Amenability of semigroups and the Ore condition for semigroup rings

TL;DR

This paper investigates the relationship between amenability of certain monoids and the Ore condition in their associated monoid rings, showing that the converse of a known implication does not hold in general, including for some groups.

Contribution

It demonstrates that the converse of the known implication between amenability and Ore condition fails even for groups, and analyzes specific cases like free metabelian groups and Thompson's group F.

Findings

The converse implication is false for free metabelian groups.

Thompson's group F's monoid does not satisfy Ore condition despite potential amenability.

The Ore condition cannot be used to establish amenability for Thompson's group F.

Abstract

Let $M$ be a cancellative monoid. It is known~\cite{Ta54} that if $M$ is left amenable then the monoid ring $K[M]$ satisfies Ore condition, that is, there exist nontrivial common right multiples for the elements of this ring. In~\cite{Don10} Donnelly shows that a partial converse to this statement is true. Namely, if the monoid $\mathbb Z^{+}[M]$ of all elements of $\mathbb Z[M]$ with positive coefficients has nonzero common right multiples, then $M$ is left amenable. He asks whether the converse is true for this particular statement. We show that the converse is false even for the case of groups. If $M$ is a free metabelian group, then $M$ is amenable but the Ore condition fails for $\mathbb Z^{+}[M]$. Besides, we study the case of the monoid $M$ of positive elements of R.\,Thompson's group $F$. The amenability problem for it is a famous open question. It is equivalent to left amenability of the monoid $M$. We show that for this case the monoid $\mathbb Z^{+}[M]$ does not satisfy Ore condition. That is, even if $F$ is amenable, this cannot be shown using the above sufficient condition.