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  "body": "\n## Definition\n\nLet $A$ be an algebra.\nThe algebra $A$ is *abelian* if the following holds.\nFor every $n \\ge 1$, every $(n+1)$-ary [[term operation|N9M2N27END]] $t$ of $A$, all elements $a, b \\in A$ and all tuples $\\bar{c}, \\bar{d} \\in A^n$:\n$$t(a, \\bar{c}) = t(a, \\bar{d}) \\implies t(b, \\bar{c}) = t(b, \\bar{d}).$$\n\n## Remarks\n\nIn the language of [[commutator theory|6DSWPHSWQW]], $A$ is abelian if and only if $[1_A, 1_A] = 0_A$, where $1_A$ and $0_A$ are the largest and the least congruences of $A$.\nExample: every module $M$ over a ring $R$ is abelian.\nEach term operation has the form $t(x, \\bar{y}) = r x + s_1 y_1 + \\cdots + s_n y_n$ with $r, s_i \\in R$.\nSo $t(a, \\bar{c}) = t(a, \\bar{d})$ means $\\sum_i s_i c_i = \\sum_i s_i d_i$, and this gives $t(b, \\bar{c}) = t(b, \\bar{d})$ for every $b$.\nNon-example: a lattice $L$ with elements $0 \u003c 1$ is not abelian.\nTake $t(x, y) = x \\wedge y$, $a = 0$, $c = 0$ and $d = 1$: then $t(0, 0) = 0 = t(0, 1)$, but $t(1, 0) = 0 \\ne 1 = t(1, 1)$.\nA group is abelian in this sense exactly when it is commutative.\n\n## Explanations\n\n### L1 human\n\nAn algebra is abelian when the effect of changing some inputs of a term operation never depends on the value of another input.\n",
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