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  "body": "\n## Definition\n\nLet $\\mathcal{V}$ be a [[variety|AJNZJD8442]] of algebras and let $n \\ge 0$.\n*Gumm terms* for $\\mathcal{V}$ are ternary terms $d_0, d_1, \\dots, d_n$ and $p$ such that $\\mathcal{V}$ satisfies the following [[identities|PKD440G2XW]]:\n\n1. $d_0(x, y, z) = x$.\n2. $d_i(x, y, x) = x$ for every $i \\le n$.\n3. $d_i(x, x, z) = d_{i+1}(x, x, z)$ for every even $i \u003c n$.\n4. $d_i(x, z, z) = d_{i+1}(x, z, z)$ for every odd $i \u003c n$.\n5. $d_n(x, z, z) = p(x, z, z)$.\n6. $p(x, x, z) = z$.\n\n## Remarks\n\nA variety has Gumm terms if and only if it is a [[congruence-modular variety|QAQAS3JWXW]].\nExample: if $p$ is a [[Mal'cev term|Y203PPTS61]], then $n = 0$, $d_0 = x$ and $p$ are Gumm terms, since $p(x, z, z) = x = d_0(x, z, z)$ and $p(x, x, z) = z$.\nExample: lattices have Gumm terms with $n = 1$, $d_0 = x$, $d_1 = (x \\wedge y) \\vee (y \\wedge z) \\vee (x \\wedge z)$ and $p(x, y, z) = z$.\nHere $d_0(x, x, z) = x = d_1(x, x, z)$, $d_1(x, z, z) = z = p(x, z, z)$, and $p(x, x, z) = z$.\nThese examples show the two extreme cases: a pure Mal'cev step, and a chain of [[Jónsson terms|3M8Y76E4AG]] with $p$ the third projection.\nNon-example: with $n = 0$, the terms $d_0 = x$ and $p = x$ are not Gumm terms in a nontrivial variety, since clause 6 reads $x = z$.\n\n## Explanations\n\n### L1 human\n\nGumm terms are a chain of three-input formulas, ending in one Mal'cev-like formula, that witness modularity of all congruence lattices by equations alone.\n",
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