We develop an effective and natural approach to interpret any semigroup admitting a special language of greedy normal forms as an automaton semigroup, namely the semigroup generated by a Mealy automaton encoding the behaviour of such a language of greedy normal forms under one-sided multiplication. The framework embraces many of the well-known classes of (automatic) semigroups: free semigroups, free commutative semigroups, trace or divisibility monoids, braid or Artin - Tits or Krammer or Garside monoids, Baumslag - Solitar semigroups, etc. Like plactic monoids or Chinese monoids, some neither left- nor right-cancellative automatic semigroups are also investigated, as well as some residually finite variations of the bicyclic monoid. It provides what appears to be the first known connection from a class of automatic semigroups to a class of automaton semigroups. It is worthwhile noting that, "being an automatic semigroup" and "being an automaton semigroup" become dual properties in a very automata-theoretical sense. Quadratic rewriting systems and associated tilings appear as the cornerstone of our construction.
@InProceedings{picantin:LIPIcs.ICALP.2019.124, author = {Picantin, Matthieu}, title = {{Automatic Semigroups vs Automaton Semigroups}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {124:1--124:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-109-2}, ISSN = {1868-8969}, year = {2019}, volume = {132}, editor = {Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.124}, URN = {urn:nbn:de:0030-drops-107004}, doi = {10.4230/LIPIcs.ICALP.2019.124}, annote = {Keywords: Mealy machine, semigroup, rewriting system, automaticity, self-similarity} }
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