Abstract. A semigroup variety is called {\it modular} [{\it 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 $\le2$, and in the class of varieties satisfying an identity of the form $x_1x_2 \cdots x_n=x_{1\pi }x_{2\pi }\cdots x_{n\pi }$, where $\pi $ is a permutation on the set $\{1,2,\ldots,n\} $ with $1\pi\not=1$ and $n\pi\not=n$. It turns out that every lower-modular variety is modular in all these three classes. Moreover, for varieties of index $\le2$, 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.

AMS Subject Classification (1991): 20M07, 08B15

Keyword(s): Semigroup, variety, lattice of varieties, [lower-, upper-]modular element, neutral element

Received October 16, 2007, and in revised form February 14, 2008. (Registered under 6031/2009.)