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

TL;DR

This paper demonstrates that dropping finiteness in algebraic structures means certain weak near unanimity equations only imply trivial conditions, showing no single nontrivial linear Maltsev condition applies universally to all idempotent algebras with Taylor terms.

Contribution

It proves that no nontrivial linear one-equality Maltsev condition holds in all idempotent algebras with Taylor terms, especially when finiteness is not assumed.

Findings

Weak near unanimity equations imply only trivial conditions in infinite cases.

No single equation can characterize all idempotent algebras with Taylor terms.

Olšák's minimal conditions cannot be reduced to one equation.

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.