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  "body": "\n## Statement\n\nLet $A$ be a finite algebra and let $d \\ge 2$ be an integer.\nAssume that $A$ has a [[near-unanimity term|4CRH8NNFEF]] of arity $d + 1$.\nLet $n \\ge 1$ and let $f \\colon A^n \\to A$ be an operation.\nThen $f$ is a [[term operation|N9M2N27END]] of $A$ if and only if every [[subalgebra|7CHG6SD365]] of the [[direct power|WJ73XYBNE4]] $A^d$ is closed under $f$, applied coordinatewise.\n\n## Remarks\n\nEvery term operation preserves every subalgebra of every power, so only the converse needs work.\nThe idea is interpolation.\nThe operation $f$ agrees with a term operation on each set of at most $d$ points of $A^n$.\nThe reason is that $f$ preserves the subalgebra of $A^d$ generated by the coordinate tuples of those points.\nThe near-unanimity term then glues such local interpolations, by induction on the number $k \u003e d$ of points.\nGiven $k$ points, pick $d + 1$ of them, $p_1, \\dots, p_{d + 1}$, and by induction take $d + 1$ terms, the $i$-th agreeing with $f$ on all $k$ points except possibly $p_i$.\nAt each point at most one of these terms disagrees with $f$, so the near-unanimity term applied to them agrees with $f$ on all $k$ points.\nSince $A^n$ is finite, one reaches a term that agrees with $f$ on all of $A^n$.\nFor a [[majority term|9879ZMW5AA]], so $d = 2$, the theorem says that the term operations are the operations that preserve all subalgebras of $A^2$.\nFor example, a finite lattice $L$ has the majority term $(x \\wedge y) \\vee (y \\wedge z) \\vee (x \\wedge z)$, so an operation on $L$ is a lattice term operation exactly when it preserves all sublattices of $L^2$.\nFor an infinite algebra the same argument only shows that $f$ agrees with some term operation on each finite subset of $A^n$.\n\n## Explanations\n\n### L1 human\n\nTo test whether an operation can be written with the operations of an algebra, one would like a finite list of conditions to check.\nThe theorem gives such a list when the algebra has a near-unanimity term.\nIt is enough to check that the operation respects every subalgebra of a fixed small power $A^d$.\n\n### L2 human\n\nThe theorem is a strong form of the statement that the clone of term operations of $A$ equals the set of polymorphisms of the subalgebras of $A^d$.\nIn the language of the Galois connection between operations and relations, the term clone of $A$ is determined by relations of arity at most $d$.\nIt also shows that two finite algebras on the same set with near-unanimity terms have the same term operations once they have the same subalgebras of $A^d$.\nIn an arithmetical variety a Pixley term $p$ gives the majority term $p(x, p(x, y, z), z)$, so the theorem applies there with $d = 2$.\n",
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  "name": "Baker–Pixley theorem",
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