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  "body": "\n## Definition\n\nLet $\\mathcal{V}$ be a [[variety|AJNZJD8442]] of algebras and let $n \\ge 1$.\n*Day terms* for $\\mathcal{V}$ are quaternary terms $m_0, m_1, \\dots, m_n$ such that $\\mathcal{V}$ satisfies the following [[identities|PKD440G2XW]]:\n\n1. $m_0(x, y, z, u) = x$ and $m_n(x, y, z, u) = u$.\n2. $m_i(x, y, y, x) = x$ for every $i \\le n$.\n3. $m_i(x, x, u, u) = m_{i+1}(x, x, u, u)$ for every even $i \u003c n$.\n4. $m_i(x, y, y, u) = m_{i+1}(x, y, y, u)$ for every odd $i \u003c n$.\n\n## Remarks\n\nDay's theorem says that $\\mathcal{V}$ is a [[congruence-modular variety|QAQAS3JWXW]] if and only if $\\mathcal{V}$ has Day terms for some $n$.\nExample: if $p$ is a [[Mal'cev term|Y203PPTS61]] for $\\mathcal{V}$, then $m_0 = x$, $m_1 = p(u, z, y)$, $m_2 = u$ are Day terms.\nThe checks are $m_1(x, y, y, x) = p(x, y, y) = x$, $m_1(x, x, u, u) = p(u, u, x) = x = m_0(x, x, u, u)$, and $m_1(x, y, y, u) = p(u, y, y) = u = m_2(x, y, y, u)$.\nFor groups this gives $m_1(x, y, z, u) = u z^{-1} y$.\nNon-example: with $n = 1$, the terms $m_0 = x$ and $m_1 = u$ are not Day terms in any nontrivial variety.\nClause 3 for $i = 0$ reads $x = u$, which holds only in one-element algebras.\n\n## Explanations\n\n### L1 human\n\nDay terms are a chain of four-input formulas that witness, by equations alone, that a variety has modular congruence lattices.\n",
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