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  "body": "\n## Definition\n\nLet $\\sigma$ be an algebraic signature.\nA class $K$ of $\\sigma$-algebras is a *prevariety* if it satisfies these three conditions.\n\n1. If $A \\in K$ and $B$ is [[isomorphic|PZMGQMDN25]] to $A$, then $B \\in K$.\n2. If $A \\in K$ and $B$ is a [[subalgebra|7CHG6SD365]] of $A$, then $B \\in K$.\n3. If $(A_i)_{i \\in I}$ is a family of members of $K$, then the [[direct product|WJ73XYBNE4]] $\\prod_{i \\in I} A_i$ is in $K$; for $I = \\emptyset$ this product is the one-element algebra.\n\n## Remarks\n\nIn operator notation, $K$ is a prevariety exactly when $\\mathbf{ISP}(K) = K$, and $\\mathbf{ISP}(K)$ is the prevariety generated by $K$.\n\nExample: residually finite groups form a prevariety.\nA subgroup of a residually finite group is residually finite, by restricting the finite quotients.\nA product of residually finite groups is residually finite.\nIndeed, a nontrivial element has a nontrivial coordinate.\nCompose the projection to that coordinate with a finite quotient that separates this coordinate from $1$.\nThis prevariety is not a [[quasivariety|WRXEKT4NQS]], since it is not closed under ultraproducts.\nA nonprincipal ultraproduct of the groups $\\Z / p\\Z$ over the primes $p$ is a nontrivial torsion-free divisible abelian group, which has no proper subgroup of finite index.\n\nNon-example: the class of fields fails condition 3, since $\\Q \\times \\Q$ has the zero divisor $(1, 0)$.\n",
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