A collection of cancellative, singly aligned, non-embeddable monoids

By classical results of Malcev, cancellative monoids need not be group-embeddable. In this paper, we describe and give presentations for and study an infinite family ${\mathcal{M}}_n$ of cancellative monoids which are not group-embeddable, originating from Malcev’s original work. We show that ${\mathcal{M}}_n$ is singly aligned for $n\ge 2$, owing to applications in the study of ${\mathrm{C}}^{\ast }$-algebras by Brix, Bruce and Dor-On. We finish by showing that ${\mathcal{M}}_1$ is not singly aligned, but is $2$-aligned.