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  "body": "\n## Definition\n\nLet $\\sigma$ be a signature with only function symbols and let $X$ be a set of variables.\nAn *identity*, or *equation*, over $\\sigma$ and $X$ is an ordered pair $(s, t)$ of [[terms|6D3WA7N2W4]] over $\\sigma$ and $X$, written $s \\approx t$.\nAn algebra $A$ of signature $\\sigma$ *satisfies* $s \\approx t$, written $A \\models s \\approx t$, if $s^A[\\alpha] = t^A[\\alpha]$ for every map $\\alpha \\colon X \\to A$, where $t^A[\\alpha]$ is the [[value|DH30WTA26T]] of $t$.\nA class $\\mathcal{K}$ of algebras satisfies $s \\approx t$ if every member of $\\mathcal{K}$ does.\n\n## Remarks\n\nThe symbol $\\approx$ separates the formal pair from the equality $=$ of elements.\nAs a first-order sentence, $s \\approx t$ stands for the universal closure $\\forall x_1 \\cdots \\forall x_n \\, (s = t)$, where $x_1, \\ldots, x_n$ are the variables of $s$ and $t$.\n\nExample: the group $(\\Z, +, -, 0)$ satisfies $x + y \\approx y + x$, since $a + b = b + a$ for all integers $a, b$.\n\nNon-example: the same group does not satisfy $x + x \\approx x$.\nAt $\\alpha(x) = 1$ the two sides take the values $2$ and $1$.\n",
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