Non-locality of the meet levels of the Trotter-Weil Hierarchy

Abstract

We prove that the meet level $m$ of the Trotter-Weil, $\mathsf{V}_m$ is not local for all $m \geq 1$, as conjectured in a paper by Kufleitner and Lauser. In order to show this, we explicitly provide a language whose syntactic semigroup is in $L \mathsf{V}_m$ and not in $\mathsf{V}_m*\mathsf{D}$.