  In this paper, we work on the pro-nilpotent group topology of a free group.
First we investigate the closure of the product of finitely many subgroups of a
free group in the pro-nilpotent group topology. We present an algorithm for the
calculation of the closure in the pro-nilpotent group topology of the product
of finitely many finitely generated subgroups of a free group. Then we deduce
that the nil-closure of a rational subset is computable. We also prove that the
pseudovarieties V malcev G_nil, for every decidable pseudovariety of monoids V,
and J * G_nil are decidable.
