Day terms

Remarks

Day's theorem says that $\mathcal{V}$ is a congruence-modular variety if and only if $\mathcal{V}$ has Day terms for some $n$. Example: if $p$ is a Mal'cev term for $\mathcal{V}$, then $m_0 = x$, $m_1 = p(u, z, y)$, $m_2 = u$ are Day terms. The 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 $m_1(x, y, y, u) = p(u, y, y) = u = m_2(x, y, y, u)$. For groups this gives $m_1(x, y, z, u) = u z^{-1} y$. Non-example: with $n = 1$, the terms $m_0 = x$ and $m_1 = u$ are not Day terms in any nontrivial variety. Clause 3 for $i = 0$ reads $x = u$, which holds only in one-element algebras.