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  "body": "\n## Summary\n\nThe class operators $H$, $S$ and $P$ act on classes of algebras of one signature.\nThey give the homomorphic images, the subalgebras and the direct products of the members of the class.\nThey are the three closure operations that appear in Birkhoff's characterisation of varieties.\nRules for composing them, such as $SH \\le HS$ and $PS \\le SP$, lead to Tarski's formula $V(K) = HSP(K)$ for the variety generated by a class $K$.\n\n## Explanations\n\n### L1 human\n\nStart with some algebras, for example the single group $\\Z$.\nThe operator $S$ adds all subgroups, such as $2\\Z$.\nThe operator $H$ adds all homomorphic images, such as $\\Z / 6\\Z$.\nThe operator $P$ adds all direct products, such as $\\Z \\times \\Z$.\nDoing all three, in the order $P$, then $S$, then $H$, gives every abelian group.\n",
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    "name": "Class operators H S P",
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        "url": "https://en.wikipedia.org/wiki/Variety_(universal_algebra)"
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  "name": "Class operators H S P",
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