  We prove that every monoid $\mathrm{Mon}\langle
a,b:a^{\alpha}b^{\beta}a^{\gamma}b^{\delta}=b\rangle$ 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.

Group Theory