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  "body": "\n## Definition\n\nLet $\\sigma$ be an algebraic signature and let $X$ be a set of variables.\nA *quasi-identity* of type $\\sigma$ is an expression\n$$s_1 \\approx t_1 \\wedge \\cdots \\wedge s_n \\approx t_n \\rightarrow s \\approx t,$$\nwhere $n \\ge 0$ and $s_1, t_1, \\ldots, s_n, t_n, s, t$ are [[terms|3YYKGWJJND]] of type $\\sigma$ over $X$.\nA $\\sigma$-algebra $A$ *satisfies* this quasi-identity if every assignment $h \\colon X \\to A$ with $h(s_i) = h(t_i)$ for all $i \\le n$ also has $h(s) = h(t)$.\nHere $h(u)$ denotes the value of the term $u$ in $A$ under $h$.\n\n## Remarks\n\nA quasi-identity with $n = 0$ is an [[identity|PKD440G2XW]] $s \\approx t$.\n\nExample: the right cancellation law $x \\cdot z \\approx y \\cdot z \\rightarrow x \\approx y$ is satisfied by every group.\nIndeed, if $a z = b z$ in a group, then multiplying on the right by $z^{-1}$ gives $a = b$.\n\nNon-example: the monoid $(\\Z, \\cdot, 1)$ does not satisfy this quasi-identity.\nThe assignment $x \\mapsto 1$, $y \\mapsto 2$, $z \\mapsto 0$ satisfies the hypothesis $1 \\cdot 0 = 2 \\cdot 0$ but not the conclusion $1 = 2$.\n\nIn first-order terms, a quasi-identity stands for the universal sentence $\\forall \\bar x \\, (s_1 = t_1 \\wedge \\cdots \\wedge s_n = t_n \\rightarrow s = t)$.\nAnother example is $x^2 \\approx 1 \\rightarrow x \\approx 1$, which a group satisfies exactly when it has no element of order $2$.\nThis property cannot be expressed by identities, since the class of such groups is not closed under quotients: the additive group $\\Z$ has it, and its quotient $\\Z / 2\\Z$ does not.\n",
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