We completely classify all neutral or costandard elements in the lattice $\mathbb{ MON}$ of all monoid varieties. Further, we prove that an arbitrary upper-modular element of $\mathbb{ MON}$ except the variety of all monoids is either a completely regular or a commutative variety. Finally, we verify that all commutative varieties of monoids are codistributive elements of $\mathbb{ MON}$. Thus, the problems of describing codistributive or upper-modular elements of $\mathbb{ MON}$ are completely reduced to the completely regular case.
Categories: math.GR