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Automaton semigroup constructions

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Abstract

The aim of this paper is to investigate whether the class of automaton semigroups is closed under certain semigroup constructions. We prove that the free product of two automaton semigroups that contain left identities is again an automaton semigroup. We also show that the class of automaton semigroups is closed under the combined operation of ‘free product followed by adjoining an identity’. We present an example of a free product of finite semigroups that we conjecture is not an automaton semigroup. Turning to wreath products, we consider two slight generalizations of the concept of an automaton semigroup, and show that a wreath product of an automaton monoid and a finite monoid arises as a generalized automaton semigroup in both senses. We also suggest a potential counterexample that would show that a wreath product of an automaton monoid and a finite monoid is not necessarily an automaton monoid in the usual sense.

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Acknowledgments

The first author’s research was funded by an EPSRC grant EP/H011978/1. The second author’s was supported by an FCT Ciência 2008 fellowship and later by an FCT Investigador fellowship, and by the FCT project PEst-OE/MAT/UI0297/2014.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Mathematical Institute, North Haugh, St Andrews, Fife, KY16 9SS, Scotland, UK

    Tara Brough

  2. Centro de Matemática, Faculdade de Ciências, Universidade do Porto, Rua do Campo Alegre 687, 4169–007, Porto, Portugal

    Alan J. Cain

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  1. Tara Brough
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  2. Alan J. Cain
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Correspondence to Tara Brough.

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Communicated by Benjamin Steinberg.

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Brough, T., Cain, A.J. Automaton semigroup constructions. Semigroup Forum 90, 763–774 (2015). https://doi.org/10.1007/s00233-014-9632-x

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  • DOI: https://doi.org/10.1007/s00233-014-9632-x

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