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 ra...

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Bibliographic Details
Published in:Journal of algebra 2017-06, Vol.480, p.332-345
Main Authors: Almeida, J., Shahzamanian, M.H., Steinberg, B.
Format: Article
Language:English
Subjects:
Automata
Block groups
Mal'cev products
Monoids
Profinite topologies
Pseudovarieties
Rational languages
Semidirect products
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Summary: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.
ISSN:0021-8693
1090-266X
DOI:10.1016/j.jalgebra.2017.03.009