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 G(nil)-kernel of a finite monoid is computable and hence pseudovarieties of the form V (sic) G(nil) have decidable membership problem, for every decidable pseudovariety of monoids V. Finally, we prove that the semidirect product J * G(nil) has a decidable membership problem