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  "body": "\n## Summary\n\nA [[variety|AJNZJD8442]] is residually small when the sizes of its [[subdirectly irreducible|DASF939439]] members are bounded by a cardinal.\nBy Birkhoff's subdirect representation theorem, every algebra of a variety is a subdirect product of subdirectly irreducible members, so these members control the whole variety.\nA residually small variety has, up to isomorphism, only a set of subdirectly irreducible members; otherwise it is called residually large.\nResidual smallness is related to the existence of enough injective algebras in a variety.\n\n## Explanations\n\n### L1 human\n\nEvery algebra of a variety can be broken down into irreducible pieces that cannot be split further.\nIn a residually small variety these pieces never get too big: there is a fixed size that none of them reaches.\nAbelian groups form such a variety: their irreducible pieces are the cyclic groups $\\Z/p^n\\Z$ and the Prüfer groups $\\Z(p^\\infty)$, all countable.\nThe variety of all groups is not residually small, because there are simple groups of every infinite size.\n",
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