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

TL;DR

This paper introduces an infinite family of cancellative monoids that are not group-embeddable, explores their algebraic properties such as alignment, and relates these findings to applications in $ ext{C}^*$-algebra studies.

Contribution

It describes and provides presentations for the monoids $ ext{M}_n$, demonstrating their non-embeddability and alignment properties, extending Malcev's classical results.

Findings

$ ext{M}_n$ are singly aligned for $n \\geq 2$

$ ext{M}_1$ is not singly aligned but is 2-aligned

The monoids relate to $ ext{C}^*$-algebra applications

Abstract

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 $n \geq 2$, owing to applications in the study of $\mathrm{C}^*$-algebras by Brix, Bruce and Dor-On. We finish by showing that $\mathcal{M}_1$ is not singly aligned, but is $2$-aligned.