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  "body": "\n## Definition\n\nLet $A$ be an algebra and let $n \\ge 1$.\nAn operation $p \\colon A^n \\to A$ is an $n$-ary *polynomial operation* of $A$ if the following holds.\nThere are $m \\ge 0$, an $(n + m)$-ary [[term operation|N9M2N27END]] $t^A$ of $A$, and elements $c_1, \\ldots, c_m \\in A$ such that\n$$p(a_1, \\ldots, a_n) = t^A(a_1, \\ldots, a_n, c_1, \\ldots, c_m)$$\nfor all $a_1, \\ldots, a_n \\in A$.\n\n## Remarks\n\nEquivalently, the polynomial operations of $A$ are the term operations of the algebra $A_A$ obtained from $A$ by adding a constant symbol for each element of $A$.\nEvery term operation is a polynomial operation, with $m = 0$.\nAn equivalence relation on $A$ is a [[congruence|DCSXXT4SER]] exactly when it is preserved by every unary polynomial operation of $A$.\nFor a commutative ring $R$, the polynomial operations of $R$ are the [[polynomial functions|1G0B73WPX8]] $R^n \\to R$ given by polynomials with coefficients in $R$.\n\nExample: in the group $(\\Z, +, -, 0)$, the map $p(a) = 2a + 5$ is a polynomial operation.\nTake $t = (x_1 + x_1) + x_2$ and $c_1 = 5$; then $t^A(a, 5) = 2a + 5$.\n\nNon-example: on $(\\Z, +, -, 0)$ the map $a \\mapsto a^2$ is not a polynomial operation.\nBy induction on terms, every unary polynomial operation of this group has the form $a \\mapsto k a + c$ with $k, c \\in \\Z$.\nThe map $a \\mapsto a^2$ is not of this form, since its values $0, 1, 4$ at $0, 1, 2$ are not in arithmetic progression.\n",
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