $\\mathcal{K}$.\nA set $\\Sigma$ of identities over $\\sigma$ and $X$ is an *equational theory* if $\\Sigma = \\operatorname{Id}(\\mathcal{K})$ for some class $\\mathcal{K}$.\n\n## Remarks\n\nEvery equational theory contains $t \\approx t$ for every [[term|6D3WA7N2W4]] $t$, is closed under symmetry and transitivity, under applying an operation symbol to identities, and under substitution of terms for variables.\nBy Birkhoff's completeness theorem, these closure properties characterise equational theories.\nFor a set $\\Sigma$ of identities, the equational theory of the class of all algebras satisfying $\\Sigma$ is the set of consequences of $\\Sigma$.\n\nExample: for $\\mathcal{K}$ the class of all commutative semigroups, $\\operatorname{Id}(\\mathcal{K})$ contains $x(yz) \\approx (zy)x$, since both sides equal $xyz$ in every commutative semigroup.\n\nNon-example: the set $\\set{x y \\approx y x}$ is not an equational theory.\nEvery set of the form $\\operatorname{Id}(\\mathcal{K})$ contains the identity $x \\approx x$, which is not in this set.\n",
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