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  "body": "\n## Summary\n\nA quasi-identity is a law of the form \"if these finitely many equations between terms hold, then this further equation holds\".\nIt is required for all values of the variables.\nIt generalizes an [[identity|PKD440G2XW]], which is the case with no hypotheses.\nQuasi-identities are exactly the axioms of [[quasivarieties|WRXEKT4NQS]], just as identities are the axioms of varieties.\nThey are the strict universal [[Horn clauses|QR92EZDTKF]] of an algebraic language.\n\n## Explanations\n\n### L1 human\n\nAn identity such as $x \\cdot y = y \\cdot x$ must hold for every choice of $x$ and $y$.\nA quasi-identity only demands its conclusion when some equations already hold, like a conditional rule.\nThe cancellation law \"if $x z = y z$ then $x = y$\" is a typical example.\nIt holds in every group, since one can multiply by $z^{-1}$ on the right.\nIt fails in the integers under multiplication, because $1 \\cdot 0 = 2 \\cdot 0$ while $1 \\ne 2$.\n",
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