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We prove that every monoid $\mathrm{Mon}\langlea,b:a^{\alpha}b^{\beta}a^{\gamma}b^{\delta}=b\rangle$ admits a finite completerewriting system. Furthermore we prove that $\mathrm{Mon}\langlea,b:ab^2a^2b^2=b\rangle$ is non-hopfian, providing an example of a finitelypresented non-residually finite monoid with linear Dehn function.

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