Abstract:
The notion of $I$-element of a lattice is introduced, where $I$ is an arbitrary lattice identity. This notion generalizes practically all types of special elements of lattices considered earlier. It is proved that, if a semigroup variety is an $I$-element of the lattice of all semigroup varieties for some nontrivial lattice identity $I$ and is different from the variety of all semigroups, then it is a periodic variety. It is established that the converse is not true.
Keywords:semigroup, variety, lattice of varieties, special elements of a lattice.