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Closures of regular languages for profinite topologies

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Abstract

The Pin-Reutenauer algorithm gives a method, that can be viewed as a descriptive procedure, to compute the closure in the free group of a regular language with respect to the Hall topology. A similar descriptive procedure is shown to hold for the pseudovariety \(\mathsf{A}\) of aperiodic semigroups, where the closure is taken in the free aperiodic \(\omega \)-semigroup. It is inherited by a subpseudovariety of a given pseudovariety if both of them enjoy the property of being full. The pseudovariety \(\mathsf{A}\), as well as some of its subpseudovarieties are shown to be full. The interest in such descriptions stems from the fact that, for each of the main pseudovarieties \(\mathsf{V}\) in our examples, the closures of two regular languages are disjoint if and only if the languages can be separated by a language whose syntactic semigroup lies in \(\mathsf{V}\). In the cases of \(\mathsf{A}\) and of the pseudovariety \(\mathsf{DA}\) of semigroups in which all regular elements are idempotents, this is a new result.

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Acknowledgments

Work partly supported by the AutoMathA programme of the European Science Foundation and by the Pessoa French-Portuguese project Egide-Grices 11113YM, Automata, profinite semigroups and symbolic dynamics. The work of the first two authors was supported, in part, by the European Regional Development Fund, through the programme COMPETE, and by the Portuguese Government through FCT – Fundação para a Ciência e a Tecnologia, respectively under the projects PEst-C/MAT/UI0144/2013 and PEst-OE/MAT/UI0013/2014. The work of the first two authors was also partly supported by the FCT and the project PTDC/MAT/65481/2006 which was partly funded by the European Community Fund FEDER, and by the INTAS grant #99-1224. The work of the third author was partly supported by ANR 2010 BLAN 0202 01 FREC.

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Authors and Affiliations

  1. Centro de Matemática e Departamento de Matemática, Faculdade de Ciências, Universidade do Porto, Rua do Campo Alegre, 687, 4169-007 , Porto, Portugal

    J. Almeida

  2. Centro de Matemática e Departamento de Matemática e Aplicações, Universidade do Minho, Campus de Gualtar, 4700-320 , Braga, Portugal

    J. C. Costa

  3. LaBRI, Université de Bordeaux & CNRS UMR 5800, 351 cours de la Libération, 33405 , Talence Cedex, France

    M. Zeitoun

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  1. J. Almeida
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  2. J. C. Costa
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  3. M. Zeitoun
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Correspondence to J. Almeida.

Additional information

Communicated by Jean-Eric Pin.

Dedicated to the memory of John M. Howie.

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Almeida, J., Costa, J.C. & Zeitoun, M. Closures of regular languages for profinite topologies. Semigroup Forum 89, 20–40 (2014). https://doi.org/10.1007/s00233-014-9574-3

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