Ternary discriminator

Remarks

Example: on a field $F$ with $q$ elements, the polynomial $s(x, y, z) = x + (z - x)\bigl(1 - (x - y)^{q - 1}\bigr)$ gives the discriminator. Check: if $x = y$ then $(x - y)^{q - 1} = 0$ and $s = z$; if $x \neq y$ then $(x - y)^{q - 1} = 1$ and $s = x$. Non-example: on $\set{0, 1}$ the lattice term $(x \wedge y) \vee (y \wedge z) \vee (x \wedge z)$ is not the discriminator, since at $(0, 1, 1)$ it gives $1$ while the discriminator gives $0$. The discriminator satisfies $t(x, y, y) = x$, $t(x, x, y) = y$ and $t(x, y, x) = x$, so it is a Pixley term operation.