We work on the pro-nilpotent group topology of a free group. We investigate the closure of the product of finitely many subgroups of a free group in the pro-nilpotent group. Then we present an algorithm for the calculation of the closure in the pro-nilpotent group topology of rational subsets of a free group.

At another work, I define a congruence $ \eta^{\ast}$ on semigroups. For the finite semigroups $S$, $ \eta^{\ast}$ is the smallest congruence relation such that $S/ \eta^{\ast}$ is a nilpotent semigroup. In order to study the congruence relation $ \eta^{\ast}$ on finite semigroups, we define a $\textbf{CS}$-diagonal finite regular Rees matrix semigroup. I prove that, if $S$ is a $\textbf{CS}$-diagonal finite regular Rees matrix semigroup then $S/ \eta^{\ast}$ is inverse. Also, if $S$ is a completely regular finite semigroup, then $S/ \eta^{\ast}$ is a Clifford semigroup. I show that, for every non-null principal factor $A/B$ of $S$, there is a special principal factor $C/D$ such that every element of $A\setminus B$ is $ \eta^{\ast}$-equivalent with some element of $C\setminus D$. I call the principal factor $C/D$, the $ \eta^{\ast}$-root of $A/B$. All $ \eta^{\ast}$-roots are $\textbf{CS}$-diagonal.  If certain elements of $S$ act in the special way on the $\textbf{R}$-classes of a $\textbf{CS}$-diagonal principal factor then it is not an $ \eta^{\ast}$-root. Some of these results are also expressed in terms of pseudovarieties of semigroups. I submitted this work at International journal.