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On the structure theory of partial automaton semigroups

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Abstract

We study automaton structures, i.e., groups, monoids and semigroups generated by an automaton, which, in this context, means a deterministic finite-state letter-to-letter transducer. Instead of considering only complete automata, we specifically investigate semigroups generated by partial automata. First, we show that the class of semigroups generated by partial automata coincides with the class of semigroups generated by complete automata if and only if the latter class is closed under removing a previously adjoined zero, which is an open problem in (complete) automaton semigroup theory stated by Cain. Then, we show that no semidirect product (and, thus, also no direct product) of an arbitrary semigroup with a (non-trivial) subsemigroup of the free monogenic semigroup is an automaton semigroup. Finally, we concentrate on inverse semigroups generated by invertible but partial automata, which we call automaton-inverse semigroups, and show that any inverse automaton semigroup can be generated by such an automaton (showing that automaton-inverse semigroups and inverse automaton semigroups coincide).

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Notes

  1. Note, however, that their paper considers slightly different inverse semigroups: they are generated by arbitrary subsets of states of an automaton, while, in this paper, the inverse semigroups are always generated by all states.

  2. For readers familiar with syntactic semigroups: \(B_2\) is the syntactic semigroup of \((qp)^+\).

  3. In fact, this is what was stated in [9, Proposition 1].

  4. A word x is a prefix of another word y if there is some word z such that \(y = xz\).

  5. Cross diagrams seem to increase in usage lately. They seem to have been introduced in [1] where the authors connect them to the square diagrams of [12].

  6. Recall that an element s of a semigroup has torsion if there are \(i, j > 0\) with \(i \ne j\) such that \(s^i = s^j\).

  7. In fact, we can find a suitable \(r_j\) for every \(r_i \in Y_i\).

  8. For example, we can choose \(i = 2 \omega \) where \(\omega \) is the smallest exponent such that \(s^\omega \) is idempotent. Then i satisfies the condition because of \(s^{2 \omega - 1} s^\omega = s^{2 \omega - 1}\).

  9. We reserve the term injective for total functions.

  10. The modification is inspired by [16, Fig. 8]; see also [7, Example 2].

  11. In fact, it is an anti-isomorphism.

  12. See, e.g., [14, Theorem 5.1.1, p. 145].

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Authors and Affiliations

  1. Universitá degli Studi Niccoló Cusano, Via Don Carlo Gnocchi, 3, 00166, Rome, Italy

    Daniele D’Angeli

  2. Department of Mathematics, Politecnico di Milano, Piazza Leonardo da Vinci, 32, 20133, Milan, Italy

    Emanuele Rodaro

  3. Institut für Formale Methoden der Informatik (FMI), Universität Stuttgart, Universitätsstraße 38, 70569, Stuttgart, Germany

    Jan Philipp Wächter

Authors
  1. Daniele D’Angeli
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  2. Emanuele Rodaro
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  3. Jan Philipp Wächter
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Correspondence to Emanuele Rodaro.

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Communicated by Mikhail Volkov.

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Daniele D’Angeli was supported by the Austrian Science Fund Project FWF P24028-N18 and FWF P29355-N35.

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D’Angeli, D., Rodaro, E. & Wächter, J.P. On the structure theory of partial automaton semigroups. Semigroup Forum 101, 51–76 (2020). https://doi.org/10.1007/s00233-020-10114-5

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