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Reducibility of pointlike problems

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Abstract

We show that the pointlike and the idempotent pointlike problems are reducible with respect to natural signatures in the following cases: the pseudovariety of all finite semigroups in which the order of every subgroup is a product of elements of a fixed set \(\pi \) of primes; the pseudovariety of all finite semigroups in which every regular \(\mathcal J\)-class is the product of a rectangular band by a group from a fixed pseudovariety of groups that is reducible for the pointlike problem, respectively graph reducible. Allowing only trivial groups, we obtain \(\omega \)-reducibility of the pointlike and idempotent pointlike problems, respectively for the pseudovarieties of all finite aperiodic semigroups (\(\mathsf{A}\)) and of all finite semigroups in which all regular elements are idempotents (\(\mathsf{DA}\)).

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Acknowledgments

Work partly supported by the Pessoa French-Portuguese project “Separation in automata theory: algebraic, logical, and combinatorial aspects”. The work of the first author was also partially supported by CMUP (UID/MAT/00144/2013), which is funded by FCT (Portugal) with national (MEC) and European structural funds through the programs FEDER, under the partnership agreement PT2020. The work of the second author was also partially supported by the European Regional Development Fund, through the program COMPETE, and by the Portuguese Government through FCT under the project PEst-OE/MAT/UI0013/2014. 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. Univ. Bordeaux, LaBRI, UMR 5800, 33400, Talence, France

    M. Zeitoun

Authors
  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.

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Communicated by Benjamin Steinberg.

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Almeida, J., Costa, J.C. & Zeitoun, M. Reducibility of pointlike problems. Semigroup Forum 94, 325–335 (2017). https://doi.org/10.1007/s00233-015-9769-2

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