Special elements of the lattice of monoid varieties

We completely classify all neutral and costandard elements in the lattice MON of all monoid varieties. Further, we prove that an arbitrary upper-modular element of 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 MON . Thus, the problems of describing codistributive or upper-modular elements of MON are completely reduced to the completely regular case.