## Boris M. Vernikov

*On modular elements of the lattice of semigroup varieties *

Comment.Math.Univ.Carolin. 48,4 (2007) 595-606. **Abstract:**A semigroup variety is called {modular} if it is a modular element of the lattice of all semigroup varieties. We obtain a strong necessary condition for a semigroup variety to be modular. In particular, we prove that every modular nil-variety may be given by 0-reduced identities and substitutive identities only. (An identity $u=v$ is called {substitutive} if the words $u$ and $v$ depend on the same letters and $v$ may be obtained from $u$ by renaming of letters.) We completely determine all commutative modular varieties and obtain an essential information about modular varieties satisfying a permutable identity.

**Keywords:** semigroup, variety, nil-variety, 0-reduced identity, substitutive identity, permutable identity, lattice of subvarieties, modular element of a lattice, upper-modular element of a lattice

**AMS Subject Classification:** Primary 20M07; Secondary 08B15