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

TL;DR

This paper proves that the meet levels of the Trotter-Weil hierarchy are not local for all levels, providing explicit languages to demonstrate this non-locality, thus resolving a conjecture in the field.

Contribution

It establishes the non-locality of all meet levels of the Trotter-Weil hierarchy, confirming a conjecture and providing explicit examples of languages with specific semigroup properties.

Findings

Meet levels are not local for all m ≥ 1.

Explicit languages are constructed with specific semigroup properties.

Confirms the conjecture by Kufleitner and Lauser.

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 $\mathsf{V}_m*\mathsf{D}$.