Finitely presented monoids with linear Dehn function need not have   regular cross-sections
  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. 
Alan J. Cain and Victor Maltcev
cs.FL math.GR