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  "body": "\n## Definition\n\nLet $\\mathcal{V}$ be a [[variety|AJNZJD8442]] of algebras.\nThe variety $\\mathcal{V}$ is *residually small* if there is a cardinal $\\kappa$ such that every [[subdirectly irreducible algebra|DASF939439]] in $\\mathcal{V}$ has fewer than $\\kappa$ elements.\nA variety that is not residually small is *residually large*.\n\n## Remarks\n\nEquivalently, the subdirectly irreducible members of $\\mathcal{V}$ form a set up to isomorphism.\nThe variety is residually finite when $\\kappa$ can be chosen with every subdirectly irreducible member finite, and it has finite residual bound when $\\kappa$ can be chosen finite.\n\nExample: the variety of abelian groups is residually small.\nIts subdirectly irreducible members are the groups $\\Z/p^n\\Z$ and $\\Z(p^\\infty)$ for primes $p$ and $n \\ge 1$, which are all countable, so $\\kappa = \\aleph_1$ works.\n\nNon-example: the variety of all groups is residually large.\nFor an infinite set $X$, the group of even permutations of $X$ that move only finitely many points is [[simple|9CJCXNYNK8]] and has $\\abs{X}$ elements.\nA nontrivial simple algebra is subdirectly irreducible, so no cardinal $\\kappa$ bounds the sizes.\n",
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