Closed Subgroups in Pro-V Topologies and The Extension Problem for Inverse Automata

S. Margolis, M. Sapir, P. Weil

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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 languageEnglish
Pages (from-to)405-445
Number of pages41
JournalInternational Journal of Algebra and Computation
Volume11
Issue number4
DOIs
StatePublished - 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.

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