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  "body": "\n## Definition\n\nLet $\\sigma$ be an algebraic signature.\nLet $K$ be a class of $\\sigma$-algebras.\nThen $K$ is a *quasivariety* if there is a set $\\Sigma$ of [[quasi-identities|94GVY3APPQ]] of type $\\sigma$ with the following property.\nAn algebra of type $\\sigma$ is in $K$ exactly when it satisfies every member of $\\Sigma$.\n\n## Remarks\n\nExample: the class of cancellative semigroups is a quasivariety.\nIt is defined by the associative law, which is a quasi-identity with no hypotheses, and by the two quasi-identities $z x \\approx z y \\rightarrow x \\approx y$ and $x z \\approx y z \\rightarrow x \\approx y$.\n\nNon-example: the class of fields, in the signature $(+, -, \\cdot, 0, 1)$, is not a quasivariety.\nEvery quasivariety contains the one-element algebra, since all terms take the same value there, so every conclusion $s \\approx t$ holds.\nBut a field needs $0 \\ne 1$.\nAlso, every quasivariety is closed under direct products, and $\\Q \\times \\Q$ is not a field.\n\nEach [[variety|AJNZJD8442]] is a quasivariety, defined by identities, which are quasi-identities with no hypotheses.\nThe smallest quasivariety containing a class $K$ is denoted $Q(K)$ or $\\mathbf{ISPP_U}(K)$, by [[Mal'cev's characterization|DFNM3BGMAY]].\n",
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