Lower-modular elements of the lattice of semigroup varieties. II

A semigroup variety is called 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. We classify all lower-modular varieties in the class of varieties of semigroups with a completely regular power, in the class of varieties of index <= 2, and in the class of varieties satisfying an identity of the form x(1)x(2) ... x(n) = x(1 pi)x(2 pi) ... x(n pi), where pi is a permutation on the set {1, 2,..., n} with 1 pi (SIC) 1 and n pi (SIC) n. It turns out that every lower-modular variety is modular in all these three classes. Moreover, for varieties of index <= 2, the properties of being lower-modular, modular and neutral are equivalent. We completely determine also all semigroup varieties that are both upper-modular and lower-modular. It turns out that all such varieties are neutral.