Siggers term

Remarks

Example: in the variety of semilattices, $s = x_1 \wedge \cdots \wedge x_6$ is a Siggers term. Check: it is idempotent, and both sides of the second identity equal $x \wedge y \wedge z$, by commutativity, associativity and idempotence. Non-example: in the variety of sets with no operations, the only terms are variables, and no variable $x_i$ is a Siggers term. The two sides of the second identity differ in every place, so for $s = x_i$ it would read $x \approx y$, $x \approx z$ or $y \approx z$. Every Siggers term is a Taylor term: substituting $z = x$, $y = x$ or $x = y$ in the second identity gives, for each place, an identity in two variables that differs in that place. A $4$-ary variant, an idempotent term with $s(a, r, e, a) \approx s(r, a, r, e)$, is also used; a finite algebra has a Taylor term exactly when it has a term of this kind.