A collection of cancellative, singly aligned, non-embeddable monoids
Milo Edwardes, Daniel Heath
TL;DR
The paper addresses the embeddability problem for cancellative monoids by constructing the infinite family of monoids $\mathcal{M}_n$ arising from Malcev’s embeddability conditions, each of which is cancellative and not group-embeddable. It provides presentations $\mathcal{M}_n = \mathbf{Mon}\langle X_n \mid \rho_n\rangle$ with two-letter relations, develops left normal forms using the sets $L_n$, $R_n$, $P_n$, and $Q_n$, and shows the associated Cayley graphs are acyclic. A central result is that $\mathcal{M}_n$ is singly aligned for $n \ge 2$, connecting to $C^*$-algebra considerations, while $\mathcal{M}_1$ fails to be singly aligned but is $2$-aligned. The work thus extends Malcev’s classic example to a broad family, delineating when intersections of right ideals are principal and when they require multiple generators, with implications for algebraic structures associated to semigroups and monoids.
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.