<p>By classical results of Malcev, cancellative monoids need not be group-embeddable. We describe, give presentations for and study an infinite family <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="233_2025_10509_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="31" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathcal {M}}_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">M</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> of cancellative monoids which are not group-embeddable, originating from Malcev’s work. We show that <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="233_2025_10509_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="31" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathcal {M}}_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">M</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> is singly aligned for <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="233_2025_10509_Article_IEq3.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$n \ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, owing to applications in the study of <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="233_2025_10509_Article_IEq4.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">$\textrm{C}^*$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mtext>C</mtext> <mo>∗</mo> </msup> </math></EquationSource> </InlineEquation>-algebras by Brix, Bruce and Dor-On. We finish by showing that <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="233_2025_10509_Article_IEq5.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="28" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathcal {M}}_1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">M</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> is not singly aligned, but 2-aligned.</p>

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

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2025年2月28日

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Milo Edwardes,
Daniel Heath

摘要

By classical results of Malcev, cancellative monoids need not be group-embeddable. We describe, give presentations for and study an infinite family ${\mathcal {M}}_n$ $M_{n}$ of cancellative monoids which are not group-embeddable, originating from Malcev’s work. We show that ${\mathcal {M}}_n$ $M_{n}$ is singly aligned for $n \ge 2$ $n \geq 2$ , owing to applications in the study of $\textrm{C}^*$ $C^{*}$ -algebras by Brix, Bruce and Dor-On. We finish by showing that ${\mathcal {M}}_1$ $M_{1}$ is not singly aligned, but 2-aligned.