Codistributive elements of the lattice of semigroup varieties

We prove that if a semigroup variety is a codistributive element of the lattice SEM of all semigroup varieties then it either coincides with the variety of all semigroups or is a variety of semigroups with completely regular square. We completely classify strongly permutative varieties that are codistributive elements of SEM. We prove that a semigroup variety is a costandard element of the lattice SEM if and only if it is a neutral element of this lattice. In view of results obtained earlier, this gives a complete description of costandard elements of the lattice SEM.