Remark

Let E be a monoid and x an element of E. If x is left invertible it is left cancellable; for, if $x'$ is a left inverse of $x$, then

and $\gamma_x$ is injective. In particular, if x is left invertible, the left and right translations by x are bijective. Conversely, suppose that $\gamma_x$ is bijective; there exists $x' \in E$ such that $xx' = \gamma_x(x') = e$; then $\gamma_x(x'x) = (xx')x = x = \gamma_x(e)$ and hence $x'x = e$, so that x is invertible. We see similarly that if $\delta_x$ is bijective, x is invertible.