Lower-Modular Elements of the Lattice of Semigroup Varieties

We call a semigroup variety modular [upper-modular, lower-modular, neutral] if it is a modular [respectively upper-modular, lower-modular, neutral] element of the lattice of all semigroup varieties. It is proved that if V is a lower-modular variety then either V coincides with the variety of all semigroups or V is periodic and the greatest nil-subvariety of V may be given by 0-reduced identities only. We completely determine all commutative lower-modular varieties. In particular, it turns out that a commutative variety is lower-modular if and only if it is neutral. A number of corollaries of these results are obtained.