The partially ordered monoid of semigroup varieties under wreath product

The decomposition of the ordered monoid of semigroup varieties under wreath product into a five-element semilattice of its subsemigroups is obtained. One of these subsemigroups is the one-element and consists of the only variety of all trivial semigroups. The second one is an ideal with the zero multiplication consisting of all overcommutative varieties. The third one is the free semigroup of continuum rank consisting of all non-trivial periodic group varieties. The fourth one is the countable semilattice of finite nilpotent subsemigroups $T_{jm}$ ($m\ge1$, $0\le j\le m$). The fifth one is a semigroup without idempotents containing a subsemigroup isomorphic to a free semigroup of continuum rank. This semigroup satisfies neither right nor left cancellation law. It is proved that $T_{jm}$ are lattice intervals of the lattice of all semigroup varieties. The greatest variety in the semigroup $T_{jm}$ is the non-zero idempotent of monoid of all semigroup varieties. The description of all idempotents of this monoid is known. The equational description for the least variety in $T_{jm}$ is found. In conclusion, the indices of nilpotence of semigroups $T_{0m}$ ($m\ge1$) are calculated. In particular, we obtain that the indices of nilpotence of $T_{jm}$ are not bounded.