Taylor term does not imply any nontrivial linear one-equality Maltsev condition

Abstract

It is known that any finite idempotent algebra that satisfies a nontrivial Maltsev condition must satisfy the linear one-equality Maltsev condition (a variant of the term discovered by Siggers and refined by Kearnes, Marković, and McKenzie):

$$\begin{aligned} t(r,a,r,e)\approx t(a,r,e,a). \end{aligned}$$

We show that if we drop the finiteness assumption, the k-ary weak near unanimity equations imply only trivial linear one-equality Maltsev conditions for every $k\ge 3$. From this it follows that there is no nontrivial linear one-equality condition that would hold in all idempotent algebras having Taylor terms. Miroslav Olšák has recently shown that there is a weakest nontrivial strong Maltsev condition for idempotent algebras. Olšák has found several such (mutually equivalent) conditions consisting of two or more equations. Our result shows that Olšák’s equation systems cannot be compressed into just one equation.