The pro-nilpotent group topology on a free group

In this paper, we study the pro-nilpotent group topology on a free group. First we describe the closure of the product of finitely many finitely generated subgroups of a free group in the pro-nilpotent group topology and then present an algorithm to compute it. We deduce that the nil-closure of a rational subset of a free group is an effectively constructible rational subset and hence has decidable membership. We also prove that the Gnil-kernel of a finite monoid is computable and hence pseudovarieties of the form VⓜGnil have decidable membership problem, for every decidable pseudovariety of monoids V. Finally, we prove that the semidirect product J⁎Gnil has a decidable membership problem.