## Abstract

We relate the problem of computing the closure of a finitely generated subgroup of the free group in the pro-V topology, where V is a pseudovariety of finite groups, with an extension problem for inverse automata which can be stated as follows: given partial one-to-one maps on a finite set, can they be extended into permutations generating a group in V? The two problems are equivalent when V is extension-closed. Turning to practical computations, we modify Ribes and Zalesskiǐ's algorithm to compute the pro-p closure of a finitely generated subgroup of the free group in polynomial time, and to effectively compute its pro-nilpotent closure. Finally, we apply our results to a problem in finite monoid theory, the membership problem in pseudovarieties of inverse monoids which are Mal'cev products of semilattices and a pseudovariety of groups.

Original language | English |
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Pages (from-to) | 405-445 |

Number of pages | 41 |

Journal | International Journal of Algebra and Computation |

Volume | 11 |

Issue number | 4 |

DOIs | |

State | Published - 2001 |

### Bibliographical note

Funding Information:∗The rst and third authors received support from the French-Israeli CNRS-MOSA project Algorithmic problems on automata and semigroups. yThe rst and the second authors were supported by the NSF grant DMS-9623284. zAll three authors were supported by the Center for Communication and Information Sciences, University of Nebraska-Lincoln.