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 M. Siggers and refined by K. Kearnes, P. Markovi\'c, and R. McKenzie): \[ t(r,a,r,e)\approx t(a,r,e,a). \] 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 $k\geq 3$. From this it follows that there is no nontrivial linear one-equality condition that would hold in all idempotent algebras having Taylor terms. Miroslav Ol\v{s}\'ak has recently shown that there is a weakest nontrivial strong Maltsev condition for idempotent algebras. Ol\v{s}\'ak has found several such (mutually equivalent) conditions consisting of two or more equations. Our result shows that Ol\v{s}\'ak's equation systems can't be compressed into just one equation.