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  "body": "\n## Statement\n\nLet $A$ be a finite nonempty set.\nFor a set $F$ of finitary operations on $A$, let $\\operatorname{Inv}(F)$ be the set of finitary relations on $A$ preserved by every member of $F$.\nFor a set $\\Gamma$ of finitary relations on $A$, let $\\operatorname{Pol}(\\Gamma)$ be the set of operations that are [[polymorphisms|A0K9EZRM7E]] of all members of $\\Gamma$.\nThen:\n\n1. $\\operatorname{Pol}(\\operatorname{Inv}(F))$ is the [[clone|PTKR703G6A]] generated by $F$.\n2. $\\operatorname{Inv}(\\operatorname{Pol}(\\Gamma))$ is the [[relational clone|MC25MTN2N7]] generated by $\\Gamma$.\n3. Consequently, $C \\mapsto \\operatorname{Inv}(C)$ is an inclusion-reversing bijection from the clones on $A$ to the relational clones on $A$, with inverse $\\Gamma \\mapsto \\operatorname{Pol}(\\Gamma)$.\n\n## Remarks\n\nThe maps $\\operatorname{Pol}$ and $\\operatorname{Inv}$ come from the relation \"$f$ preserves $R$\" between operations and relations, so they always form a [[Galois connection|Q5JHEPRYQ6]]; the theorem identifies its closed sets.\nFor part 1, take an $n$-ary $f \\in \\operatorname{Pol}(\\operatorname{Inv}(F))$ and the relation $R$ of arity $\\abs{A}^n$ formed by the columns of all $n$-ary operations in the clone generated by $F$.\nThis $R$ is preserved by $F$ and contains the columns of the $n$ projections.\nSo $f$ applied to these columns gives the column of $f$, which lies in $R$, and $f$ is in the clone.\nPart 2 is proved in a similar way, with a relation built from $\\operatorname{Pol}(\\Gamma)$ and a primitive positive definition of it from $\\Gamma$.\nThe result is due to Geiger and, independently, to Bodnarčuk, Kalužnin, Kotov and Romov.\nFor example, on $\\set{0, 1}$ the clone of monotone operations is $\\operatorname{Pol}(\\set{\\le})$.\nFiniteness of $A$ is needed: on an infinite set, the closed sets are clones that are locally closed, and some clones are not of the form $\\operatorname{Pol}(\\Gamma)$.\n\n## Explanations\n\n### L1 human\n\nA family of operations can be described in two ways: by listing operations that generate it, or by listing the relations that all of them respect.\nOn a finite set, the theorem says that the second description is always possible and exact.\nEvery clone is the set of all operations that respect some family of relations.\nDually, a family of relations closed under the natural constructions is the set of all relations respected by some operations.\n\n### L2 human\n\nThe theorem turns questions about clones into questions about relations, which are often easier to handle.\nMaximal clones, for example, are of the form $\\operatorname{Pol}(\\set{\\rho})$ for a single relation $\\rho$, and [[Rosenberg's classification|GF1RK91VED]] lists those relations.\nIn constraint satisfaction, adding a primitive positive definable relation to a template does not change the complexity.\nSo the complexity of a finite template depends only on its polymorphism clone.\nThe same reasoning underlies the [[Baker–Pixley theorem|97SRYBF41G]], which bounds the arity of the relations needed for a clone with a near-unanimity operation.\n",
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  "name": "Pol–Inv Galois connection",
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