Pixley term

Remarks

The first two identities say that $p$ is a Mal'cev term. The term $m(x, y, z) = p(x, p(x, y, z), z)$ is then a majority term: $m(x, x, z) = p(x, z, z) = x$, $m(x, y, x) = p(x, x, x) = x$ and $m(x, z, z) = p(x, x, z) = z$. Pixley's theorem says that $\mathcal{V}$ has a Pixley term if and only if $\mathcal{V}$ is an arithmetical variety. Example: for Boolean algebras, $p(x, y, z) = (x \wedge \neg y) \vee (x \wedge z) \vee (\neg y \wedge z)$ is a Pixley term. Indeed $p(x, y, y) = (x \wedge \neg y) \vee (x \wedge y) \vee 0 = x$, $p(x, x, y) = 0 \vee (x \wedge y) \vee (\neg x \wedge y) = y$, and $p(x, y, x) = (x \wedge \neg y) \vee x \vee (\neg y \wedge x) = x$. Non-example: for groups, the Mal'cev term $x y^{-1} z$ is not a Pixley term. In the group $\Z$, the third identity would read $x - y + x = x$, which fails for $x \ne y$.