Cancellable elements of the lattice of monoid varieties

The set of all cancellable elements of the lattice of semigroup varieties has recently been shown to be countably infinite. But the description of all cancellable elements of the lattice $\mathbb{MON}$ of monoid varieties remains unknown. This problem is addressed in the present article. The first example of a monoid variety with modular but non-distributive subvariety lattice is first exhibited. Then a necessary condition of the modularity of an element in $\mathbb{MON}$ is established. These results play a crucial role in the complete description of all cancellable elements of the lattice $\mathbb{MON}$. It turns out that there are precisely five such elements.