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  "body": "\n## Definition\n\nLet $\\mathcal{V}$ be a [[variety|AJNZJD8442]] of signature $\\sigma$ and let $\\mathcal{W}$ be a variety of signature $\\tau$.\nAn *interpretation* of $\\mathcal{V}$ in $\\mathcal{W}$ is a map $f \\mapsto t_f$ with the following properties.\nIt assigns to each $n$-ary operation symbol $f$ of $\\sigma$ a $\\tau$-term $t_f(x_1, \\dots, x_n)$.\nFor $A$ in $\\mathcal{W}$, let $A^{*}$ be the algebra on the universe of $A$ where each $f$ acts as the [[term operation|N9M2N27END]] $t_f^{A}$.\nThen $A^{*}$ belongs to $\\mathcal{V}$ for every $A$ in $\\mathcal{W}$.\nThe variety $\\mathcal{V}$ is *interpretable* in $\\mathcal{W}$ if an interpretation of $\\mathcal{V}$ in $\\mathcal{W}$ exists.\n\n## Remarks\n\nExample: let $\\mathcal{V}$ be the variety of Boolean algebras, with operations $\\vee, \\wedge, \\neg, 0, 1$, and let $\\mathcal{W}$ be the variety of Boolean rings.\nThe assignment $x \\vee y \\mapsto x + y + xy$, $x \\wedge y \\mapsto xy$, $\\neg x \\mapsto 1 + x$, $0 \\mapsto 0$, $1 \\mapsto 1$ is an interpretation.\nIn every Boolean ring these term operations satisfy the Boolean algebra identities.\nNon-example: let $\\mathcal{V}$ be the variety of semigroups and $\\mathcal{W}$ the variety of groups, and assign to the semigroup product the group term $x y^{-1}$.\nThis is not an interpretation, since in the group $(\\Z, +)$ the operation $a - b$ is not associative: $(1 - 1) - 1 = -1$ but $1 - (1 - 1) = 1$.\nIt suffices to check the defining identities of $\\mathcal{V}$ on the free algebras of $\\mathcal{W}$.\n\n## Explanations\n\n### L1 human\n\nAn interpretation writes each operation of one variety as a formula in the operations of another, in a way that keeps all the laws true.\n",
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