Finitely presented monoids with linear Dehn function need not have regular cross-sections

TL;DR

This paper demonstrates that finitely presented monoids with linear Dehn functions can lack regular cross-sections, highlighting a key difference from finitely presented groups where such regular cross-sections always exist.

Contribution

It provides the first example of a finitely presented monoid with linear Dehn function that does not have a regular cross-section, expanding understanding of monoid properties.

Findings

Finitely presented monoids with linear Dehn functions may lack regular cross-sections.

Contrasts with finitely presented groups where linear Dehn functions imply regular cross-sections.

Strengthens the distinction between monoid and group properties regarding Dehn functions.

Abstract

This paper shows that a finitely presented monoid with linear Dehn function need not have a regular cross-section, strengthening the previously-known result that such a monoid need not be presented by a finite complete string rewriting system, and contrasting the fact that finitely presented groups with linear Dehn function always have regular cross-sections.