Taylor term

Remarks

Example: a semilattice term $x \wedge y$ is a Taylor term of the variety of semilattices. Check: $x \wedge x \approx x$, and the identity $x \wedge y \approx y \wedge x$ has $x$ on the left and $y$ on the right in place $1$. Swapping the names of $x$ and $y$ in it gives the required identity for place $2$. Non-example: in the variety of sets with no operations, the term $x_1$ is idempotent but fails clause 2 for $i = 1$. An identity with $x_1$ on both sides reads $u_1 \approx v_1$, which is false for $u_1 = x$ and $v_1 = y$. Every majority term and every Mal'cev term is a Taylor term. For a majority term $m$, the identity $m(x, x, y) \approx m(y, x, x)$ serves places $1$ and $3$, and $m(x, y, x) \approx m(x, x, y)$ handles place $2$ after renaming.