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

Abstract By classical results of Malcev, cancellative monoids need not be group-embeddable. We describe, give presentations for and study an infinite family $${\mathcal {M}}_n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>M</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:math> of cancellative monoids which are not group-embeddable, originating from Malcev’s work. We show that $${\mathcal {M}}_n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>M</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:math> is singly aligned for $$n \ge 2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> , owing to applications in the study of $$\textrm{C}^*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mtext>C</mml:mtext> <mml:mo>∗</mml:mo> </mml:msup> </mml:math> -algebras by Brix, Bruce and Dor-On. We finish by showing that $${\mathcal {M}}_1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>M</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> is not singly aligned, but 2-aligned.