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  "body": "\n## Definition\n\nLet $K$ be a class of algebras of a fixed signature $\\sigma$.\n\n1. $I(K)$ is the class of algebras [[isomorphic|PZMGQMDN25]] to a member of $K$.\n2. $H(K)$ is the class of algebras $B$ such that there is a surjective [[homomorphism|KBXSZC6BV1]] $A \\to B$ for some $A \\in K$.\n3. $S(K)$ is the class of algebras isomorphic to a [[subalgebra|7CHG6SD365]] of a member of $K$.\n4. $P(K)$ is the class of algebras isomorphic to a [[direct product|WJ73XYBNE4]] $\\prod_{i \\in I} A_i$ with all $A_i \\in K$ and $I$ any index set.\n\nFor operators $O_1, O_2$, the composite $O_1 O_2$ is defined by $O_1 O_2(K) = O_1(O_2(K))$.\nOne writes $O_1 \\le O_2$ when $O_1(K) \\subseteq O_2(K)$ for every class $K$.\nA class $K$ is *closed under* $O$ when $O(K) \\subseteq K$.\n\n## Remarks\n\nThe empty product, with $I = \\emptyset$, is the one-element algebra, so $P(K)$ always contains a trivial algebra.\nExample: for $K = \\set{\\Z}$ in the signature of groups, $\\Z / 2\\Z \\in H(K)$, by reduction modulo $2$, and $\\Z^2 \\in P(K)$.\nNon-example: $\\Z / 2\\Z \\notin S(K)$, because every nontrivial subgroup of $\\Z$ is infinite.\nEach of $H$, $S$ and $P$ is extensive, monotone and idempotent on classes closed under $I$.\nThe standard composition rules are $SH \\le HS$, $PS \\le SP$ and $PH \\le HP$.\n",
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    "name": "Class operators H S P",
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        "locator": "Birkhoff's variety theorem",
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