Let $S$ be the free semigroup on the set $\{x_1,\ldots ,x_n\}$, where $n$ is a positive integer. Suppose that $\mu=\mu (x_1,\ldots ,x_n)$ and $\nu = \nu (x_1,\ldots ,x_n)$ are two elements in $S$. We say that $\mu=\nu$ is a *semigroup law* in a group $G$ if for every $n$-tuple $(g_1,\ldots , g_n)$ of elements of $G$, we have $\mu (g_1,\ldots , g_n)=\nu (g_1,\ldots , g_n)$.

For example an Abelian group satisfies the semigroup law $\mu=\nu$ where $\mu (x_1,x_2)=x_1x_2$ and $\nu (x_1,x_2)=x_2x_1$. And similarly for nilpotent groups of class $c$ we can find a finite sequence of semigroup laws $\mu_1=\nu_1,\ldots,\mu_c=\nu_c$.

Now I am looking for semigroup laws that solvable groups are expressible by them.