On the lattice of subvarieties of the wreath product the variety of semilattices and the variety of semigroups with zero multiplication

It is known that the monoid wreath product of any semigroup varieties that are atoms in the lattice of all semigroup varieties mays have a finite as well as an infinite lattice of subvarieties. If this lattice is finite, then as a rule it has at most eleven elements. This was proved in a paper of the author in 2007. The exclusion is the monoid wreath product $\mathbf{Sl}\mathrm w\mathbf N_2$ of the variety of semilattices and the variety of semigroups with zero multiplication. The number of elements of the lattice $L(\mathbf{Sl}\mathrm w\mathbf N_2)$ of subvarieties of $\mathbf{Sl}\mathrm w\mathbf N_2$ is still unknown. In our paper, we show that the lattice $L(\mathbf{Sl}\mathrm w\mathbf N_2)$ contains no less than 33 elements. In addition, we give some exponential upper bound of the cardinality of this lattice.