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.
Categories: math.GR