Monoids $\mathrm{Mon}\langle a,b:a^αb^βa^γb^δ=b\rangle$ admit finite complete rewriting systems

We prove that every monoid $\mathrm{Mon}\langle a,b:a^αb^βa^γb^δ=b\rangle$ admits a finite complete rewriting system. Furthermore we prove that $\mathrm{Mon}\langle a,b:ab^2a^2b^2=b\rangle$ is non-hopfian, providing an example of a finitely presented non-residually finite monoid with linear Dehn function.

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