Monoids $\mathrm{Mon}⟨a,b:a^{\alpha }{b}^{\beta }{a}^{\gamma }{b}^{\delta }=b⟩$ admit finite complete rewriting systems

Abstract

We prove that every monoid $\mathrm{Mon}⟨a,b:a^{\alpha }{b}^{\beta }{a}^{\gamma }{b}^{\delta }=b⟩$ admits a finite complete rewriting system. Furthermore we prove that $\mathrm{Mon}⟨a,b:a{b}^2{a}^2{b}^2=b⟩$ is non-hopfian, providing an example of a finitely presented non-residually finite monoid with linear Dehn function.